[
  {
    "path": ".gitattributes",
    "content": "# Auto detect text files and perform LF normalization\n* text=auto\n\n# Custom for Visual Studio\n*.cs     diff=csharp\n\n# Standard to msysgit\n*.doc\t diff=astextplain\n*.DOC\t diff=astextplain\n*.docx diff=astextplain\n*.DOCX diff=astextplain\n*.dot  diff=astextplain\n*.DOT  diff=astextplain\n*.pdf  diff=astextplain\n*.PDF\t diff=astextplain\n*.rtf\t diff=astextplain\n*.RTF\t diff=astextplain\n"
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  {
    "path": ".gitignore",
    "content": "# Byte-compiled / optimized / DLL files\n__pycache__/\n*.py[cod]\n\n# C extensions\n*.so\n\n# Distribution / packaging\n.Python\nenv/\nbuild/\ndevelop-eggs/\ndist/\ndownloads/\neggs/\nlib/\nlib64/\nparts/\nsdist/\nvar/\n*.egg-info/\n.installed.cfg\n*.egg\n\n# PyInstaller\n#  Usually these files are written by a python script from a template\n#  before PyInstaller builds the exe, so as to inject date/other infos into it.\n*.manifest\n*.spec\n\n# Installer logs\npip-log.txt\npip-delete-this-directory.txt\n\n# Unit test / coverage reports\nhtmlcov/\n.tox/\n.coverage\n.cache\nnosetests.xml\ncoverage.xml\n\n# Translations\n*.mo\n*.pot\n\n# Django stuff:\n*.log\n\n# PyBuilder\ntarget/\n\n# =========================\n# Operating System Files\n# =========================\n\n# OSX\n# =========================\n\n.DS_Store\n.AppleDouble\n.LSOverride\n\n# Thumbnails\n._*\n\n# Files that might appear on external disk\n.Spotlight-V100\n.Trashes\n\n# Directories potentially created on remote AFP share\n.AppleDB\n.AppleDesktop\nNetwork Trash Folder\nTemporary Items\n.apdisk\n\n# Windows\n# =========================\n\n# Windows image file caches\nThumbs.db\nehthumbs.db\n\n# Folder config file\nDesktop.ini\n\n# Recycle Bin used on file shares\n$RECYCLE.BIN/\n\n# Windows Installer files\n*.cab\n*.msi\n*.msm\n*.msp\n\n# Windows shortcuts\n*.lnk\npyTempNets.v12.suo\n*.png\n*.TMP\npyTempNet.egg-info/requires.txt\npyTempNet.egg-info/SOURCES.txt\npyTempNet.egg-info/dependency_links.txt\npyTempNet.egg-info/requires.txt\n*.suo\npyTempNet.egg-info/top_level.txt\npyTempNet.egg-info/requires.txt\npyTempNet.egg-info/requires.txt\npyTempNet.egg-info/dependency_links.txt\npyTempNet.egg-info/SOURCES.txt\n.idea/\n.eggs/\nprof/*\n"
  },
  {
    "path": "DESCRIPTION.rst",
    "content": "pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models."
  },
  {
    "path": "LICENSE.txt",
    "content": "                    GNU AFFERO GENERAL PUBLIC LICENSE\n                       Version 3, 19 November 2007\n\n Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>\n Everyone is permitted to copy and distribute verbatim copies\n of this license document, but changing it is not allowed.\n\n                            Preamble\n\n  The GNU Affero General Public License is a free, copyleft license for\nsoftware and other kinds of works, specifically designed to ensure\ncooperation with the community in the case of network server software.\n\n  The licenses for most software and other practical works are designed\nto take away your freedom to share and change the works.  By contrast,\nour General Public Licenses are intended to guarantee your freedom to\nshare and change all versions of a program--to make sure it remains free\nsoftware for all its users.\n\n  When we speak of free software, we are referring to freedom, not\nprice.  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    "path": "README.md",
    "content": "<img src=\"https://github.com/IngoScholtes/pathpy/blob/master/pathpy_logo.png\" width=\"300\" alt=\"pathpy logo\" />\n\n**Note**: This is the repository of an old version of pathpy. It will soon be replaced by [pathpy 3](https://github.com/pathpy/pathpy), which has a new [home on gitHub](https://github.com/pathpy/pathpy).\n\n# Introduction\n\n`pathpy` is an OpenSource python package for the modeling and analysis of pathways and temporal networks\nusing **higher-order** and **multi-order** graphical models.\n\nThe package is specifically tailored to analyze sequential data which capture multiple observations of short, independent paths\nobserved in an underlying graph topology. Examples for such data include user click streams in information networks,\nbiological pathways, or traces of information propagating in social media. Unifying the analysis of pathways and temporal networks,\n`pathpy` also supports the extraction of time-respecting paths from time-stamped network data. It extends (and will eventually supersede)\nthe package [`pyTempnets`](https://github.com/IngoScholtes/pyTempNets).\n\n`pathpy` facilitates the analysis of temporal correlations in such sequential data. It uses a principled model selection\ntechnique to infer higher-order graphical representations that capture both topological and temporal\ncharacteristics of time-resolved relational data. It specifically allows to answer the question whether a (first-order) network\nabstraction of such data is justified, or whether higher-order network abstractions are needed.\n\nThe theoretical foundation of this package, **higher-order network models**, has been developed in the following research works:\n\n1. I Scholtes: [When is a network a network? Multi-Order Graphical Model Selection in Pathways and Temporal Networks](https://arxiv.org/abs/1702.05499), to appear in KDD'17, arXiv:1702.05499\n2. I Scholtes, N Wider, A Garas: [Higher-Order Aggregate Networks in the Analysis of Temporal Networks: Path structures and centralities](http://dx.doi.org/10.1140/epjb/e2016-60663-0), The European Physical Journal B, 89:61, March 2016\n3. I Scholtes, N Wider, R Pfitzner, A Garas, CJ Tessone, F Schweitzer: [Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks](http://www.nature.com/ncomms/2014/140924/ncomms6024/full/ncomms6024.html), Nature Communications, 5, September 2014\n4. R Pfitzner, I Scholtes, A Garas, CJ Tessone, F Schweitzer: [Betweenness preference: Quantifying correlations in the topological dynamics of temporal networks](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.198701), Phys Rev Lett, 110(19), 198701, May 2013\n\n`pathpy` extends this approach towards **multi-layer graphical models** that capture temporal correlations in pathways at multiple length scales simultaneously. An illustrative example for\na collection of pathways (left) and a multi-order graphical representation of these pathways is shown below. All mathematical details of the framework can be found in this [recent research paper](https://arxiv.org/abs/1702.05499).\n\n[![Watch promotional video](https://img.youtube.com/vi/CxJkVrD2ZlM/0.jpg)](https://www.youtube.com/watch?v=CxJkVrD2ZlM)\n\n<img src=\"https://github.com/IngoScholtes/pathpy/blob/master/multiorder.png\" width=\"500\" alt=\"Illustration of Multi-Order Model\" />\n\n# Download and installation\n\n`pathpy` is pure python code. It has no platform-specific dependencies and should thus work on all platforms. It builds on `numpy` and `scipy`. The latest version of `pathpy` can be installed by typing:\n\n`> pip install git+git://github.com/IngoScholtes/pathpy.git`\n\n`pathpy` currently requires python 3.x. We are planning to make the next version backwards compatible with python 2.x. \n\n# Tutorial\n\nA [comprehensive educational tutorial](https://ingoscholtes.github.io/pathpy/tutorial.html) which shows how you can use `pathpy` to analyze data on pathways and temporal networks is [available online](https://ingoscholtes.github.io/pathpy/tutorial.html).\nMoreover, a tutorial which illustrates the abstraction of **higher-order networks** in the modeling of dynamical processes in temporal networks is [available here](https://www.sg.ethz.ch/team/people/ischoltes/research-insights/temporal-networks-demo/). The\nlatter tutorial is based on the predecessor library [`pyTempNets`](https://github.com/IngoScholtes/pyTempNets). Most of its features have been ported to `pathpy`.\n\n# Documentation\n\nThe code is fully documented via docstrings which are accessible through python's built-in help system. Just type `help(SYMBOL_NAME)` to see the documentation of a class or method. A [reference manual is available here](https://ingoscholtes.github.io/pathpy/hierarchy.html).\n\n# Releases and Versioning\n\nThe first public beta release of pathpy (released February 17 2017) is [v1.0-beta](https://github.com/IngoScholtes/pathpy/releases/tag/v1.0-beta.1). Following versions are named MAJOR.MINOR.PATCH according to [semantic versioning](http://semver.org/). The date of each release is encoded in the PATCH version.\n\n# Acknowledgements\n\nThe research behind this data analysis framework was funded by the Swiss State Secretariat for Education, Research and Innovation [(Grant C14.0036)](https://www.sg.ethz.ch/projects/seri-information-spaces/). The development of this package was generously supported by the [MTEC Foundation](http://www.mtec.ethz.ch/research/support/MTECFoundation.html) in the context of the project [The Influence of Interaction Patterns on Success in Socio-Technical Systems: From Theory to Practice](https://www.sg.ethz.ch/projects/mtec-interaction-patterns/).\n\n# Contributors\n\n[Ingo Scholtes](http://www.ingoscholtes.net) (project lead, development)  \n[Luca Verginer](http://www.verginer.eu/about/) (development, testing)  \n\n# Past Contributors\nRoman Cattaneo (development)  \nNicolas Wider (testing)  \n\n# Copyright\n\npathpy is licensed under the [GNU Affero General Public License](https://choosealicense.com/licenses/agpl-3.0/).\n\n(c) Copyright ETH Zürich, Chair of Systems Design, 2015-2018\n"
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Private\n# class members and static file members will be hidden unless the\n# EXTRACT_PRIVATE respectively EXTRACT_STATIC tags are set to YES.\n# Note: This will also disable the warnings about undocumented members that are\n# normally produced when WARNINGS is set to YES.\n# The default value is: NO.\n\nEXTRACT_ALL            = NO\n\n# If the EXTRACT_PRIVATE tag is set to YES all private members of a class will\n# be included in the documentation.\n# The default value is: NO.\n\nEXTRACT_PRIVATE        = NO\n\n# If the EXTRACT_PACKAGE tag is set to YES all members with package or internal\n# scope will be included in the documentation.\n# The default value is: NO.\n\nEXTRACT_PACKAGE        = NO\n\n# If the EXTRACT_STATIC tag is set to YES all static members of a file will be\n# included in the documentation.\n# The default value is: NO.\n\nEXTRACT_STATIC         = NO\n\n# If the EXTRACT_LOCAL_CLASSES tag is set to YES classes (and structs) defined\n# locally in source files will be included in the documentation. If set to NO\n# only classes defined in header files are included. Does not have any effect\n# for Java sources.\n# The default value is: YES.\n\nEXTRACT_LOCAL_CLASSES  = YES\n\n# This flag is only useful for Objective-C code. When set to YES local methods,\n# which are defined in the implementation section but not in the interface are\n# included in the documentation. If set to NO only methods in the interface are\n# included.\n# The default value is: NO.\n\nEXTRACT_LOCAL_METHODS  = NO\n\n# If this flag is set to YES, the members of anonymous namespaces will be\n# extracted and appear in the documentation as a namespace called\n# 'anonymous_namespace{file}', where file will be replaced with the base name of\n# the file that contains the anonymous namespace. By default anonymous namespace\n# are hidden.\n# The default value is: NO.\n\nEXTRACT_ANON_NSPACES   = NO\n\n# If the HIDE_UNDOC_MEMBERS tag is set to YES, doxygen will hide all\n# undocumented members inside documented classes or files. If set to NO these\n# members will be included in the various overviews, but no documentation\n# section is generated. This option has no effect if EXTRACT_ALL is enabled.\n# The default value is: NO.\n\nHIDE_UNDOC_MEMBERS     = NO\n\n# If the HIDE_UNDOC_CLASSES tag is set to YES, doxygen will hide all\n# undocumented classes that are normally visible in the class hierarchy. If set\n# to NO these classes will be included in the various overviews. This option has\n# no effect if EXTRACT_ALL is enabled.\n# The default value is: NO.\n\nHIDE_UNDOC_CLASSES     = NO\n\n# If the HIDE_FRIEND_COMPOUNDS tag is set to YES, doxygen will hide all friend\n# (class|struct|union) declarations. If set to NO these declarations will be\n# included in the documentation.\n# The default value is: NO.\n\nHIDE_FRIEND_COMPOUNDS  = NO\n\n# If the HIDE_IN_BODY_DOCS tag is set to YES, doxygen will hide any\n# documentation blocks found inside the body of a function. If set to NO these\n# blocks will be appended to the function's detailed documentation block.\n# The default value is: NO.\n\nHIDE_IN_BODY_DOCS      = NO\n\n# The INTERNAL_DOCS tag determines if documentation that is typed after a\n# \\internal command is included. If the tag is set to NO then the documentation\n# will be excluded. Set it to YES to include the internal documentation.\n# The default value is: NO.\n\nINTERNAL_DOCS          = NO\n\n# If the CASE_SENSE_NAMES tag is set to NO then doxygen will only generate file\n# names in lower-case letters. If set to YES upper-case letters are also\n# allowed. This is useful if you have classes or files whose names only differ\n# in case and if your file system supports case sensitive file names. Windows\n# and Mac users are advised to set this option to NO.\n# The default value is: system dependent.\n\nCASE_SENSE_NAMES       = YES\n\n# If the HIDE_SCOPE_NAMES tag is set to NO then doxygen will show members with\n# their full class and namespace scopes in the documentation. If set to YES the\n# scope will be hidden.\n# The default value is: NO.\n\nHIDE_SCOPE_NAMES       = NO\n\n# If the SHOW_INCLUDE_FILES tag is set to YES then doxygen will put a list of\n# the files that are included by a file in the documentation of that file.\n# The default value is: YES.\n\nSHOW_INCLUDE_FILES     = YES\n\n# If the SHOW_GROUPED_MEMB_INC tag is set to YES then Doxygen will add for each\n# grouped member an include statement to the documentation, telling the reader\n# which file to include in order to use the member.\n# The default value is: NO.\n\nSHOW_GROUPED_MEMB_INC  = NO\n\n# If the FORCE_LOCAL_INCLUDES tag is set to YES then doxygen will list include\n# files with double quotes in the documentation rather than with sharp brackets.\n# The default value is: NO.\n\nFORCE_LOCAL_INCLUDES   = NO\n\n# If the INLINE_INFO tag is set to YES then a tag [inline] is inserted in the\n# documentation for inline members.\n# The default value is: YES.\n\nINLINE_INFO            = YES\n\n# If the SORT_MEMBER_DOCS tag is set to YES then doxygen will sort the\n# (detailed) documentation of file and class members alphabetically by member\n# name. If set to NO the members will appear in declaration order.\n# The default value is: YES.\n\nSORT_MEMBER_DOCS       = YES\n\n# If the SORT_BRIEF_DOCS tag is set to YES then doxygen will sort the brief\n# descriptions of file, namespace and class members alphabetically by member\n# name. If set to NO the members will appear in declaration order. Note that\n# this will also influence the order of the classes in the class list.\n# The default value is: NO.\n\nSORT_BRIEF_DOCS        = NO\n\n# If the SORT_MEMBERS_CTORS_1ST tag is set to YES then doxygen will sort the\n# (brief and detailed) documentation of class members so that constructors and\n# destructors are listed first. If set to NO the constructors will appear in the\n# respective orders defined by SORT_BRIEF_DOCS and SORT_MEMBER_DOCS.\n# Note: If SORT_BRIEF_DOCS is set to NO this option is ignored for sorting brief\n# member documentation.\n# Note: If SORT_MEMBER_DOCS is set to NO this option is ignored for sorting\n# detailed member documentation.\n# The default value is: NO.\n\nSORT_MEMBERS_CTORS_1ST = NO\n\n# If the SORT_GROUP_NAMES tag is set to YES then doxygen will sort the hierarchy\n# of group names into alphabetical order. If set to NO the group names will\n# appear in their defined order.\n# The default value is: NO.\n\nSORT_GROUP_NAMES       = NO\n\n# If the SORT_BY_SCOPE_NAME tag is set to YES, the class list will be sorted by\n# fully-qualified names, including namespaces. If set to NO, the class list will\n# be sorted only by class name, not including the namespace part.\n# Note: This option is not very useful if HIDE_SCOPE_NAMES is set to YES.\n# Note: This option applies only to the class list, not to the alphabetical\n# list.\n# The default value is: NO.\n\nSORT_BY_SCOPE_NAME     = NO\n\n# If the STRICT_PROTO_MATCHING option is enabled and doxygen fails to do proper\n# type resolution of all parameters of a function it will reject a match between\n# the prototype and the implementation of a member function even if there is\n# only one candidate or it is obvious which candidate to choose by doing a\n# simple string match. By disabling STRICT_PROTO_MATCHING doxygen will still\n# accept a match between prototype and implementation in such cases.\n# The default value is: NO.\n\nSTRICT_PROTO_MATCHING  = NO\n\n# The GENERATE_TODOLIST tag can be used to enable ( YES) or disable ( NO) the\n# todo list. This list is created by putting \\todo commands in the\n# documentation.\n# The default value is: YES.\n\nGENERATE_TODOLIST      = YES\n\n# The GENERATE_TESTLIST tag can be used to enable ( YES) or disable ( NO) the\n# test list. This list is created by putting \\test commands in the\n# documentation.\n# The default value is: YES.\n\nGENERATE_TESTLIST      = YES\n\n# The GENERATE_BUGLIST tag can be used to enable ( YES) or disable ( NO) the bug\n# list. This list is created by putting \\bug commands in the documentation.\n# The default value is: YES.\n\nGENERATE_BUGLIST       = YES\n\n# The GENERATE_DEPRECATEDLIST tag can be used to enable ( YES) or disable ( NO)\n# the deprecated list. This list is created by putting \\deprecated commands in\n# the documentation.\n# The default value is: YES.\n\nGENERATE_DEPRECATEDLIST= YES\n\n# The ENABLED_SECTIONS tag can be used to enable conditional documentation\n# sections, marked by \\if <section_label> ... \\endif and \\cond <section_label>\n# ... \\endcond blocks.\n\nENABLED_SECTIONS       =\n\n# The MAX_INITIALIZER_LINES tag determines the maximum number of lines that the\n# initial value of a variable or macro / define can have for it to appear in the\n# documentation. If the initializer consists of more lines than specified here\n# it will be hidden. Use a value of 0 to hide initializers completely. The\n# appearance of the value of individual variables and macros / defines can be\n# controlled using \\showinitializer or \\hideinitializer command in the\n# documentation regardless of this setting.\n# Minimum value: 0, maximum value: 10000, default value: 30.\n\nMAX_INITIALIZER_LINES  = 30\n\n# Set the SHOW_USED_FILES tag to NO to disable the list of files generated at\n# the bottom of the documentation of classes and structs. If set to YES the list\n# will mention the files that were used to generate the documentation.\n# The default value is: YES.\n\nSHOW_USED_FILES        = YES\n\n# Set the SHOW_FILES tag to NO to disable the generation of the Files page. This\n# will remove the Files entry from the Quick Index and from the Folder Tree View\n# (if specified).\n# The default value is: YES.\n\nSHOW_FILES             = YES\n\n# Set the SHOW_NAMESPACES tag to NO to disable the generation of the Namespaces\n# page. This will remove the Namespaces entry from the Quick Index and from the\n# Folder Tree View (if specified).\n# The default value is: YES.\n\nSHOW_NAMESPACES        = YES\n\n# The FILE_VERSION_FILTER tag can be used to specify a program or script that\n# doxygen should invoke to get the current version for each file (typically from\n# the version control system). Doxygen will invoke the program by executing (via\n# popen()) the command command input-file, where command is the value of the\n# FILE_VERSION_FILTER tag, and input-file is the name of an input file provided\n# by doxygen. Whatever the program writes to standard output is used as the file\n# version. For an example see the documentation.\n\nFILE_VERSION_FILTER    =\n\n# The LAYOUT_FILE tag can be used to specify a layout file which will be parsed\n# by doxygen. The layout file controls the global structure of the generated\n# output files in an output format independent way. To create the layout file\n# that represents doxygen's defaults, run doxygen with the -l option. You can\n# optionally specify a file name after the option, if omitted DoxygenLayout.xml\n# will be used as the name of the layout file.\n#\n# Note that if you run doxygen from a directory containing a file called\n# DoxygenLayout.xml, doxygen will parse it automatically even if the LAYOUT_FILE\n# tag is left empty.\n\nLAYOUT_FILE            =\n\n# The CITE_BIB_FILES tag can be used to specify one or more bib files containing\n# the reference definitions. This must be a list of .bib files. The .bib\n# extension is automatically appended if omitted. This requires the bibtex tool\n# to be installed. See also http://en.wikipedia.org/wiki/BibTeX for more info.\n# For LaTeX the style of the bibliography can be controlled using\n# LATEX_BIB_STYLE. To use this feature you need bibtex and perl available in the\n# search path. Do not use file names with spaces, bibtex cannot handle them. See\n# also \\cite for info how to create references.\n\nCITE_BIB_FILES         =\n\n#---------------------------------------------------------------------------\n# Configuration options related to warning and progress messages\n#---------------------------------------------------------------------------\n\n# The QUIET tag can be used to turn on/off the messages that are generated to\n# standard output by doxygen. If QUIET is set to YES this implies that the\n# messages are off.\n# The default value is: NO.\n\nQUIET                  = NO\n\n# The WARNINGS tag can be used to turn on/off the warning messages that are\n# generated to standard error ( stderr) by doxygen. If WARNINGS is set to YES\n# this implies that the warnings are on.\n#\n# Tip: Turn warnings on while writing the documentation.\n# The default value is: YES.\n\nWARNINGS               = YES\n\n# If the WARN_IF_UNDOCUMENTED tag is set to YES, then doxygen will generate\n# warnings for undocumented members. If EXTRACT_ALL is set to YES then this flag\n# will automatically be disabled.\n# The default value is: YES.\n\nWARN_IF_UNDOCUMENTED   = YES\n\n# If the WARN_IF_DOC_ERROR tag is set to YES, doxygen will generate warnings for\n# potential errors in the documentation, such as not documenting some parameters\n# in a documented function, or documenting parameters that don't exist or using\n# markup commands wrongly.\n# The default value is: YES.\n\nWARN_IF_DOC_ERROR      = YES\n\n# This WARN_NO_PARAMDOC option can be enabled to get warnings for functions that\n# are documented, but have no documentation for their parameters or return\n# value. If set to NO doxygen will only warn about wrong or incomplete parameter\n# documentation, but not about the absence of documentation.\n# The default value is: NO.\n\nWARN_NO_PARAMDOC       = NO\n\n# The WARN_FORMAT tag determines the format of the warning messages that doxygen\n# can produce. The string should contain the $file, $line, and $text tags, which\n# will be replaced by the file and line number from which the warning originated\n# and the warning text. Optionally the format may contain $version, which will\n# be replaced by the version of the file (if it could be obtained via\n# FILE_VERSION_FILTER)\n# The default value is: $file:$line: $text.\n\nWARN_FORMAT            = \"$file:$line: $text\"\n\n# The WARN_LOGFILE tag can be used to specify a file to which warning and error\n# messages should be written. If left blank the output is written to standard\n# error (stderr).\n\nWARN_LOGFILE           =\n\n#---------------------------------------------------------------------------\n# Configuration options related to the input files\n#---------------------------------------------------------------------------\n\n# The INPUT tag is used to specify the files and/or directories that contain\n# documented source files. You may enter file names like myfile.cpp or\n# directories like /usr/src/myproject. Separate the files or directories with\n# spaces.\n# Note: If this tag is empty the current directory is searched.\n\nINPUT                  = ../pathpy\n\n# This tag can be used to specify the character encoding of the source files\n# that doxygen parses. Internally doxygen uses the UTF-8 encoding. Doxygen uses\n# libiconv (or the iconv built into libc) for the transcoding. See the libiconv\n# documentation (see: http://www.gnu.org/software/libiconv) for the list of\n# possible encodings.\n# The default value is: UTF-8.\n\nINPUT_ENCODING         = UTF-8\n\n# If the value of the INPUT tag contains directories, you can use the\n# FILE_PATTERNS tag to specify one or more wildcard patterns (like *.cpp and\n# *.h) to filter out the source-files in the directories. If left blank the\n# following patterns are tested:*.c, *.cc, *.cxx, *.cpp, *.c++, *.java, *.ii,\n# *.ixx, *.ipp, *.i++, *.inl, *.idl, *.ddl, *.odl, *.h, *.hh, *.hxx, *.hpp,\n# *.h++, *.cs, *.d, *.php, *.php4, *.php5, *.phtml, *.inc, *.m, *.markdown,\n# *.md, *.mm, *.dox, *.py, *.f90, *.f, *.for, *.tcl, *.vhd, *.vhdl, *.ucf,\n# *.qsf, *.as and *.js.\n\nFILE_PATTERNS          = *.py\n\n# The RECURSIVE tag can be used to specify whether or not subdirectories should\n# be searched for input files as well.\n# The default value is: NO.\n\nRECURSIVE              = YES\n\n# The EXCLUDE tag can be used to specify files and/or directories that should be\n# excluded from the INPUT source files. This way you can easily exclude a\n# subdirectory from a directory tree whose root is specified with the INPUT tag.\n#\n# Note that relative paths are relative to the directory from which doxygen is\n# run.\n\nEXCLUDE                =\n\n# The EXCLUDE_SYMLINKS tag can be used to select whether or not files or\n# directories that are symbolic links (a Unix file system feature) are excluded\n# from the input.\n# The default value is: NO.\n\nEXCLUDE_SYMLINKS       = NO\n\n# If the value of the INPUT tag contains directories, you can use the\n# EXCLUDE_PATTERNS tag to specify one or more wildcard patterns to exclude\n# certain files from those directories.\n#\n# Note that the wildcards are matched against the file with absolute path, so to\n# exclude all test directories for example use the pattern */test/*\n\nEXCLUDE_PATTERNS       =\n\n# The EXCLUDE_SYMBOLS tag can be used to specify one or more symbol names\n# (namespaces, classes, functions, etc.) that should be excluded from the\n# output. The symbol name can be a fully qualified name, a word, or if the\n# wildcard * is used, a substring. Examples: ANamespace, AClass,\n# AClass::ANamespace, ANamespace::*Test\n#\n# Note that the wildcards are matched against the file with absolute path, so to\n# exclude all test directories use the pattern */test/*\n\nEXCLUDE_SYMBOLS        =\n\n# The EXAMPLE_PATH tag can be used to specify one or more files or directories\n# that contain example code fragments that are included (see the \\include\n# command).\n\nEXAMPLE_PATH           =\n\n# If the value of the EXAMPLE_PATH tag contains directories, you can use the\n# EXAMPLE_PATTERNS tag to specify one or more wildcard pattern (like *.cpp and\n# *.h) to filter out the source-files in the directories. If left blank all\n# files are included.\n\nEXAMPLE_PATTERNS       =\n\n# If the EXAMPLE_RECURSIVE tag is set to YES then subdirectories will be\n# searched for input files to be used with the \\include or \\dontinclude commands\n# irrespective of the value of the RECURSIVE tag.\n# The default value is: NO.\n\nEXAMPLE_RECURSIVE      = NO\n\n# The IMAGE_PATH tag can be used to specify one or more files or directories\n# that contain images that are to be included in the documentation (see the\n# \\image command).\n\nIMAGE_PATH             =\n\n# The INPUT_FILTER tag can be used to specify a program that doxygen should\n# invoke to filter for each input file. Doxygen will invoke the filter program\n# by executing (via popen()) the command:\n#\n# <filter> <input-file>\n#\n# where <filter> is the value of the INPUT_FILTER tag, and <input-file> is the\n# name of an input file. Doxygen will then use the output that the filter\n# program writes to standard output. If FILTER_PATTERNS is specified, this tag\n# will be ignored.\n#\n# Note that the filter must not add or remove lines; it is applied before the\n# code is scanned, but not when the output code is generated. If lines are added\n# or removed, the anchors will not be placed correctly.\n\nINPUT_FILTER           =\n\n# The FILTER_PATTERNS tag can be used to specify filters on a per file pattern\n# basis. Doxygen will compare the file name with each pattern and apply the\n# filter if there is a match. The filters are a list of the form: pattern=filter\n# (like *.cpp=my_cpp_filter). See INPUT_FILTER for further information on how\n# filters are used. If the FILTER_PATTERNS tag is empty or if none of the\n# patterns match the file name, INPUT_FILTER is applied.\n\nFILTER_PATTERNS        =\n\n# If the FILTER_SOURCE_FILES tag is set to YES, the input filter (if set using\n# INPUT_FILTER ) will also be used to filter the input files that are used for\n# producing the source files to browse (i.e. when SOURCE_BROWSER is set to YES).\n# The default value is: NO.\n\nFILTER_SOURCE_FILES    = NO\n\n# The FILTER_SOURCE_PATTERNS tag can be used to specify source filters per file\n# pattern. A pattern will override the setting for FILTER_PATTERN (if any) and\n# it is also possible to disable source filtering for a specific pattern using\n# *.ext= (so without naming a filter).\n# This tag requires that the tag FILTER_SOURCE_FILES is set to YES.\n\nFILTER_SOURCE_PATTERNS =\n\n# If the USE_MDFILE_AS_MAINPAGE tag refers to the name of a markdown file that\n# is part of the input, its contents will be placed on the main page\n# (index.html). This can be useful if you have a project on for instance GitHub\n# and want to reuse the introduction page also for the doxygen output.\n\nUSE_MDFILE_AS_MAINPAGE =\n\n#---------------------------------------------------------------------------\n# Configuration options related to source browsing\n#---------------------------------------------------------------------------\n\n# If the SOURCE_BROWSER tag is set to YES then a list of source files will be\n# generated. Documented entities will be cross-referenced with these sources.\n#\n# Note: To get rid of all source code in the generated output, make sure that\n# also VERBATIM_HEADERS is set to NO.\n# The default value is: NO.\n\nSOURCE_BROWSER         = NO\n\n# Setting the INLINE_SOURCES tag to YES will include the body of functions,\n# classes and enums directly into the documentation.\n# The default value is: NO.\n\nINLINE_SOURCES         = NO\n\n# Setting the STRIP_CODE_COMMENTS tag to YES will instruct doxygen to hide any\n# special comment blocks from generated source code fragments. Normal C, C++ and\n# Fortran comments will always remain visible.\n# The default value is: YES.\n\nSTRIP_CODE_COMMENTS    = YES\n\n# If the REFERENCED_BY_RELATION tag is set to YES then for each documented\n# function all documented functions referencing it will be listed.\n# The default value is: NO.\n\nREFERENCED_BY_RELATION = NO\n\n# If the REFERENCES_RELATION tag is set to YES then for each documented function\n# all documented entities called/used by that function will be listed.\n# The default value is: NO.\n\nREFERENCES_RELATION    = NO\n\n# If the REFERENCES_LINK_SOURCE tag is set to YES and SOURCE_BROWSER tag is set\n# to YES, then the hyperlinks from functions in REFERENCES_RELATION and\n# REFERENCED_BY_RELATION lists will link to the source code. Otherwise they will\n# link to the documentation.\n# The default value is: YES.\n\nREFERENCES_LINK_SOURCE = YES\n\n# If SOURCE_TOOLTIPS is enabled (the default) then hovering a hyperlink in the\n# source code will show a tooltip with additional information such as prototype,\n# brief description and links to the definition and documentation. Since this\n# will make the HTML file larger and loading of large files a bit slower, you\n# can opt to disable this feature.\n# The default value is: YES.\n# This tag requires that the tag SOURCE_BROWSER is set to YES.\n\nSOURCE_TOOLTIPS        = YES\n\n# If the USE_HTAGS tag is set to YES then the references to source code will\n# point to the HTML generated by the htags(1) tool instead of doxygen built-in\n# source browser. The htags tool is part of GNU's global source tagging system\n# (see http://www.gnu.org/software/global/global.html). You will need version\n# 4.8.6 or higher.\n#\n# To use it do the following:\n# - Install the latest version of global\n# - Enable SOURCE_BROWSER and USE_HTAGS in the config file\n# - Make sure the INPUT points to the root of the source tree\n# - Run doxygen as normal\n#\n# Doxygen will invoke htags (and that will in turn invoke gtags), so these\n# tools must be available from the command line (i.e. in the search path).\n#\n# The result: instead of the source browser generated by doxygen, the links to\n# source code will now point to the output of htags.\n# The default value is: NO.\n# This tag requires that the tag SOURCE_BROWSER is set to YES.\n\nUSE_HTAGS              = NO\n\n# If the VERBATIM_HEADERS tag is set the YES then doxygen will generate a\n# verbatim copy of the header file for each class for which an include is\n# specified. Set to NO to disable this.\n# See also: Section \\class.\n# The default value is: YES.\n\nVERBATIM_HEADERS       = YES\n\n#---------------------------------------------------------------------------\n# Configuration options related to the alphabetical class index\n#---------------------------------------------------------------------------\n\n# If the ALPHABETICAL_INDEX tag is set to YES, an alphabetical index of all\n# compounds will be generated. Enable this if the project contains a lot of\n# classes, structs, unions or interfaces.\n# The default value is: YES.\n\nALPHABETICAL_INDEX     = YES\n\n# The COLS_IN_ALPHA_INDEX tag can be used to specify the number of columns in\n# which the alphabetical index list will be split.\n# Minimum value: 1, maximum value: 20, default value: 5.\n# This tag requires that the tag ALPHABETICAL_INDEX is set to YES.\n\nCOLS_IN_ALPHA_INDEX    = 5\n\n# In case all classes in a project start with a common prefix, all classes will\n# be put under the same header in the alphabetical index. The IGNORE_PREFIX tag\n# can be used to specify a prefix (or a list of prefixes) that should be ignored\n# while generating the index headers.\n# This tag requires that the tag ALPHABETICAL_INDEX is set to YES.\n\nIGNORE_PREFIX          =\n\n#---------------------------------------------------------------------------\n# Configuration options related to the HTML output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_HTML tag is set to YES doxygen will generate HTML output\n# The default value is: YES.\n\nGENERATE_HTML          = YES\n\n# The HTML_OUTPUT tag is used to specify where the HTML docs will be put. If a\n# relative path is entered the value of OUTPUT_DIRECTORY will be put in front of\n# it.\n# The default directory is: html.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_OUTPUT            = .\n\n# The HTML_FILE_EXTENSION tag can be used to specify the file extension for each\n# generated HTML page (for example: .htm, .php, .asp).\n# The default value is: .html.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_FILE_EXTENSION    = .html\n\n# The HTML_HEADER tag can be used to specify a user-defined HTML header file for\n# each generated HTML page. If the tag is left blank doxygen will generate a\n# standard header.\n#\n# To get valid HTML the header file that includes any scripts and style sheets\n# that doxygen needs, which is dependent on the configuration options used (e.g.\n# the setting GENERATE_TREEVIEW). It is highly recommended to start with a\n# default header using\n# doxygen -w html new_header.html new_footer.html new_stylesheet.css\n# YourConfigFile\n# and then modify the file new_header.html. See also section \"Doxygen usage\"\n# for information on how to generate the default header that doxygen normally\n# uses.\n# Note: The header is subject to change so you typically have to regenerate the\n# default header when upgrading to a newer version of doxygen. For a description\n# of the possible markers and block names see the documentation.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_HEADER            =\n\n# The HTML_FOOTER tag can be used to specify a user-defined HTML footer for each\n# generated HTML page. If the tag is left blank doxygen will generate a standard\n# footer. See HTML_HEADER for more information on how to generate a default\n# footer and what special commands can be used inside the footer. See also\n# section \"Doxygen usage\" for information on how to generate the default footer\n# that doxygen normally uses.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_FOOTER            =\n\n# The HTML_STYLESHEET tag can be used to specify a user-defined cascading style\n# sheet that is used by each HTML page. It can be used to fine-tune the look of\n# the HTML output. If left blank doxygen will generate a default style sheet.\n# See also section \"Doxygen usage\" for information on how to generate the style\n# sheet that doxygen normally uses.\n# Note: It is recommended to use HTML_EXTRA_STYLESHEET instead of this tag, as\n# it is more robust and this tag (HTML_STYLESHEET) will in the future become\n# obsolete.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_STYLESHEET        =\n\n# The HTML_EXTRA_STYLESHEET tag can be used to specify an additional user-\n# defined cascading style sheet that is included after the standard style sheets\n# created by doxygen. Using this option one can overrule certain style aspects.\n# This is preferred over using HTML_STYLESHEET since it does not replace the\n# standard style sheet and is therefor more robust against future updates.\n# Doxygen will copy the style sheet file to the output directory. For an example\n# see the documentation.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_EXTRA_STYLESHEET  =\n\n# The HTML_EXTRA_FILES tag can be used to specify one or more extra images or\n# other source files which should be copied to the HTML output directory. Note\n# that these files will be copied to the base HTML output directory. Use the\n# $relpath^ marker in the HTML_HEADER and/or HTML_FOOTER files to load these\n# files. In the HTML_STYLESHEET file, use the file name only. Also note that the\n# files will be copied as-is; there are no commands or markers available.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_EXTRA_FILES       =\n\n# The HTML_COLORSTYLE_HUE tag controls the color of the HTML output. Doxygen\n# will adjust the colors in the stylesheet and background images according to\n# this color. Hue is specified as an angle on a colorwheel, see\n# http://en.wikipedia.org/wiki/Hue for more information. For instance the value\n# 0 represents red, 60 is yellow, 120 is green, 180 is cyan, 240 is blue, 300\n# purple, and 360 is red again.\n# Minimum value: 0, maximum value: 359, default value: 220.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_COLORSTYLE_HUE    = 220\n\n# The HTML_COLORSTYLE_SAT tag controls the purity (or saturation) of the colors\n# in the HTML output. For a value of 0 the output will use grayscales only. A\n# value of 255 will produce the most vivid colors.\n# Minimum value: 0, maximum value: 255, default value: 100.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_COLORSTYLE_SAT    = 100\n\n# The HTML_COLORSTYLE_GAMMA tag controls the gamma correction applied to the\n# luminance component of the colors in the HTML output. Values below 100\n# gradually make the output lighter, whereas values above 100 make the output\n# darker. The value divided by 100 is the actual gamma applied, so 80 represents\n# a gamma of 0.8, The value 220 represents a gamma of 2.2, and 100 does not\n# change the gamma.\n# Minimum value: 40, maximum value: 240, default value: 80.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_COLORSTYLE_GAMMA  = 80\n\n# If the HTML_TIMESTAMP tag is set to YES then the footer of each generated HTML\n# page will contain the date and time when the page was generated. Setting this\n# to NO can help when comparing the output of multiple runs.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_TIMESTAMP         = YES\n\n# If the HTML_DYNAMIC_SECTIONS tag is set to YES then the generated HTML\n# documentation will contain sections that can be hidden and shown after the\n# page has loaded.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_DYNAMIC_SECTIONS  = NO\n\n# With HTML_INDEX_NUM_ENTRIES one can control the preferred number of entries\n# shown in the various tree structured indices initially; the user can expand\n# and collapse entries dynamically later on. Doxygen will expand the tree to\n# such a level that at most the specified number of entries are visible (unless\n# a fully collapsed tree already exceeds this amount). So setting the number of\n# entries 1 will produce a full collapsed tree by default. 0 is a special value\n# representing an infinite number of entries and will result in a full expanded\n# tree by default.\n# Minimum value: 0, maximum value: 9999, default value: 100.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nHTML_INDEX_NUM_ENTRIES = 100\n\n# If the GENERATE_DOCSET tag is set to YES, additional index files will be\n# generated that can be used as input for Apple's Xcode 3 integrated development\n# environment (see: http://developer.apple.com/tools/xcode/), introduced with\n# OSX 10.5 (Leopard). To create a documentation set, doxygen will generate a\n# Makefile in the HTML output directory. Running make will produce the docset in\n# that directory and running make install will install the docset in\n# ~/Library/Developer/Shared/Documentation/DocSets so that Xcode will find it at\n# startup. See http://developer.apple.com/tools/creatingdocsetswithdoxygen.html\n# for more information.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nGENERATE_DOCSET        = NO\n\n# This tag determines the name of the docset feed. A documentation feed provides\n# an umbrella under which multiple documentation sets from a single provider\n# (such as a company or product suite) can be grouped.\n# The default value is: Doxygen generated docs.\n# This tag requires that the tag GENERATE_DOCSET is set to YES.\n\nDOCSET_FEEDNAME        = \"Doxygen generated docs\"\n\n# This tag specifies a string that should uniquely identify the documentation\n# set bundle. This should be a reverse domain-name style string, e.g.\n# com.mycompany.MyDocSet. Doxygen will append .docset to the name.\n# The default value is: org.doxygen.Project.\n# This tag requires that the tag GENERATE_DOCSET is set to YES.\n\nDOCSET_BUNDLE_ID       = org.doxygen.Project\n\n# The DOCSET_PUBLISHER_ID tag specifies a string that should uniquely identify\n# the documentation publisher. This should be a reverse domain-name style\n# string, e.g. com.mycompany.MyDocSet.documentation.\n# The default value is: org.doxygen.Publisher.\n# This tag requires that the tag GENERATE_DOCSET is set to YES.\n\nDOCSET_PUBLISHER_ID    = org.doxygen.Publisher\n\n# The DOCSET_PUBLISHER_NAME tag identifies the documentation publisher.\n# The default value is: Publisher.\n# This tag requires that the tag GENERATE_DOCSET is set to YES.\n\nDOCSET_PUBLISHER_NAME  = Publisher\n\n# If the GENERATE_HTMLHELP tag is set to YES then doxygen generates three\n# additional HTML index files: index.hhp, index.hhc, and index.hhk. The\n# index.hhp is a project file that can be read by Microsoft's HTML Help Workshop\n# (see: http://www.microsoft.com/en-us/download/details.aspx?id=21138) on\n# Windows.\n#\n# The HTML Help Workshop contains a compiler that can convert all HTML output\n# generated by doxygen into a single compiled HTML file (.chm). Compiled HTML\n# files are now used as the Windows 98 help format, and will replace the old\n# Windows help format (.hlp) on all Windows platforms in the future. Compressed\n# HTML files also contain an index, a table of contents, and you can search for\n# words in the documentation. The HTML workshop also contains a viewer for\n# compressed HTML files.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nGENERATE_HTMLHELP      = NO\n\n# The CHM_FILE tag can be used to specify the file name of the resulting .chm\n# file. You can add a path in front of the file if the result should not be\n# written to the html output directory.\n# This tag requires that the tag GENERATE_HTMLHELP is set to YES.\n\nCHM_FILE               =\n\n# The HHC_LOCATION tag can be used to specify the location (absolute path\n# including file name) of the HTML help compiler ( hhc.exe). If non-empty\n# doxygen will try to run the HTML help compiler on the generated index.hhp.\n# The file has to be specified with full path.\n# This tag requires that the tag GENERATE_HTMLHELP is set to YES.\n\nHHC_LOCATION           =\n\n# The GENERATE_CHI flag controls if a separate .chi index file is generated (\n# YES) or that it should be included in the master .chm file ( NO).\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTMLHELP is set to YES.\n\nGENERATE_CHI           = NO\n\n# The CHM_INDEX_ENCODING is used to encode HtmlHelp index ( hhk), content ( hhc)\n# and project file content.\n# This tag requires that the tag GENERATE_HTMLHELP is set to YES.\n\nCHM_INDEX_ENCODING     =\n\n# The BINARY_TOC flag controls whether a binary table of contents is generated (\n# YES) or a normal table of contents ( NO) in the .chm file.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTMLHELP is set to YES.\n\nBINARY_TOC             = NO\n\n# The TOC_EXPAND flag can be set to YES to add extra items for group members to\n# the table of contents of the HTML help documentation and to the tree view.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTMLHELP is set to YES.\n\nTOC_EXPAND             = NO\n\n# If the GENERATE_QHP tag is set to YES and both QHP_NAMESPACE and\n# QHP_VIRTUAL_FOLDER are set, an additional index file will be generated that\n# can be used as input for Qt's qhelpgenerator to generate a Qt Compressed Help\n# (.qch) of the generated HTML documentation.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nGENERATE_QHP           = NO\n\n# If the QHG_LOCATION tag is specified, the QCH_FILE tag can be used to specify\n# the file name of the resulting .qch file. The path specified is relative to\n# the HTML output folder.\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQCH_FILE               =\n\n# The QHP_NAMESPACE tag specifies the namespace to use when generating Qt Help\n# Project output. For more information please see Qt Help Project / Namespace\n# (see: http://qt-project.org/doc/qt-4.8/qthelpproject.html#namespace).\n# The default value is: org.doxygen.Project.\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQHP_NAMESPACE          = org.doxygen.Project\n\n# The QHP_VIRTUAL_FOLDER tag specifies the namespace to use when generating Qt\n# Help Project output. For more information please see Qt Help Project / Virtual\n# Folders (see: http://qt-project.org/doc/qt-4.8/qthelpproject.html#virtual-\n# folders).\n# The default value is: doc.\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQHP_VIRTUAL_FOLDER     = doc\n\n# If the QHP_CUST_FILTER_NAME tag is set, it specifies the name of a custom\n# filter to add. For more information please see Qt Help Project / Custom\n# Filters (see: http://qt-project.org/doc/qt-4.8/qthelpproject.html#custom-\n# filters).\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQHP_CUST_FILTER_NAME   =\n\n# The QHP_CUST_FILTER_ATTRS tag specifies the list of the attributes of the\n# custom filter to add. For more information please see Qt Help Project / Custom\n# Filters (see: http://qt-project.org/doc/qt-4.8/qthelpproject.html#custom-\n# filters).\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQHP_CUST_FILTER_ATTRS  =\n\n# The QHP_SECT_FILTER_ATTRS tag specifies the list of the attributes this\n# project's filter section matches. Qt Help Project / Filter Attributes (see:\n# http://qt-project.org/doc/qt-4.8/qthelpproject.html#filter-attributes).\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQHP_SECT_FILTER_ATTRS  =\n\n# The QHG_LOCATION tag can be used to specify the location of Qt's\n# qhelpgenerator. If non-empty doxygen will try to run qhelpgenerator on the\n# generated .qhp file.\n# This tag requires that the tag GENERATE_QHP is set to YES.\n\nQHG_LOCATION           =\n\n# If the GENERATE_ECLIPSEHELP tag is set to YES, additional index files will be\n# generated, together with the HTML files, they form an Eclipse help plugin. To\n# install this plugin and make it available under the help contents menu in\n# Eclipse, the contents of the directory containing the HTML and XML files needs\n# to be copied into the plugins directory of eclipse. The name of the directory\n# within the plugins directory should be the same as the ECLIPSE_DOC_ID value.\n# After copying Eclipse needs to be restarted before the help appears.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nGENERATE_ECLIPSEHELP   = NO\n\n# A unique identifier for the Eclipse help plugin. When installing the plugin\n# the directory name containing the HTML and XML files should also have this\n# name. Each documentation set should have its own identifier.\n# The default value is: org.doxygen.Project.\n# This tag requires that the tag GENERATE_ECLIPSEHELP is set to YES.\n\nECLIPSE_DOC_ID         = org.doxygen.Project\n\n# If you want full control over the layout of the generated HTML pages it might\n# be necessary to disable the index and replace it with your own. The\n# DISABLE_INDEX tag can be used to turn on/off the condensed index (tabs) at top\n# of each HTML page. A value of NO enables the index and the value YES disables\n# it. Since the tabs in the index contain the same information as the navigation\n# tree, you can set this option to YES if you also set GENERATE_TREEVIEW to YES.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nDISABLE_INDEX          = NO\n\n# The GENERATE_TREEVIEW tag is used to specify whether a tree-like index\n# structure should be generated to display hierarchical information. If the tag\n# value is set to YES, a side panel will be generated containing a tree-like\n# index structure (just like the one that is generated for HTML Help). For this\n# to work a browser that supports JavaScript, DHTML, CSS and frames is required\n# (i.e. any modern browser). Windows users are probably better off using the\n# HTML help feature. Via custom stylesheets (see HTML_EXTRA_STYLESHEET) one can\n# further fine-tune the look of the index. As an example, the default style\n# sheet generated by doxygen has an example that shows how to put an image at\n# the root of the tree instead of the PROJECT_NAME. Since the tree basically has\n# the same information as the tab index, you could consider setting\n# DISABLE_INDEX to YES when enabling this option.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nGENERATE_TREEVIEW      = NO\n\n# The ENUM_VALUES_PER_LINE tag can be used to set the number of enum values that\n# doxygen will group on one line in the generated HTML documentation.\n#\n# Note that a value of 0 will completely suppress the enum values from appearing\n# in the overview section.\n# Minimum value: 0, maximum value: 20, default value: 4.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nENUM_VALUES_PER_LINE   = 4\n\n# If the treeview is enabled (see GENERATE_TREEVIEW) then this tag can be used\n# to set the initial width (in pixels) of the frame in which the tree is shown.\n# Minimum value: 0, maximum value: 1500, default value: 250.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nTREEVIEW_WIDTH         = 250\n\n# When the EXT_LINKS_IN_WINDOW option is set to YES doxygen will open links to\n# external symbols imported via tag files in a separate window.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nEXT_LINKS_IN_WINDOW    = NO\n\n# Use this tag to change the font size of LaTeX formulas included as images in\n# the HTML documentation. When you change the font size after a successful\n# doxygen run you need to manually remove any form_*.png images from the HTML\n# output directory to force them to be regenerated.\n# Minimum value: 8, maximum value: 50, default value: 10.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nFORMULA_FONTSIZE       = 10\n\n# Use the FORMULA_TRANPARENT tag to determine whether or not the images\n# generated for formulas are transparent PNGs. Transparent PNGs are not\n# supported properly for IE 6.0, but are supported on all modern browsers.\n#\n# Note that when changing this option you need to delete any form_*.png files in\n# the HTML output directory before the changes have effect.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nFORMULA_TRANSPARENT    = YES\n\n# Enable the USE_MATHJAX option to render LaTeX formulas using MathJax (see\n# http://www.mathjax.org) which uses client side Javascript for the rendering\n# instead of using prerendered bitmaps. Use this if you do not have LaTeX\n# installed or if you want to formulas look prettier in the HTML output. When\n# enabled you may also need to install MathJax separately and configure the path\n# to it using the MATHJAX_RELPATH option.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nUSE_MATHJAX            = NO\n\n# When MathJax is enabled you can set the default output format to be used for\n# the MathJax output. See the MathJax site (see:\n# http://docs.mathjax.org/en/latest/output.html) for more details.\n# Possible values are: HTML-CSS (which is slower, but has the best\n# compatibility), NativeMML (i.e. MathML) and SVG.\n# The default value is: HTML-CSS.\n# This tag requires that the tag USE_MATHJAX is set to YES.\n\nMATHJAX_FORMAT         = HTML-CSS\n\n# When MathJax is enabled you need to specify the location relative to the HTML\n# output directory using the MATHJAX_RELPATH option. The destination directory\n# should contain the MathJax.js script. For instance, if the mathjax directory\n# is located at the same level as the HTML output directory, then\n# MATHJAX_RELPATH should be ../mathjax. The default value points to the MathJax\n# Content Delivery Network so you can quickly see the result without installing\n# MathJax. However, it is strongly recommended to install a local copy of\n# MathJax from http://www.mathjax.org before deployment.\n# The default value is: http://cdn.mathjax.org/mathjax/latest.\n# This tag requires that the tag USE_MATHJAX is set to YES.\n\nMATHJAX_RELPATH        = http://cdn.mathjax.org/mathjax/latest\n\n# The MATHJAX_EXTENSIONS tag can be used to specify one or more MathJax\n# extension names that should be enabled during MathJax rendering. For example\n# MATHJAX_EXTENSIONS = TeX/AMSmath TeX/AMSsymbols\n# This tag requires that the tag USE_MATHJAX is set to YES.\n\nMATHJAX_EXTENSIONS     =\n\n# The MATHJAX_CODEFILE tag can be used to specify a file with javascript pieces\n# of code that will be used on startup of the MathJax code. See the MathJax site\n# (see: http://docs.mathjax.org/en/latest/output.html) for more details. For an\n# example see the documentation.\n# This tag requires that the tag USE_MATHJAX is set to YES.\n\nMATHJAX_CODEFILE       =\n\n# When the SEARCHENGINE tag is enabled doxygen will generate a search box for\n# the HTML output. The underlying search engine uses javascript and DHTML and\n# should work on any modern browser. Note that when using HTML help\n# (GENERATE_HTMLHELP), Qt help (GENERATE_QHP), or docsets (GENERATE_DOCSET)\n# there is already a search function so this one should typically be disabled.\n# For large projects the javascript based search engine can be slow, then\n# enabling SERVER_BASED_SEARCH may provide a better solution. It is possible to\n# search using the keyboard; to jump to the search box use <access key> + S\n# (what the <access key> is depends on the OS and browser, but it is typically\n# <CTRL>, <ALT>/<option>, or both). Inside the search box use the <cursor down\n# key> to jump into the search results window, the results can be navigated\n# using the <cursor keys>. Press <Enter> to select an item or <escape> to cancel\n# the search. The filter options can be selected when the cursor is inside the\n# search box by pressing <Shift>+<cursor down>. Also here use the <cursor keys>\n# to select a filter and <Enter> or <escape> to activate or cancel the filter\n# option.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_HTML is set to YES.\n\nSEARCHENGINE           = YES\n\n# When the SERVER_BASED_SEARCH tag is enabled the search engine will be\n# implemented using a web server instead of a web client using Javascript. There\n# are two flavours of web server based searching depending on the\n# EXTERNAL_SEARCH setting. When disabled, doxygen will generate a PHP script for\n# searching and an index file used by the script. When EXTERNAL_SEARCH is\n# enabled the indexing and searching needs to be provided by external tools. See\n# the section \"External Indexing and Searching\" for details.\n# The default value is: NO.\n# This tag requires that the tag SEARCHENGINE is set to YES.\n\nSERVER_BASED_SEARCH    = NO\n\n# When EXTERNAL_SEARCH tag is enabled doxygen will no longer generate the PHP\n# script for searching. Instead the search results are written to an XML file\n# which needs to be processed by an external indexer. Doxygen will invoke an\n# external search engine pointed to by the SEARCHENGINE_URL option to obtain the\n# search results.\n#\n# Doxygen ships with an example indexer ( doxyindexer) and search engine\n# (doxysearch.cgi) which are based on the open source search engine library\n# Xapian (see: http://xapian.org/).\n#\n# See the section \"External Indexing and Searching\" for details.\n# The default value is: NO.\n# This tag requires that the tag SEARCHENGINE is set to YES.\n\nEXTERNAL_SEARCH        = NO\n\n# The SEARCHENGINE_URL should point to a search engine hosted by a web server\n# which will return the search results when EXTERNAL_SEARCH is enabled.\n#\n# Doxygen ships with an example indexer ( doxyindexer) and search engine\n# (doxysearch.cgi) which are based on the open source search engine library\n# Xapian (see: http://xapian.org/). See the section \"External Indexing and\n# Searching\" for details.\n# This tag requires that the tag SEARCHENGINE is set to YES.\n\nSEARCHENGINE_URL       =\n\n# When SERVER_BASED_SEARCH and EXTERNAL_SEARCH are both enabled the unindexed\n# search data is written to a file for indexing by an external tool. With the\n# SEARCHDATA_FILE tag the name of this file can be specified.\n# The default file is: searchdata.xml.\n# This tag requires that the tag SEARCHENGINE is set to YES.\n\nSEARCHDATA_FILE        = searchdata.xml\n\n# When SERVER_BASED_SEARCH and EXTERNAL_SEARCH are both enabled the\n# EXTERNAL_SEARCH_ID tag can be used as an identifier for the project. This is\n# useful in combination with EXTRA_SEARCH_MAPPINGS to search through multiple\n# projects and redirect the results back to the right project.\n# This tag requires that the tag SEARCHENGINE is set to YES.\n\nEXTERNAL_SEARCH_ID     =\n\n# The EXTRA_SEARCH_MAPPINGS tag can be used to enable searching through doxygen\n# projects other than the one defined by this configuration file, but that are\n# all added to the same external search index. Each project needs to have a\n# unique id set via EXTERNAL_SEARCH_ID. The search mapping then maps the id of\n# to a relative location where the documentation can be found. The format is:\n# EXTRA_SEARCH_MAPPINGS = tagname1=loc1 tagname2=loc2 ...\n# This tag requires that the tag SEARCHENGINE is set to YES.\n\nEXTRA_SEARCH_MAPPINGS  =\n\n#---------------------------------------------------------------------------\n# Configuration options related to the LaTeX output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_LATEX tag is set to YES doxygen will generate LaTeX output.\n# The default value is: YES.\n\nGENERATE_LATEX         = NO\n\n# The LATEX_OUTPUT tag is used to specify where the LaTeX docs will be put. If a\n# relative path is entered the value of OUTPUT_DIRECTORY will be put in front of\n# it.\n# The default directory is: latex.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_OUTPUT           = latex\n\n# The LATEX_CMD_NAME tag can be used to specify the LaTeX command name to be\n# invoked.\n#\n# Note that when enabling USE_PDFLATEX this option is only used for generating\n# bitmaps for formulas in the HTML output, but not in the Makefile that is\n# written to the output directory.\n# The default file is: latex.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_CMD_NAME         = latex\n\n# The MAKEINDEX_CMD_NAME tag can be used to specify the command name to generate\n# index for LaTeX.\n# The default file is: makeindex.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nMAKEINDEX_CMD_NAME     = makeindex\n\n# If the COMPACT_LATEX tag is set to YES doxygen generates more compact LaTeX\n# documents. This may be useful for small projects and may help to save some\n# trees in general.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nCOMPACT_LATEX          = NO\n\n# The PAPER_TYPE tag can be used to set the paper type that is used by the\n# printer.\n# Possible values are: a4 (210 x 297 mm), letter (8.5 x 11 inches), legal (8.5 x\n# 14 inches) and executive (7.25 x 10.5 inches).\n# The default value is: a4.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nPAPER_TYPE             = a4\n\n# The EXTRA_PACKAGES tag can be used to specify one or more LaTeX package names\n# that should be included in the LaTeX output. To get the times font for\n# instance you can specify\n# EXTRA_PACKAGES=times\n# If left blank no extra packages will be included.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nEXTRA_PACKAGES         =\n\n# The LATEX_HEADER tag can be used to specify a personal LaTeX header for the\n# generated LaTeX document. The header should contain everything until the first\n# chapter. If it is left blank doxygen will generate a standard header. See\n# section \"Doxygen usage\" for information on how to let doxygen write the\n# default header to a separate file.\n#\n# Note: Only use a user-defined header if you know what you are doing! The\n# following commands have a special meaning inside the header: $title,\n# $datetime, $date, $doxygenversion, $projectname, $projectnumber. Doxygen will\n# replace them by respectively the title of the page, the current date and time,\n# only the current date, the version number of doxygen, the project name (see\n# PROJECT_NAME), or the project number (see PROJECT_NUMBER).\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_HEADER           =\n\n# The LATEX_FOOTER tag can be used to specify a personal LaTeX footer for the\n# generated LaTeX document. The footer should contain everything after the last\n# chapter. If it is left blank doxygen will generate a standard footer.\n#\n# Note: Only use a user-defined footer if you know what you are doing!\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_FOOTER           =\n\n# The LATEX_EXTRA_FILES tag can be used to specify one or more extra images or\n# other source files which should be copied to the LATEX_OUTPUT output\n# directory. Note that the files will be copied as-is; there are no commands or\n# markers available.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_EXTRA_FILES      =\n\n# If the PDF_HYPERLINKS tag is set to YES, the LaTeX that is generated is\n# prepared for conversion to PDF (using ps2pdf or pdflatex). The PDF file will\n# contain links (just like the HTML output) instead of page references. This\n# makes the output suitable for online browsing using a PDF viewer.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nPDF_HYPERLINKS         = YES\n\n# If the LATEX_PDFLATEX tag is set to YES, doxygen will use pdflatex to generate\n# the PDF file directly from the LaTeX files. Set this option to YES to get a\n# higher quality PDF documentation.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nUSE_PDFLATEX           = YES\n\n# If the LATEX_BATCHMODE tag is set to YES, doxygen will add the \\batchmode\n# command to the generated LaTeX files. This will instruct LaTeX to keep running\n# if errors occur, instead of asking the user for help. This option is also used\n# when generating formulas in HTML.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_BATCHMODE        = NO\n\n# If the LATEX_HIDE_INDICES tag is set to YES then doxygen will not include the\n# index chapters (such as File Index, Compound Index, etc.) in the output.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_HIDE_INDICES     = NO\n\n# If the LATEX_SOURCE_CODE tag is set to YES then doxygen will include source\n# code with syntax highlighting in the LaTeX output.\n#\n# Note that which sources are shown also depends on other settings such as\n# SOURCE_BROWSER.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_SOURCE_CODE      = NO\n\n# The LATEX_BIB_STYLE tag can be used to specify the style to use for the\n# bibliography, e.g. plainnat, or ieeetr. See\n# http://en.wikipedia.org/wiki/BibTeX and \\cite for more info.\n# The default value is: plain.\n# This tag requires that the tag GENERATE_LATEX is set to YES.\n\nLATEX_BIB_STYLE        = plain\n\n#---------------------------------------------------------------------------\n# Configuration options related to the RTF output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_RTF tag is set to YES doxygen will generate RTF output. The\n# RTF output is optimized for Word 97 and may not look too pretty with other RTF\n# readers/editors.\n# The default value is: NO.\n\nGENERATE_RTF           = NO\n\n# The RTF_OUTPUT tag is used to specify where the RTF docs will be put. If a\n# relative path is entered the value of OUTPUT_DIRECTORY will be put in front of\n# it.\n# The default directory is: rtf.\n# This tag requires that the tag GENERATE_RTF is set to YES.\n\nRTF_OUTPUT             = rtf\n\n# If the COMPACT_RTF tag is set to YES doxygen generates more compact RTF\n# documents. This may be useful for small projects and may help to save some\n# trees in general.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_RTF is set to YES.\n\nCOMPACT_RTF            = NO\n\n# If the RTF_HYPERLINKS tag is set to YES, the RTF that is generated will\n# contain hyperlink fields. The RTF file will contain links (just like the HTML\n# output) instead of page references. This makes the output suitable for online\n# browsing using Word or some other Word compatible readers that support those\n# fields.\n#\n# Note: WordPad (write) and others do not support links.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_RTF is set to YES.\n\nRTF_HYPERLINKS         = NO\n\n# Load stylesheet definitions from file. Syntax is similar to doxygen's config\n# file, i.e. a series of assignments. You only have to provide replacements,\n# missing definitions are set to their default value.\n#\n# See also section \"Doxygen usage\" for information on how to generate the\n# default style sheet that doxygen normally uses.\n# This tag requires that the tag GENERATE_RTF is set to YES.\n\nRTF_STYLESHEET_FILE    =\n\n# Set optional variables used in the generation of an RTF document. Syntax is\n# similar to doxygen's config file. A template extensions file can be generated\n# using doxygen -e rtf extensionFile.\n# This tag requires that the tag GENERATE_RTF is set to YES.\n\nRTF_EXTENSIONS_FILE    =\n\n#---------------------------------------------------------------------------\n# Configuration options related to the man page output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_MAN tag is set to YES doxygen will generate man pages for\n# classes and files.\n# The default value is: NO.\n\nGENERATE_MAN           = NO\n\n# The MAN_OUTPUT tag is used to specify where the man pages will be put. If a\n# relative path is entered the value of OUTPUT_DIRECTORY will be put in front of\n# it. A directory man3 will be created inside the directory specified by\n# MAN_OUTPUT.\n# The default directory is: man.\n# This tag requires that the tag GENERATE_MAN is set to YES.\n\nMAN_OUTPUT             = man\n\n# The MAN_EXTENSION tag determines the extension that is added to the generated\n# man pages. In case the manual section does not start with a number, the number\n# 3 is prepended. The dot (.) at the beginning of the MAN_EXTENSION tag is\n# optional.\n# The default value is: .3.\n# This tag requires that the tag GENERATE_MAN is set to YES.\n\nMAN_EXTENSION          = .3\n\n# If the MAN_LINKS tag is set to YES and doxygen generates man output, then it\n# will generate one additional man file for each entity documented in the real\n# man page(s). These additional files only source the real man page, but without\n# them the man command would be unable to find the correct page.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_MAN is set to YES.\n\nMAN_LINKS              = NO\n\n#---------------------------------------------------------------------------\n# Configuration options related to the XML output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_XML tag is set to YES doxygen will generate an XML file that\n# captures the structure of the code including all documentation.\n# The default value is: NO.\n\nGENERATE_XML           = NO\n\n# The XML_OUTPUT tag is used to specify where the XML pages will be put. If a\n# relative path is entered the value of OUTPUT_DIRECTORY will be put in front of\n# it.\n# The default directory is: xml.\n# This tag requires that the tag GENERATE_XML is set to YES.\n\nXML_OUTPUT             = xml\n\n# The XML_SCHEMA tag can be used to specify a XML schema, which can be used by a\n# validating XML parser to check the syntax of the XML files.\n# This tag requires that the tag GENERATE_XML is set to YES.\n\nXML_SCHEMA             =\n\n# The XML_DTD tag can be used to specify a XML DTD, which can be used by a\n# validating XML parser to check the syntax of the XML files.\n# This tag requires that the tag GENERATE_XML is set to YES.\n\nXML_DTD                =\n\n# If the XML_PROGRAMLISTING tag is set to YES doxygen will dump the program\n# listings (including syntax highlighting and cross-referencing information) to\n# the XML output. Note that enabling this will significantly increase the size\n# of the XML output.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_XML is set to YES.\n\nXML_PROGRAMLISTING     = YES\n\n#---------------------------------------------------------------------------\n# Configuration options related to the DOCBOOK output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_DOCBOOK tag is set to YES doxygen will generate Docbook files\n# that can be used to generate PDF.\n# The default value is: NO.\n\nGENERATE_DOCBOOK       = NO\n\n# The DOCBOOK_OUTPUT tag is used to specify where the Docbook pages will be put.\n# If a relative path is entered the value of OUTPUT_DIRECTORY will be put in\n# front of it.\n# The default directory is: docbook.\n# This tag requires that the tag GENERATE_DOCBOOK is set to YES.\n\nDOCBOOK_OUTPUT         = docbook\n\n#---------------------------------------------------------------------------\n# Configuration options for the AutoGen Definitions output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_AUTOGEN_DEF tag is set to YES doxygen will generate an AutoGen\n# Definitions (see http://autogen.sf.net) file that captures the structure of\n# the code including all documentation. Note that this feature is still\n# experimental and incomplete at the moment.\n# The default value is: NO.\n\nGENERATE_AUTOGEN_DEF   = NO\n\n#---------------------------------------------------------------------------\n# Configuration options related to the Perl module output\n#---------------------------------------------------------------------------\n\n# If the GENERATE_PERLMOD tag is set to YES doxygen will generate a Perl module\n# file that captures the structure of the code including all documentation.\n#\n# Note that this feature is still experimental and incomplete at the moment.\n# The default value is: NO.\n\nGENERATE_PERLMOD       = NO\n\n# If the PERLMOD_LATEX tag is set to YES doxygen will generate the necessary\n# Makefile rules, Perl scripts and LaTeX code to be able to generate PDF and DVI\n# output from the Perl module output.\n# The default value is: NO.\n# This tag requires that the tag GENERATE_PERLMOD is set to YES.\n\nPERLMOD_LATEX          = NO\n\n# If the PERLMOD_PRETTY tag is set to YES the Perl module output will be nicely\n# formatted so it can be parsed by a human reader. This is useful if you want to\n# understand what is going on. On the other hand, if this tag is set to NO the\n# size of the Perl module output will be much smaller and Perl will parse it\n# just the same.\n# The default value is: YES.\n# This tag requires that the tag GENERATE_PERLMOD is set to YES.\n\nPERLMOD_PRETTY         = YES\n\n# The names of the make variables in the generated doxyrules.make file are\n# prefixed with the string contained in PERLMOD_MAKEVAR_PREFIX. This is useful\n# so different doxyrules.make files included by the same Makefile don't\n# overwrite each other's variables.\n# This tag requires that the tag GENERATE_PERLMOD is set to YES.\n\nPERLMOD_MAKEVAR_PREFIX =\n\n#---------------------------------------------------------------------------\n# Configuration options related to the preprocessor\n#---------------------------------------------------------------------------\n\n# If the ENABLE_PREPROCESSING tag is set to YES doxygen will evaluate all\n# C-preprocessor directives found in the sources and include files.\n# The default value is: YES.\n\nENABLE_PREPROCESSING   = YES\n\n# If the MACRO_EXPANSION tag is set to YES doxygen will expand all macro names\n# in the source code. If set to NO only conditional compilation will be\n# performed. Macro expansion can be done in a controlled way by setting\n# EXPAND_ONLY_PREDEF to YES.\n# The default value is: NO.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nMACRO_EXPANSION        = NO\n\n# If the EXPAND_ONLY_PREDEF and MACRO_EXPANSION tags are both set to YES then\n# the macro expansion is limited to the macros specified with the PREDEFINED and\n# EXPAND_AS_DEFINED tags.\n# The default value is: NO.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nEXPAND_ONLY_PREDEF     = NO\n\n# If the SEARCH_INCLUDES tag is set to YES the includes files in the\n# INCLUDE_PATH will be searched if a #include is found.\n# The default value is: YES.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nSEARCH_INCLUDES        = YES\n\n# The INCLUDE_PATH tag can be used to specify one or more directories that\n# contain include files that are not input files but should be processed by the\n# preprocessor.\n# This tag requires that the tag SEARCH_INCLUDES is set to YES.\n\nINCLUDE_PATH           =\n\n# You can use the INCLUDE_FILE_PATTERNS tag to specify one or more wildcard\n# patterns (like *.h and *.hpp) to filter out the header-files in the\n# directories. If left blank, the patterns specified with FILE_PATTERNS will be\n# used.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nINCLUDE_FILE_PATTERNS  =\n\n# The PREDEFINED tag can be used to specify one or more macro names that are\n# defined before the preprocessor is started (similar to the -D option of e.g.\n# gcc). The argument of the tag is a list of macros of the form: name or\n# name=definition (no spaces). If the definition and the \"=\" are omitted, \"=1\"\n# is assumed. To prevent a macro definition from being undefined via #undef or\n# recursively expanded use the := operator instead of the = operator.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nPREDEFINED             =\n\n# If the MACRO_EXPANSION and EXPAND_ONLY_PREDEF tags are set to YES then this\n# tag can be used to specify a list of macro names that should be expanded. The\n# macro definition that is found in the sources will be used. Use the PREDEFINED\n# tag if you want to use a different macro definition that overrules the\n# definition found in the source code.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nEXPAND_AS_DEFINED      =\n\n# If the SKIP_FUNCTION_MACROS tag is set to YES then doxygen's preprocessor will\n# remove all refrences to function-like macros that are alone on a line, have an\n# all uppercase name, and do not end with a semicolon. Such function macros are\n# typically used for boiler-plate code, and will confuse the parser if not\n# removed.\n# The default value is: YES.\n# This tag requires that the tag ENABLE_PREPROCESSING is set to YES.\n\nSKIP_FUNCTION_MACROS   = YES\n\n#---------------------------------------------------------------------------\n# Configuration options related to external references\n#---------------------------------------------------------------------------\n\n# The TAGFILES tag can be used to specify one or more tag files. For each tag\n# file the location of the external documentation should be added. The format of\n# a tag file without this location is as follows:\n# TAGFILES = file1 file2 ...\n# Adding location for the tag files is done as follows:\n# TAGFILES = file1=loc1 \"file2 = loc2\" ...\n# where loc1 and loc2 can be relative or absolute paths or URLs. See the\n# section \"Linking to external documentation\" for more information about the use\n# of tag files.\n# Note: Each tag file must have an unique name (where the name does NOT include\n# the path). If a tag file is not located in the directory in which doxygen is\n# run, you must also specify the path to the tagfile here.\n\nTAGFILES               =\n\n# When a file name is specified after GENERATE_TAGFILE, doxygen will create a\n# tag file that is based on the input files it reads. See section \"Linking to\n# external documentation\" for more information about the usage of tag files.\n\nGENERATE_TAGFILE       =\n\n# If the ALLEXTERNALS tag is set to YES all external class will be listed in the\n# class index. If set to NO only the inherited external classes will be listed.\n# The default value is: NO.\n\nALLEXTERNALS           = NO\n\n# If the EXTERNAL_GROUPS tag is set to YES all external groups will be listed in\n# the modules index. If set to NO, only the current project's groups will be\n# listed.\n# The default value is: YES.\n\nEXTERNAL_GROUPS        = YES\n\n# If the EXTERNAL_PAGES tag is set to YES all external pages will be listed in\n# the related pages index. If set to NO, only the current project's pages will\n# be listed.\n# The default value is: YES.\n\nEXTERNAL_PAGES         = YES\n\n# The PERL_PATH should be the absolute path and name of the perl script\n# interpreter (i.e. the result of 'which perl').\n# The default file (with absolute path) is: /usr/bin/perl.\n\nPERL_PATH              = /usr/bin/perl\n\n#---------------------------------------------------------------------------\n# Configuration options related to the dot tool\n#---------------------------------------------------------------------------\n\n# If the CLASS_DIAGRAMS tag is set to YES doxygen will generate a class diagram\n# (in HTML and LaTeX) for classes with base or super classes. Setting the tag to\n# NO turns the diagrams off. Note that this option also works with HAVE_DOT\n# disabled, but it is recommended to install and use dot, since it yields more\n# powerful graphs.\n# The default value is: YES.\n\nCLASS_DIAGRAMS         = YES\n\n# You can define message sequence charts within doxygen comments using the \\msc\n# command. Doxygen will then run the mscgen tool (see:\n# http://www.mcternan.me.uk/mscgen/)) to produce the chart and insert it in the\n# documentation. The MSCGEN_PATH tag allows you to specify the directory where\n# the mscgen tool resides. If left empty the tool is assumed to be found in the\n# default search path.\n\nMSCGEN_PATH            =\n\n# You can include diagrams made with dia in doxygen documentation. Doxygen will\n# then run dia to produce the diagram and insert it in the documentation. The\n# DIA_PATH tag allows you to specify the directory where the dia binary resides.\n# If left empty dia is assumed to be found in the default search path.\n\nDIA_PATH               =\n\n# If set to YES, the inheritance and collaboration graphs will hide inheritance\n# and usage relations if the target is undocumented or is not a class.\n# The default value is: YES.\n\nHIDE_UNDOC_RELATIONS   = YES\n\n# If you set the HAVE_DOT tag to YES then doxygen will assume the dot tool is\n# available from the path. This tool is part of Graphviz (see:\n# http://www.graphviz.org/), a graph visualization toolkit from AT&T and Lucent\n# Bell Labs. The other options in this section have no effect if this option is\n# set to NO\n# The default value is: NO.\n\nHAVE_DOT               = NO\n\n# The DOT_NUM_THREADS specifies the number of dot invocations doxygen is allowed\n# to run in parallel. When set to 0 doxygen will base this on the number of\n# processors available in the system. You can set it explicitly to a value\n# larger than 0 to get control over the balance between CPU load and processing\n# speed.\n# Minimum value: 0, maximum value: 32, default value: 0.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nDOT_NUM_THREADS        = 0\n\n# When you want a differently looking font n the dot files that doxygen\n# generates you can specify the font name using DOT_FONTNAME. You need to make\n# sure dot is able to find the font, which can be done by putting it in a\n# standard location or by setting the DOTFONTPATH environment variable or by\n# setting DOT_FONTPATH to the directory containing the font.\n# The default value is: Helvetica.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nDOT_FONTNAME           = Helvetica\n\n# The DOT_FONTSIZE tag can be used to set the size (in points) of the font of\n# dot graphs.\n# Minimum value: 4, maximum value: 24, default value: 10.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nDOT_FONTSIZE           = 10\n\n# By default doxygen will tell dot to use the default font as specified with\n# DOT_FONTNAME. If you specify a different font using DOT_FONTNAME you can set\n# the path where dot can find it using this tag.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nDOT_FONTPATH           =\n\n# If the CLASS_GRAPH tag is set to YES then doxygen will generate a graph for\n# each documented class showing the direct and indirect inheritance relations.\n# Setting this tag to YES will force the CLASS_DIAGRAMS tag to NO.\n# The default value is: YES.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nCLASS_GRAPH            = YES\n\n# If the COLLABORATION_GRAPH tag is set to YES then doxygen will generate a\n# graph for each documented class showing the direct and indirect implementation\n# dependencies (inheritance, containment, and class references variables) of the\n# class with other documented classes.\n# The default value is: YES.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nCOLLABORATION_GRAPH    = YES\n\n# If the GROUP_GRAPHS tag is set to YES then doxygen will generate a graph for\n# groups, showing the direct groups dependencies.\n# The default value is: YES.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nGROUP_GRAPHS           = YES\n\n# If the UML_LOOK tag is set to YES doxygen will generate inheritance and\n# collaboration diagrams in a style similar to the OMG's Unified Modeling\n# Language.\n# The default value is: NO.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nUML_LOOK               = NO\n\n# If the UML_LOOK tag is enabled, the fields and methods are shown inside the\n# class node. If there are many fields or methods and many nodes the graph may\n# become too big to be useful. The UML_LIMIT_NUM_FIELDS threshold limits the\n# number of items for each type to make the size more manageable. Set this to 0\n# for no limit. Note that the threshold may be exceeded by 50% before the limit\n# is enforced. So when you set the threshold to 10, up to 15 fields may appear,\n# but if the number exceeds 15, the total amount of fields shown is limited to\n# 10.\n# Minimum value: 0, maximum value: 100, default value: 10.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nUML_LIMIT_NUM_FIELDS   = 10\n\n# If the TEMPLATE_RELATIONS tag is set to YES then the inheritance and\n# collaboration graphs will show the relations between templates and their\n# instances.\n# The default value is: NO.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nTEMPLATE_RELATIONS     = NO\n\n# If the INCLUDE_GRAPH, ENABLE_PREPROCESSING and SEARCH_INCLUDES tags are set to\n# YES then doxygen will generate a graph for each documented file showing the\n# direct and indirect include dependencies of the file with other documented\n# files.\n# The default value is: YES.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nINCLUDE_GRAPH          = YES\n\n# If the INCLUDED_BY_GRAPH, ENABLE_PREPROCESSING and SEARCH_INCLUDES tags are\n# set to YES then doxygen will generate a graph for each documented file showing\n# the direct and indirect include dependencies of the file with other documented\n# files.\n# The default value is: YES.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nINCLUDED_BY_GRAPH      = YES\n\n# If the CALL_GRAPH tag is set to YES then doxygen will generate a call\n# dependency graph for every global function or class method.\n#\n# Note that enabling this option will significantly increase the time of a run.\n# So in most cases it will be better to enable call graphs for selected\n# functions only using the \\callgraph command.\n# The default value is: NO.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nCALL_GRAPH             = NO\n\n# If the CALLER_GRAPH tag is set to YES then doxygen will generate a caller\n# dependency graph for every global function or class method.\n#\n# Note that enabling this option will significantly increase the time of a run.\n# So in most cases it will be better to enable caller graphs for selected\n# functions only using the \\callergraph command.\n# The default value is: NO.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nCALLER_GRAPH           = NO\n\n# If the GRAPHICAL_HIERARCHY tag is set to YES then doxygen will graphical\n# hierarchy of all classes instead of a textual one.\n# The default value is: YES.\n# This tag requires that the tag HAVE_DOT is set to YES.\n\nGRAPHICAL_HIERARCHY    = YES\n\n# If the DIRECTORY_GRAPH tag is set to YES then doxygen will show the\n# dependencies a directory has on other directories in a graphical way. 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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Class List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span 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href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n</div><!-- top -->\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div class=\"header\">\n  <div class=\"headertitle\">\n<div class=\"title\">Class List</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<div class=\"textblock\">Here are the classes, structs, unions and interfaces with brief descriptions:</div><div class=\"directory\">\n<div class=\"levels\">[detail level <span onclick=\"javascript:toggleLevel(1);\">1</span><span onclick=\"javascript:toggleLevel(2);\">2</span><span onclick=\"javascript:toggleLevel(3);\">3</span>]</div><table class=\"directory\">\n<tr id=\"row_0_\" class=\"even\"><td class=\"entry\"><img id=\"arr_0_\" src=\"ftv2mlastnode.png\" alt=\"\\\" width=\"16\" height=\"22\" onclick=\"toggleFolder('0_')\"/><img src=\"ftv2ns.png\" alt=\"N\" width=\"24\" height=\"22\" /><b>pathpy</b></td><td class=\"desc\"></td></tr>\n<tr id=\"row_0_0_\"><td class=\"entry\"><img src=\"ftv2blank.png\" alt=\"&#160;\" width=\"16\" height=\"22\" /><img id=\"arr_0_0_\" src=\"ftv2mnode.png\" alt=\"o\" width=\"16\" height=\"22\" onclick=\"toggleFolder('0_0_')\"/><img src=\"ftv2ns.png\" alt=\"N\" width=\"24\" height=\"22\" /><b>HigherOrderNetwork</b></td><td class=\"desc\"></td></tr>\n<tr id=\"row_0_0_0_\" class=\"even\"><td class=\"entry\"><img src=\"ftv2blank.png\" alt=\"&#160;\" width=\"16\" height=\"22\" /><img src=\"ftv2vertline.png\" alt=\"|\" width=\"16\" height=\"22\" /><img src=\"ftv2node.png\" alt=\"o\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1EmptySCCError.html\" target=\"_self\">EmptySCCError</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_0_0_1_\"><td class=\"entry\"><img src=\"ftv2blank.png\" alt=\"&#160;\" width=\"16\" height=\"22\" /><img src=\"ftv2vertline.png\" alt=\"|\" width=\"16\" height=\"22\" /><img src=\"ftv2lastnode.png\" alt=\"\\\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\" target=\"_self\">HigherOrderNetwork</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_0_1_\" class=\"even\"><td class=\"entry\"><img src=\"ftv2blank.png\" alt=\"&#160;\" width=\"16\" height=\"22\" /><img id=\"arr_0_1_\" src=\"ftv2mnode.png\" alt=\"o\" width=\"16\" height=\"22\" onclick=\"toggleFolder('0_1_')\"/><img src=\"ftv2ns.png\" alt=\"N\" width=\"24\" height=\"22\" /><b>Log</b></td><td class=\"desc\"></td></tr>\n<tr id=\"row_0_1_0_\"><td class=\"entry\"><img src=\"ftv2blank.png\" alt=\"&#160;\" width=\"16\" height=\"22\" /><img src=\"ftv2vertline.png\" alt=\"|\" width=\"16\" height=\"22\" 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-->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>HigherOrderNetwork</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1EmptySCCError.html\">EmptySCCError</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.HigherOrderNetwork.EmptySCCError Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<div class=\"dynheader\">\nInheritance diagram for pathpy.HigherOrderNetwork.EmptySCCError:</div>\n<div class=\"dyncontent\">\n <div class=\"center\">\n  <img src=\"classpathpy_1_1HigherOrderNetwork_1_1EmptySCCError.png\" usemap=\"#pathpy.HigherOrderNetwork.EmptySCCError_map\" alt=\"\"/>\n  <map id=\"pathpy.HigherOrderNetwork.EmptySCCError_map\" name=\"pathpy.HigherOrderNetwork.EmptySCCError_map\">\n</map>\n </div></div>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">This exception is thrown whenever a non-empty strongly \nconnected component is needed, but we encounter an empty one\n</pre> </div><hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/HigherOrderNetwork.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Member List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span 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href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" 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href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#afcfdcfacef4f3beb7594a58600e833e4\">BetweennessCentrality</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a2f94ebbc8141be6f5194ec922e3b01a0\">ClosenessCentrality</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#acd9ab003f80216ed6beff9c513a7e876\">degrees</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a25d69b9cc9b7b328fbd201244e68ca95\">dof_ngrams</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a8a271893d9fb656f805e36335afca257\">dof_paths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ae425665357e88b0adf493854143a3f72\">ecount</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6eec968e72178ab2930f83928b3ca842\">edges</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a57e8494220dd4a5b2a9a45de17f9d26a\">EvCent</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a8e10f45369dff5f7ccff3bcf7e6c5b33\">getAdjacencyMatrix</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ae48d8ad635f7cf263897016d876c6fa2\">getAlgebraicConnectivity</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20092d5a4a182df408af6063a0887630\">getDistanceMatrix</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abed4839be0864210c0b5aff9376fe307\">getDistanceMatrixFirstOrder</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a3a03ec6087add6dd0b54f7264b2e21c5\">getDoF</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a556545310735cba27128afd37c59ed35\">getEigenValueGap</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#adea7800343373793dbd9688c77fb6191\">getFiedlerVectorDense</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#aa8f3ed627c16c15c877fc0316c88bdb3\">getFiedlerVectorSparse</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a69ea9c565b0d8bf7f1a2d0cb409f0e15\">getLaplacianMatrix</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a1b757112293f9093efc437ffb113df83\">getLeadingEigenvector</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abd69fc6003eb13d11390466182a63357\">getNodeNameMap</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a672cdad613e84eb0f528bbc02e7c6163\">getShortestPaths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20c4a62ca4706bdab81534332e3843fe\">getTransitionMatrix</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a1963d0e4370e3818de3cf6886bba8594\">HigherOrderNodeToPath</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab71b78f1c9ffe7a06364841572f1fee2\">HigherOrderPathToFirstOrder</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6626777ff215fde5f7d92368a407c683\">nodes</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6dea6fe6e34178adb395ad8e79403d5c\">order</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abc76633f53a0747353e7ab0e15744d94\">PageRank</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ac4c8ee9f7775478793d88680d6f99fc8\">paths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a117fe621fb02d356f6591620f9340eaa\">pathToHigherOrderNodes</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a4b71eed8268df33814725ae7832729e6\">reduceToGCC</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#af3e51491a2417e471eeb1404b44df204\">separator</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a522350b2e4a401732b64bb0acf1634ea\">successors</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab35dd7f65e3bfeb280fcd38c1e7448f7\">summary</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a91626f933af603a73f8bb39249ab6c51\">totalEdgeWeight</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a89ba7bbeb54449a87ba78eecc591fca4\">vcount</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"entry\"></td></tr>\n</table></div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: pathpy.HigherOrderNetwork.HigherOrderNetwork Class Reference</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n       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href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>HigherOrderNetwork</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">HigherOrderNetwork</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"summary\">\n<a href=\"#pub-methods\">Public Member Functions</a> &#124;\n<a href=\"#pub-static-methods\">Static Public Member Functions</a> &#124;\n<a href=\"#pub-attribs\">Public Attributes</a> &#124;\n<a href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork-members.html\">List of all members</a>  </div>\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.HigherOrderNetwork.HigherOrderNetwork Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-methods\"></a>\nPublic Member Functions</h2></td></tr>\n<tr class=\"memitem:a63d36720423ee8d6d88c5a06f4655c84\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a63d36720423ee8d6d88c5a06f4655c84\">__init__</a></td></tr>\n<tr class=\"separator:a63d36720423ee8d6d88c5a06f4655c84\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a89ba7bbeb54449a87ba78eecc591fca4\"><td class=\"memItemLeft\" align=\"right\" 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href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a117fe621fb02d356f6591620f9340eaa\">pathToHigherOrderNodes</a></td></tr>\n<tr class=\"separator:a117fe621fb02d356f6591620f9340eaa\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:abd69fc6003eb13d11390466182a63357\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abd69fc6003eb13d11390466182a63357\">getNodeNameMap</a></td></tr>\n<tr class=\"separator:abd69fc6003eb13d11390466182a63357\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a3a03ec6087add6dd0b54f7264b2e21c5\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" 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href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a2f94ebbc8141be6f5194ec922e3b01a0\">ClosenessCentrality</a></td></tr>\n<tr class=\"separator:a2f94ebbc8141be6f5194ec922e3b01a0\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a57e8494220dd4a5b2a9a45de17f9d26a\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a57e8494220dd4a5b2a9a45de17f9d26a\">EvCent</a></td></tr>\n<tr class=\"separator:a57e8494220dd4a5b2a9a45de17f9d26a\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:abc76633f53a0747353e7ab0e15744d94\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abc76633f53a0747353e7ab0e15744d94\">PageRank</a></td></tr>\n<tr class=\"separator:abc76633f53a0747353e7ab0e15744d94\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ab71b78f1c9ffe7a06364841572f1fee2\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab71b78f1c9ffe7a06364841572f1fee2\">HigherOrderPathToFirstOrder</a></td></tr>\n<tr class=\"separator:ab71b78f1c9ffe7a06364841572f1fee2\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:afcfdcfacef4f3beb7594a58600e833e4\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" 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href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab35dd7f65e3bfeb280fcd38c1e7448f7\">summary</a></td></tr>\n<tr class=\"separator:ab35dd7f65e3bfeb280fcd38c1e7448f7\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a4aec883869195967a9209655905ace52\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a4aec883869195967a9209655905ace52\">__str__</a></td></tr>\n<tr class=\"separator:a4aec883869195967a9209655905ace52\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:acd9ab003f80216ed6beff9c513a7e876\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#acd9ab003f80216ed6beff9c513a7e876\">degrees</a></td></tr>\n<tr class=\"separator:acd9ab003f80216ed6beff9c513a7e876\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a8e10f45369dff5f7ccff3bcf7e6c5b33\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a8e10f45369dff5f7ccff3bcf7e6c5b33\">getAdjacencyMatrix</a></td></tr>\n<tr class=\"separator:a8e10f45369dff5f7ccff3bcf7e6c5b33\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a20c4a62ca4706bdab81534332e3843fe\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20c4a62ca4706bdab81534332e3843fe\">getTransitionMatrix</a></td></tr>\n<tr class=\"separator:a20c4a62ca4706bdab81534332e3843fe\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a69ea9c565b0d8bf7f1a2d0cb409f0e15\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a69ea9c565b0d8bf7f1a2d0cb409f0e15\">getLaplacianMatrix</a></td></tr>\n<tr class=\"separator:a69ea9c565b0d8bf7f1a2d0cb409f0e15\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a556545310735cba27128afd37c59ed35\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a556545310735cba27128afd37c59ed35\">getEigenValueGap</a></td></tr>\n<tr class=\"separator:a556545310735cba27128afd37c59ed35\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:aa8f3ed627c16c15c877fc0316c88bdb3\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#aa8f3ed627c16c15c877fc0316c88bdb3\">getFiedlerVectorSparse</a></td></tr>\n<tr class=\"separator:aa8f3ed627c16c15c877fc0316c88bdb3\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:adea7800343373793dbd9688c77fb6191\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#adea7800343373793dbd9688c77fb6191\">getFiedlerVectorDense</a></td></tr>\n<tr class=\"separator:adea7800343373793dbd9688c77fb6191\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ae48d8ad635f7cf263897016d876c6fa2\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" 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id=\"ac4c8ee9f7775478793d88680d6f99fc8\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ac4c8ee9f7775478793d88680d6f99fc8\">paths</a></td></tr>\n<tr class=\"memdesc:ac4c8ee9f7775478793d88680d6f99fc8\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The paths object used to generate this instance. <br/></td></tr>\n<tr class=\"separator:ac4c8ee9f7775478793d88680d6f99fc8\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a6626777ff215fde5f7d92368a407c683\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a6626777ff215fde5f7d92368a407c683\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6626777ff215fde5f7d92368a407c683\">nodes</a></td></tr>\n<tr class=\"memdesc:a6626777ff215fde5f7d92368a407c683\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The nodes in this <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\">HigherOrderNetwork</a>. <br/></td></tr>\n<tr class=\"separator:a6626777ff215fde5f7d92368a407c683\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:af3e51491a2417e471eeb1404b44df204\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#af3e51491a2417e471eeb1404b44df204\">separator</a></td></tr>\n<tr class=\"memdesc:af3e51491a2417e471eeb1404b44df204\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The separator character used to label higher-order nodes.  <a href=\"#af3e51491a2417e471eeb1404b44df204\">More...</a><br/></td></tr>\n<tr class=\"separator:af3e51491a2417e471eeb1404b44df204\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a6eec968e72178ab2930f83928b3ca842\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a6eec968e72178ab2930f83928b3ca842\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6eec968e72178ab2930f83928b3ca842\">edges</a></td></tr>\n<tr class=\"memdesc:a6eec968e72178ab2930f83928b3ca842\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary containing edges as well as edge weights. <br/></td></tr>\n<tr class=\"separator:a6eec968e72178ab2930f83928b3ca842\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a522350b2e4a401732b64bb0acf1634ea\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a522350b2e4a401732b64bb0acf1634ea\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" 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paths assumption. <br/></td></tr>\n<tr class=\"separator:a8a271893d9fb656f805e36335afca257\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a25d69b9cc9b7b328fbd201244e68ca95\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a25d69b9cc9b7b328fbd201244e68ca95\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a25d69b9cc9b7b328fbd201244e68ca95\">dof_ngrams</a></td></tr>\n<tr class=\"memdesc:a25d69b9cc9b7b328fbd201244e68ca95\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The degrees of freedom of the higher-order model, under the ngram assumption. <br/></td></tr>\n<tr class=\"separator:a25d69b9cc9b7b328fbd201244e68ca95\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">Instances of this class capture a k-th-order representation of path statistics. Path statistics \ncan originate from pathway data, temporal networks, or from processes observed on top of a network topology.\n</pre> </div><h2 class=\"groupheader\">Constructor &amp; Destructor Documentation</h2>\n<a class=\"anchor\" id=\"a63d36720423ee8d6d88c5a06f4655c84\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.__init__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>separator</em> = <code>'-'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>nullModel</em> = <code>False</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>method</em> = <code>'FirstOrderTransitions'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>1000</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Generates a k-th-order representation based on the given path statistics.\n\n@param paths: An instance of class Paths, which contains the path statistics to be \n    used in the generation of the k-th order representation \n\n@param k: The order of the network representation to generate. For the default case of \n    k=1, the resulting representation corresponds to the usual (first-order) aggregate network, \n    i.e. links connect nodes and link weights are given by the frequency of each interaction. For \n    k&gt;1, a k-th order node corresponds to a sequence of k nodes. The weight of a k-th order link \n    captures the frequency of a path of length k.\n\n@param separator: The separator character to be used in higher-order node names.\n\n@param nullModel: For the default value False, link weights are generated based on the statistics of \n    paths of length k in the underlying path statistics instance. If True, link weights are generated \n    from the first-order model (k=1) based on the assumption of independent links (i.e. corresponding) \n    to a first-order Markov model.\n\n@param method: specifies how the null model link weights in the k-th order model are calculated. \n    For the default method='FirstOrderTransitions', the weight w('v_1-v_2-...v_k', 'v_2-...-v_k-v_k+1') of \n    a k-order edge is set to the transition probability T['v_k', 'v_k+1'] in the first order network.\n    For method='KOrderPi' the entry pi['v1-...-v_k'] in the stationary distribution of the \n    k-order network is used instead.\n</pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Function Documentation</h2>\n<a class=\"anchor\" id=\"a4aec883869195967a9209655905ace52\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.__str__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the default string representation of \nthis graphical model instance\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"afcfdcfacef4f3beb7594a58600e833e4\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.BetweennessCentrality </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the betweenness centralities of all nodes.\nIf the order of the higher-order network is larger than one \ncentralities calculated based on the higher-order \ntopology will automatically be projected back to first-order \nnodes.\n\n@param normalized: If set to True, betweenness centralities of \n    nodes will be scaled by the maximum value (default False)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a2f94ebbc8141be6f5194ec922e3b01a0\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.ClosenessCentrality </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the closeness centralities of all nodes.\nIf the order of the higher-order network is larger than one \ncentralities calculated based on the higher-order \ntopology will automatically be projected back to first-order \nnodes.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"acd9ab003f80216ed6beff9c513a7e876\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.degrees </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>weighted</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>mode</em> = <code>&quot;OUT&quot;</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the (weighted) degrees of nodes in the higher-order network\n\n@param weighted: If true, calculates the sum of weights for each node. If false, the \n    number of links is calculated\n\n@param mode: either \"IN\", \"OUT\", or \"TOTAL\" \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ae425665357e88b0adf493854143a3f72\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.ecount </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the number of links </pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a57e8494220dd4a5b2a9a45de17f9d26a\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.EvCent </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>projection</em> = <code>'scaled'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the eigenvector centralities of higher-order nodes. If \nthe order of the HigherOrderNetwork is larger than one, the centralities\nwill be projected to the first-order nodes. \n\n@param projection: Indicates how the projection from k-th-order nodes (v1, v2, ... , v{k-1})\n    shall be performed. For the method 'all', the eigenvector centrality of the higher-order node \n    will be added to *all* first-order nodes on the path corresponding to the higher-order node. For \n    the method 'last', the centrality of the higher-order node will only be assigned to *last* \n    first-order node v{k-1}. For the method 'scaled' (default), the eigenvector centrality of higher-order \n    nodes will be assigned proportionally to first-order nodes, i.e. each of the three nodes in the \n    third-order node (a,b,c) will receive one third of the eigenvector centrality of (a,b,c).\n@param includeSubPaths: whether or not to include subpath statistics in the calculation (default True)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a8e10f45369dff5f7ccff3bcf7e6c5b33\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getAdjacencyMatrix </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>weighted</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>transposed</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a sparse adjacency matrix of the higher-order network. By default, the entry \n    corresponding to a directed link source -&gt; target is stored in row s and column t\n    and can be accessed via A[s,t].\n    \n@param includeSubPaths: if set to True, the returned adjacency matrix will \n    account for the occurrence of links of order k (i.e. paths of length k-1)\n    as subpaths\n\n@param weighted: if set to False, the function returns a binary adjacency matrix.\n  If set to True, adjacency matrix entries will contain the weight of an edge.\n      \n@param transposed: whether to transpose the matrix or not.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ae48d8ad635f7cf263897016d876c6fa2\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getAlgebraicConnectivity </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>20</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the algebraic connectivity of the higher-order network.    \n\n@param lanczosVecs: number of Lanczos vectors to be used in the approximate\n    calculation of eigenvectors and eigenvalues. This maps to the ncv parameter \n    of scipy's underlying function eigs. \n@param maxiter: scaling factor for the number of iterations to be used in the \n    approximate calculation of eigenvectors and eigenvalues. The number of iterations \n    passed to scipy's underlying eigs function will be n*maxiter where n is the\n    number of rows/columns of the Laplacian matrix.         \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a20092d5a4a182df408af6063a0887630\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getDistanceMatrix </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates shortest path distances between all pairs of \nhigher-order nodes using the Floyd-Warshall algorithm.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"abed4839be0864210c0b5aff9376fe307\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getDistanceMatrixFirstOrder </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Projects a distance matrix from a higher-order to \nfirst-order nodes, while path lengths are calculated \nbased on the higher-order topology\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a3a03ec6087add6dd0b54f7264b2e21c5\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getDoF </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>assumption</em> = <code>&quot;paths&quot;</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the degrees of freedom (i.e. number of parameters) of \nthis k-order model. Depending on the modeling assumptions, this either\ncorresponds to the number of paths of length k in the first-order network \nor to the number of all possible k-grams. The degrees of freedom of a model \ncan be used to assess the model complexity when calculating, e.g., the \nBayesian Information Criterion (BIC).\n\n@param assumption: if set to 'paths', for the degree of freedon calculation in the BIC, \n    only paths in the first-order network topology will be considered. This is \n    needed whenever we are interested in a modeling of paths in a given network topology.\n    If set to 'ngrams' all possible n-grams will be considered, independent of whether they \n    are valid paths in the first-order network or not. The 'ngrams' and the 'paths' assumption \n    coincide if the first-order network is fully connected.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a556545310735cba27128afd37c59ed35\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getEigenValueGap </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>20</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the eigenvalue gap of the transition matrix.\n\n@param includeSubPaths: whether or not to include subpath statistics in the \n    calculation of transition probabilities.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"adea7800343373793dbd9688c77fb6191\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getFiedlerVectorDense </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\"> Returns the (dense)Fiedler vector of the higher-order network. The Fiedler \n vector can be used for a spectral bisectioning of the network.             \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"aa8f3ed627c16c15c877fc0316c88bdb3\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getFiedlerVectorSparse </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>20</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the (sparse) Fiedler vector of the higher-order network. The Fiedler \nvector can be used for a spectral bisectioning of the network.\n     \nNote that sparse linear algebra for eigenvalue problems with small eigenvalues \nis problematic in terms of numerical stability. Consider using the dense version\nof this method in this case. Note also that the sparse Fiedler vector might be scaled by \na factor (-1) compared to the dense version.\n  \n@param normalized: whether (default) or not to normalize the fiedler vector.\n  Normalization is done such that the sum of squares equals one in order to\n  get reasonable values as entries might be positive and negative.\n@param lanczosVecs: number of Lanczos vectors to be used in the approximate\n    calculation of eigenvectors and eigenvalues. This maps to the ncv parameter \n    of scipy's underlying function eigs. \n@param maxiter: scaling factor for the number of iterations to be used in the \n    approximate calculation of eigenvectors and eigenvalues. The number of iterations \n    passed to scipy's underlying eigs function will be n*maxiter where n is the \n    number of rows/columns of the Laplacian matrix.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a69ea9c565b0d8bf7f1a2d0cb409f0e15\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getLaplacianMatrix </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the transposed Laplacian matrix corresponding to the higher-order network.\n\n@param includeSubpaths: Whether or not subpath statistics shall be included in the \n    calculation of matrix weights\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a1b757112293f9093efc437ffb113df83\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n<table class=\"mlabels\">\n  <tr>\n  <td class=\"mlabels-left\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getLeadingEigenvector </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>A</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>1000</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n  </td>\n  <td class=\"mlabels-right\">\n<span class=\"mlabels\"><span class=\"mlabel\">static</span></span>  </td>\n  </tr>\n</table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Compute normalized leading eigenvector of a given matrix A.\n\n@param A: sparse matrix for which leading eigenvector will be computed\n@param normalized: wheter or not to normalize. Default is C{True}\n@param lanczosVecs: number of Lanczos vectors to be used in the approximate\n    calculation of eigenvectors and eigenvalues. This maps to the ncv parameter \n    of scipy's underlying function eigs. \n@param maxiter: scaling factor for the number of iterations to be used in the \n    approximate calculation of eigenvectors and eigenvalues. The number of iterations \n    passed to scipy's underlying eigs function will be n*maxiter where n is the \n    number of rows/columns of the Laplacian matrix.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"abd69fc6003eb13d11390466182a63357\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getNodeNameMap </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a dictionary that can be used to map \nnode nodes to matrix/vector indices\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a672cdad613e84eb0f528bbc02e7c6163\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getShortestPaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates all shortest paths between all pairs of \nhigher-order nodes using the Floyd-Warshall algorithm.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a20c4a62ca4706bdab81534332e3843fe\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.getTransitionMatrix </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a (transposed) random walk transition matrix \ncorresponding to the higher-order network.\n\n@param includeSubpaths: whether or not to include subpath statistics in the \n    transition probability calculation (default True)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a1963d0e4370e3818de3cf6886bba8594\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.HigherOrderNodeToPath </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>node</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Helper function that transforms a node in a\nhigher-order network of order k into a corresponding \npath of length k-1. For a higher-order node 'a-b-c-d' \nthis function will return ('a','b','c','d')\n\n@param node: The higher-order node to be transformed to a path.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ab71b78f1c9ffe7a06364841572f1fee2\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.HigherOrderPathToFirstOrder </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>path</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Maps a path in the higher-order network \nto a path in the first-order network. As an \nexample, the second-order path ('a-b', 'b-c', 'c-d')\nof length two is mapped to the first-order path ('a','b','c','d')\nof length four. In general, a path of length l in a network of \norder k is mapped to a path of length l+k-1 in the first-order network. \n\n@param path: The higher-order path that shall be mapped to the first-order network\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"abc76633f53a0747353e7ab0e15744d94\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.PageRank </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>alpha</em> = <code>0.85</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxIterations</em> = <code>100</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>convergenceThres</em> = <code>1.0e-6</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>projection</em> = <code>'scaled'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>includeSubPaths</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the PageRank of higher-order nodes based on a \npower iteration. If the order of the higher-order network is larger than one,\nthe PageRank calculated based on the higher-order\ntopology will automatically be projected back to first-order \nnodes.\n\n@param projection: Indicates how the projection from k-th-order nodes (v1, v2, ... , v{k-1})\n    shall be performed. For the method 'all', the pagerank value of the higher-order node \n    will be added to *all* first-order nodes on the path corresponding to the higher-order node. For \n    the method 'last', the PR value of the higher-order node will only be assigned to *last* \n    first-order node v{k-1}. For the method 'scaled' (default), the PageRank of higher-order \n    nodes will be assigned proportionally to first-order nodes, i.e. each of the three nodes in the \n    third-order node (a,b,c) will receive one third of the PageRank of (a,b,c).\n@param includeSubpaths: whether or not to use subpath statistics in the PageRank calculation\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a117fe621fb02d356f6591620f9340eaa\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.pathToHigherOrderNodes </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>path</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>None</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Helper function that transforms a path into a sequence of k-order nodes \nusing the separator character of the HigherOrderNetwork instance \n\nConsider an example path (a,b,c,d) with a separator string '-'\nFor k=1, the output will be the list of strings ['a', 'b', 'c', 'd']\nFor k=2, the output will be the list of strings ['a-b', 'b-c', 'c-d']\nFor k=3, the output will be the list of strings ['a-b-c', 'b-c-d']\netc. \n\n@param path: the path tuple to turn into a sequence of higher-order nodes \n\n@param k: the order of the representation to use (default: order of the HigherOrderNetwork instance)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a4b71eed8268df33814725ae7832729e6\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.reduceToGCC </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Reduces the higher-order network to its \nlargest (giant) strongly connected component \n(using Tarjan's algorithm)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ab35dd7f65e3bfeb280fcd38c1e7448f7\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.summary </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a string containing basic summary statistics \nof this higher-order graphical model instance\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a91626f933af603a73f8bb39249ab6c51\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.totalEdgeWeight </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the sum of all edge weights </pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a89ba7bbeb54449a87ba78eecc591fca4\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.HigherOrderNetwork.HigherOrderNetwork.vcount </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the number of nodes </pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Data Documentation</h2>\n<a class=\"anchor\" id=\"af3e51491a2417e471eeb1404b44df204\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">pathpy.HigherOrderNetwork.HigherOrderNetwork.separator</td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n\n<p>The separator character used to label higher-order nodes. </p>\n<p>For separator '-', a second-order node will be 'a-b'. </p>\n\n</div>\n</div>\n<hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/HigherOrderNetwork.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "path": "docs/classpathpy_1_1Log_1_1Log.html",
    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: pathpy.Log.Log Class Reference</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" 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<li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>Log</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html\">Log</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"summary\">\n<a href=\"#pub-static-methods\">Static Public Member Functions</a> &#124;\n<a href=\"#pub-static-attribs\">Static Public Attributes</a> &#124;\n<a href=\"classpathpy_1_1Log_1_1Log-members.html\">List of all members</a>  </div>\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.Log.Log Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-static-methods\"></a>\nStatic Public Member Functions</h2></td></tr>\n<tr class=\"memitem:a9e0171144845116732949b87d348d27d\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a9e0171144845116732949b87d348d27d\">setMinSeverity</a></td></tr>\n<tr class=\"separator:a9e0171144845116732949b87d348d27d\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a0e7ec3decada72adee6edcda4951e720\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a0e7ec3decada72adee6edcda4951e720\">setOutputStream</a></td></tr>\n<tr class=\"separator:a0e7ec3decada72adee6edcda4951e720\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a7b948e9bbdcd1ab31bf6ee1a425195f7\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a7b948e9bbdcd1ab31bf6ee1a425195f7\">add</a></td></tr>\n<tr class=\"separator:a7b948e9bbdcd1ab31bf6ee1a425195f7\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table><table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-static-attribs\"></a>\nStatic Public Attributes</h2></td></tr>\n<tr class=\"memitem:a8f7664b2a5379f9e94402117e22ed058\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a8f7664b2a5379f9e94402117e22ed058\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a8f7664b2a5379f9e94402117e22ed058\">output_stream</a> = sys.stdout</td></tr>\n<tr class=\"memdesc:a8f7664b2a5379f9e94402117e22ed058\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">the output stream to which log entries will be written <br/></td></tr>\n<tr class=\"separator:a8f7664b2a5379f9e94402117e22ed058\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a327ac21443db1980997ddb0c8ef65313\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a327ac21443db1980997ddb0c8ef65313\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a327ac21443db1980997ddb0c8ef65313\">min_severity</a> = <a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a12fb98e4cc0d9e68114e29e8f6758c22\">Severity.INFO</a></td></tr>\n<tr class=\"memdesc:a327ac21443db1980997ddb0c8ef65313\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The minimum severity level of messages to be logged. <br/></td></tr>\n<tr class=\"separator:a327ac21443db1980997ddb0c8ef65313\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">A simple logging class, that allows to select what messages should \n    be recorded in the output, and where these message should be directed.\n</pre> </div><h2 class=\"groupheader\">Member Function Documentation</h2>\n<a class=\"anchor\" id=\"a7b948e9bbdcd1ab31bf6ee1a425195f7\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n<table class=\"mlabels\">\n  <tr>\n  <td class=\"mlabels-left\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Log.Log.add </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>msg</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>severity</em> = <code><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a12fb98e4cc0d9e68114e29e8f6758c22\">Severity.INFO</a></code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n  </td>\n  <td class=\"mlabels-right\">\n<span class=\"mlabels\"><span class=\"mlabel\">static</span></span>  </td>\n  </tr>\n</table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Adds a message with the given severity to the log. This message will be written \n    to the log output stream, which by default is sys.stdout. A newline character \n    will be added to the message by default.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a9e0171144845116732949b87d348d27d\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n<table class=\"mlabels\">\n  <tr>\n  <td class=\"mlabels-left\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Log.Log.setMinSeverity </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>severity</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n  </td>\n  <td class=\"mlabels-right\">\n<span class=\"mlabels\"><span class=\"mlabel\">static</span></span>  </td>\n  </tr>\n</table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Sets the minimum sveerity level a message \nneeds to have in order to be recorded in the output stream.\nBy default, any message which has a severity of at least \nSeverity.INFO will be written to the output stream. All messages \nwith lower priority will be surpressed.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a0e7ec3decada72adee6edcda4951e720\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n<table class=\"mlabels\">\n  <tr>\n  <td class=\"mlabels-left\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Log.Log.setOutputStream </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>stream</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n  </td>\n  <td class=\"mlabels-right\">\n<span class=\"mlabels\"><span class=\"mlabel\">static</span></span>  </td>\n  </tr>\n</table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Sets the output stream to which all messages will be \n    written. By default, this is sys.stdout, but it can be \n    changed in order to redirect the log to a logfile. \n</pre> \n</div>\n</div>\n<hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/Log.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Member List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span 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href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" 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class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#ad6d71133083c6972a8400a1e4b355381\">DEBUG</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\">pathpy.Log.Severity</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#aa2e434a659bbb9f2e2c25e47fab1dc37\">ERROR</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\">pathpy.Log.Severity</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a12fb98e4cc0d9e68114e29e8f6758c22\">INFO</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\">pathpy.Log.Severity</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a0c184f8e48c1f5cb7f795d00c9205ed5\">TIMING</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\">pathpy.Log.Severity</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#ae4a13b6d9d1d2485bceb5843df6e3f40\">WARNING</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\">pathpy.Log.Severity</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n</table></div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" 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onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>Log</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\">Severity</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"summary\">\n<a href=\"#pub-static-attribs\">Static Public Attributes</a> &#124;\n<a href=\"classpathpy_1_1Log_1_1Severity-members.html\">List of all members</a>  </div>\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.Log.Severity Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<div class=\"dynheader\">\nInheritance diagram for pathpy.Log.Severity:</div>\n<div class=\"dyncontent\">\n <div class=\"center\">\n  <img src=\"classpathpy_1_1Log_1_1Severity.png\" usemap=\"#pathpy.Log.Severity_map\" alt=\"\"/>\n  <map id=\"pathpy.Log.Severity_map\" name=\"pathpy.Log.Severity_map\">\n</map>\n </div></div>\n<table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-static-attribs\"></a>\nStatic Public Attributes</h2></td></tr>\n<tr class=\"memitem:aa2e434a659bbb9f2e2c25e47fab1dc37\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"aa2e434a659bbb9f2e2c25e47fab1dc37\"></a>\nint&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#aa2e434a659bbb9f2e2c25e47fab1dc37\">ERROR</a> = 4</td></tr>\n<tr class=\"memdesc:aa2e434a659bbb9f2e2c25e47fab1dc37\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">Error messages. <br/></td></tr>\n<tr class=\"separator:aa2e434a659bbb9f2e2c25e47fab1dc37\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ae4a13b6d9d1d2485bceb5843df6e3f40\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ae4a13b6d9d1d2485bceb5843df6e3f40\"></a>\nint&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#ae4a13b6d9d1d2485bceb5843df6e3f40\">WARNING</a> = 3</td></tr>\n<tr class=\"memdesc:ae4a13b6d9d1d2485bceb5843df6e3f40\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">Warning messages. <br/></td></tr>\n<tr class=\"separator:ae4a13b6d9d1d2485bceb5843df6e3f40\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a12fb98e4cc0d9e68114e29e8f6758c22\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a12fb98e4cc0d9e68114e29e8f6758c22\"></a>\nint&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a12fb98e4cc0d9e68114e29e8f6758c22\">INFO</a> = 2</td></tr>\n<tr class=\"memdesc:a12fb98e4cc0d9e68114e29e8f6758c22\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">Informational messages (default minimum level) <br/></td></tr>\n<tr class=\"separator:a12fb98e4cc0d9e68114e29e8f6758c22\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a0c184f8e48c1f5cb7f795d00c9205ed5\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a0c184f8e48c1f5cb7f795d00c9205ed5\"></a>\nint&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a0c184f8e48c1f5cb7f795d00c9205ed5\">TIMING</a> = 1</td></tr>\n<tr class=\"memdesc:a0c184f8e48c1f5cb7f795d00c9205ed5\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">Messages regarding timing and performance. <br/></td></tr>\n<tr class=\"separator:a0c184f8e48c1f5cb7f795d00c9205ed5\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ad6d71133083c6972a8400a1e4b355381\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ad6d71133083c6972a8400a1e4b355381\"></a>\nint&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#ad6d71133083c6972a8400a1e4b355381\">DEBUG</a> = 0</td></tr>\n<tr class=\"memdesc:ad6d71133083c6972a8400a1e4b355381\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">Debug messages (really verbose) <br/></td></tr>\n<tr class=\"separator:ad6d71133083c6972a8400a1e4b355381\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">An enumeration that can be used to indicate \n    the severity of log messages, and which can be \n    used tpo filter messages based on severities.\n</pre> </div><hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/Log.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Member List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" 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inherited members.</p>\n<table class=\"directory\">\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a670778608688e4926328cf2d851b7d6d\">__init__</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html\">pathpy.MarkovSequence.MarkovSequence</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a0e794225267c8195f091f5ed452d34e6\">estimateOrder</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html\">pathpy.MarkovSequence.MarkovSequence</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a422f6b70f888eaab6a6f942158355072\">fitMarkovModel</a></td><td class=\"entry\"><a class=\"el\" 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class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html\">pathpy.MarkovSequence.MarkovSequence</a></td><td class=\"entry\"></td></tr>\n</table></div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: pathpy.MarkovSequence.MarkovSequence Class Reference</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" 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onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>MarkovSequence</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html\">MarkovSequence</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"summary\">\n<a href=\"#pub-methods\">Public Member Functions</a> &#124;\n<a href=\"#pub-attribs\">Public Attributes</a> &#124;\n<a href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence-members.html\">List of all members</a>  </div>\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.MarkovSequence.MarkovSequence Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-methods\"></a>\nPublic Member Functions</h2></td></tr>\n<tr class=\"memitem:a670778608688e4926328cf2d851b7d6d\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a670778608688e4926328cf2d851b7d6d\">__init__</a></td></tr>\n<tr class=\"separator:a670778608688e4926328cf2d851b7d6d\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a422f6b70f888eaab6a6f942158355072\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a422f6b70f888eaab6a6f942158355072\">fitMarkovModel</a></td></tr>\n<tr class=\"separator:a422f6b70f888eaab6a6f942158355072\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ab820bbcfb5569ef05e361aa478c521f6\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ab820bbcfb5569ef05e361aa478c521f6\">getLikelihood</a></td></tr>\n<tr class=\"separator:ab820bbcfb5569ef05e361aa478c521f6\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a975922931ec471f436c4d340830a7ca3\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a975922931ec471f436c4d340830a7ca3\">getBIC</a></td></tr>\n<tr class=\"separator:a975922931ec471f436c4d340830a7ca3\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:aba5377e966e3bfb9c4c736e782863484\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#aba5377e966e3bfb9c4c736e782863484\">getAIC</a></td></tr>\n<tr class=\"separator:aba5377e966e3bfb9c4c736e782863484\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a0e794225267c8195f091f5ed452d34e6\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a0e794225267c8195f091f5ed452d34e6\">estimateOrder</a></td></tr>\n<tr class=\"separator:a0e794225267c8195f091f5ed452d34e6\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table><table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-attribs\"></a>\nPublic Attributes</h2></td></tr>\n<tr class=\"memitem:ac0cbbe436a3938f2ff94d313c72c4e67\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ac0cbbe436a3938f2ff94d313c72c4e67\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ac0cbbe436a3938f2ff94d313c72c4e67\">sequence</a></td></tr>\n<tr class=\"memdesc:ac0cbbe436a3938f2ff94d313c72c4e67\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The sequence to be modeled. <br/></td></tr>\n<tr class=\"separator:ac0cbbe436a3938f2ff94d313c72c4e67\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ac0d2ff028f2c2c88349555527e44a898\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ac0d2ff028f2c2c88349555527e44a898\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ac0d2ff028f2c2c88349555527e44a898\">P</a></td></tr>\n<tr class=\"memdesc:ac0d2ff028f2c2c88349555527e44a898\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The transition probabilities of higher-order Markov chains. <br/></td></tr>\n<tr class=\"separator:ac0d2ff028f2c2c88349555527e44a898\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a77ca53bfcfb5458a8834b8b6392422ce\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a77ca53bfcfb5458a8834b8b6392422ce\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a77ca53bfcfb5458a8834b8b6392422ce\">states</a></td></tr>\n<tr class=\"memdesc:a77ca53bfcfb5458a8834b8b6392422ce\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">the set of states of higher-order Markov chains <br/></td></tr>\n<tr class=\"separator:a77ca53bfcfb5458a8834b8b6392422ce\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">Instances of this class can be used to fit \n    standard higher-order Markov models for \n    sequences generated from concatenated paths </pre> </div><h2 class=\"groupheader\">Constructor &amp; Destructor Documentation</h2>\n<a class=\"anchor\" id=\"a670778608688e4926328cf2d851b7d6d\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MarkovSequence.MarkovSequence.__init__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>sequence</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Generates a Markov model for a sequence, given \nas a single list of strings\n</pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Function Documentation</h2>\n<a class=\"anchor\" id=\"a0e794225267c8195f091f5ed452d34e6\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MarkovSequence.MarkovSequence.estimateOrder </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrder</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>method</em> = <code>'BIC'</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Estimates the optimal order of a Markov model\n    based on Likelihood, BIC or AIC </pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a422f6b70f888eaab6a6f942158355072\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MarkovSequence.MarkovSequence.fitMarkovModel </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Generates a k-th order Markov model \n    for the underlying sequence\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"aba5377e966e3bfb9c4c736e782863484\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MarkovSequence.MarkovSequence.getAIC </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>m</em> = <code>1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the Aikake Information Criterion\n    assuming a k-th order Markov model </pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a975922931ec471f436c4d340830a7ca3\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MarkovSequence.MarkovSequence.getBIC </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>m</em> = <code>1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the Bayesian Information Criterion\n    assuming a k-th order Markov model </pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ab820bbcfb5569ef05e361aa478c521f6\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MarkovSequence.MarkovSequence.getLikelihood </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>log</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the likelihood of the sequence \nassuming a k-th order Markov model \n</pre> \n</div>\n</div>\n<hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/MarkovSequence.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "path": "docs/classpathpy_1_1MultiOrderModel_1_1MultiOrderModel-members.html",
    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Member List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span 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onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>MultiOrderModel</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">MultiOrderModel</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.MultiOrderModel.MultiOrderModel Member List</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n\n<p>This is the complete list of members for <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a>, 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href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ab56ae5b47770b09178ae5aa49f695d17\">getLayerLikelihood</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#acceb5eabd7d1cb8a8856a485a29fc5f8\">getLikelihood</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#abc678904e6dd23fc36bace35f8c8b651\">layers</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a0518e905c00b8c3a2df5cce509084fb8\">likeliHoodRatioTest</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ad77293d316dbc4264e07d33f15c43f55\">maxOrder</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#adf751f249355e9a26e8062050567cf54\">paths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a58f11a90bea210c70f12eeac3af53d65\">summary</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a93118aa6719067efdbe8b38ef85a578a\">T</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#aaa9b2f2852c4ae513e5d42a96008c030\">testNetworkAssumption</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"entry\"></td></tr>\n</table></div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: pathpy.MultiOrderModel.MultiOrderModel Class Reference</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" 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onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>MultiOrderModel</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\">MultiOrderModel</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"summary\">\n<a href=\"#pub-methods\">Public Member Functions</a> &#124;\n<a href=\"#pub-attribs\">Public Attributes</a> &#124;\n<a href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel-members.html\">List of all members</a>  </div>\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.MultiOrderModel.MultiOrderModel Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-methods\"></a>\nPublic Member Functions</h2></td></tr>\n<tr class=\"memitem:ac8dc89b42d8e51906ce15124da699409\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ac8dc89b42d8e51906ce15124da699409\">__init__</a></td></tr>\n<tr class=\"separator:ac8dc89b42d8e51906ce15124da699409\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a58f11a90bea210c70f12eeac3af53d65\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a58f11a90bea210c70f12eeac3af53d65\">summary</a></td></tr>\n<tr class=\"separator:a58f11a90bea210c70f12eeac3af53d65\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:af69183cc68e6b8aae85cce91341dbf44\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#af69183cc68e6b8aae85cce91341dbf44\">__str__</a></td></tr>\n<tr class=\"separator:af69183cc68e6b8aae85cce91341dbf44\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:acceb5eabd7d1cb8a8856a485a29fc5f8\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#acceb5eabd7d1cb8a8856a485a29fc5f8\">getLikelihood</a></td></tr>\n<tr class=\"separator:acceb5eabd7d1cb8a8856a485a29fc5f8\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a7882e4dcbe9ec932e863e20d6b49a4ed\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a7882e4dcbe9ec932e863e20d6b49a4ed\">factorial</a></td></tr>\n<tr class=\"separator:a7882e4dcbe9ec932e863e20d6b49a4ed\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ab56ae5b47770b09178ae5aa49f695d17\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ab56ae5b47770b09178ae5aa49f695d17\">getLayerLikelihood</a></td></tr>\n<tr class=\"separator:ab56ae5b47770b09178ae5aa49f695d17\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a219e12dca2b474515d74c65c4ad15c69\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a219e12dca2b474515d74c65c4ad15c69\">getDegreesOfFreedom</a></td></tr>\n<tr class=\"separator:a219e12dca2b474515d74c65c4ad15c69\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a0518e905c00b8c3a2df5cce509084fb8\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a0518e905c00b8c3a2df5cce509084fb8\">likeliHoodRatioTest</a></td></tr>\n<tr class=\"separator:a0518e905c00b8c3a2df5cce509084fb8\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a866b2b0b96f4bafa594b2a8d5a64efbf\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a866b2b0b96f4bafa594b2a8d5a64efbf\">estimateOrder</a></td></tr>\n<tr class=\"separator:a866b2b0b96f4bafa594b2a8d5a64efbf\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:aaa9b2f2852c4ae513e5d42a96008c030\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#aaa9b2f2852c4ae513e5d42a96008c030\">testNetworkAssumption</a></td></tr>\n<tr class=\"separator:aaa9b2f2852c4ae513e5d42a96008c030\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table><table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-attribs\"></a>\nPublic Attributes</h2></td></tr>\n<tr class=\"memitem:abc678904e6dd23fc36bace35f8c8b651\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"abc678904e6dd23fc36bace35f8c8b651\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#abc678904e6dd23fc36bace35f8c8b651\">layers</a></td></tr>\n<tr class=\"memdesc:abc678904e6dd23fc36bace35f8c8b651\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary containing the layers of HigherOrderNetworks, where layers[k] contains the network of order k. <br/></td></tr>\n<tr class=\"separator:abc678904e6dd23fc36bace35f8c8b651\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ad77293d316dbc4264e07d33f15c43f55\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ad77293d316dbc4264e07d33f15c43f55\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ad77293d316dbc4264e07d33f15c43f55\">maxOrder</a></td></tr>\n<tr class=\"memdesc:ad77293d316dbc4264e07d33f15c43f55\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">the maximum order of this multi-order model <br/></td></tr>\n<tr class=\"separator:ad77293d316dbc4264e07d33f15c43f55\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:adf751f249355e9a26e8062050567cf54\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"adf751f249355e9a26e8062050567cf54\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#adf751f249355e9a26e8062050567cf54\">paths</a></td></tr>\n<tr class=\"memdesc:adf751f249355e9a26e8062050567cf54\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">the paths object from which this multi-order model was created <br/></td></tr>\n<tr class=\"separator:adf751f249355e9a26e8062050567cf54\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a93118aa6719067efdbe8b38ef85a578a\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a93118aa6719067efdbe8b38ef85a578a\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a93118aa6719067efdbe8b38ef85a578a\">T</a></td></tr>\n<tr class=\"memdesc:a93118aa6719067efdbe8b38ef85a578a\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">a dictionary of transition matrices for all layers of the model <br/></td></tr>\n<tr class=\"separator:a93118aa6719067efdbe8b38ef85a578a\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">Instances of this class represent a hierarchy of \n    higher-order networks which collectively represent \n    a multi-order model for path statistics. </pre> </div><h2 class=\"groupheader\">Constructor &amp; Destructor Documentation</h2>\n<a class=\"anchor\" id=\"ac8dc89b42d8e51906ce15124da699409\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.__init__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrder</em> = <code>1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Generates a hierarchy of higher-order\nmodels for the given path statistics, \nup to a given maximum order \n\n@param paths: the paths instance for which the model should be created \n@param maxOrder: the maximum order of the multi-order model\n</pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Function Documentation</h2>\n<a class=\"anchor\" id=\"af69183cc68e6b8aae85cce91341dbf44\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.__str__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the default string representation of \nthis multi-order model instance\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a866b2b0b96f4bafa594b2a8d5a64efbf\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.estimateOrder </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrder</em> = <code>None</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>significanceThreshold</em> = <code>0.01</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Selects the optimal maximum order of a multi-order network model for the \nobserved paths, based on a likelihood ratio test with p-value threshold of p\nBy default, all orders up to the maximum order of the multi-order model will be tested. \n\n@param paths: The path statistics for which to perform the order selection\n\n@param maxOrder: The maximum order up to which the multi-order model shall be tested.        \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a7882e4dcbe9ec932e863e20d6b49a4ed\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.factorial </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>n</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>log</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Caclulates (or approximates) the (log of the) factorial n!. The function applies Stirling's approximation if n&gt;20.\n\n@param n: computes factorial of n\n@param log: whether or not to return the (natural) logarithm of the factorial        \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a219e12dca2b474515d74c65c4ad15c69\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.getDegreesOfFreedom </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrder</em> = <code>None</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>assumption</em> = <code>&quot;paths&quot;</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the degrees of freedom of the model based on \ndifferent assumptions, and taking into account layers up to \na maximum order. \n\n@param: maxOrder: the maximum order up to which model layers shall be \n    taken into account\n\n@param assumption: if set to 'paths', for the degree of freedom calculation \n    only paths in the first-order network topology will be considered. This is \n    needed whenever we model paths in a *given* network topology.\n    If set to 'ngrams' all possible n-grams will be considered, independent of whether they \n    are valid paths in the first-order network or not. The 'ngrams' and the 'paths' assumption \n    coincide if the first-order network is fully connected, i.e. if all possible paths actually occur.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ab56ae5b47770b09178ae5aa49f695d17\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.getLayerLikelihood </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>l</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>considerLongerPaths</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>log</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the (log-)likelihood of the first l layers of a multi-order network model\nusing all observed paths of (at least) length l\n\n@param paths: the path statistics for which to calculate the layer likelihood\n\n@param l: The minimum length of paths for which the likelihood shall be calculated.\n    Paths of length l (and possibly longer) will be used to calculate the likelihood \n    of model layers for all orders up to l\n\n@param considerLongerPaths: whether or not to include paths longer than l\n    in the calculation of the likelihood. In general, when calculating the likelihood\n    of a multi-order model which combines orders from 1 to l, this should be set to \n    true only for the value of l that corresponds to the largest order in the model.\n\n@param log: whether to compute Log-Likelihood (default: True)\n\n@returns: the (log-)likelihood of the model layer given the path statistics\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"acceb5eabd7d1cb8a8856a485a29fc5f8\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.getLikelihood </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrder</em> = <code>None</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>log</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the likelihood of a multi-order\nnetwork model up to a maximum order maxOrder based on all \npath statistics.\n\n@param paths: the path statistics to be used in the likelihood \n    calculation\n\n@param maxOrder: the maximum layer order to take into \n    account for the likelihood calculation. For the default \n    value None, all orders will be used for the \n    likelihood calculation. \n\n@log: Whether or not to return the log likelihood (default: True)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a0518e905c00b8c3a2df5cce509084fb8\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.likeliHoodRatioTest </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrderNull</em> = <code>0</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxOrder</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>assumption</em> = <code>'<a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#adf751f249355e9a26e8062050567cf54\">paths</a>'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>significanceThreshold</em> = <code>0.01</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Performs a likelihood-ratio test between two multi-order models with given maximum orders, where maxOrderNull serves \nas null hypothesis and maxOrder serves as alternative hypothesis. The null hypothesis is rejected if the p-value for \nthe observed paths under the null hypothesis is smaller than the given significance threshold.\n\nApplying this test makes the assumption that we have nested models, i.e. that the null model is contained\nas a special case in the parameter space of the more complex model. If we assume that the path constraint holds, \nthis is not true for the test of the first- against the zero-order model (since some sequences of the zero order model \ncannot be generated in the first-order model). However, since the set of possible higher-order transitions is generated \nbased on the first-order model, the nestedness property holds for all higher order models.\n\n@param paths: the path data to be used in the liklihood ratio test\n@param maxOrderNull: maximum order of the multi-order model \nto be used as a null hypothesis\n@param maxOrder: maximum order of the multi-order model to be used as \nalternative hypothesis\n@param assumption: paths or ngrams\n@param significanceThreshold: the threshold for the p-value \nbelow which to accept the alternative hypothesis\n@returns: a tuple of the format (reject, p) which captures whether or \nnot the null hypothesis is rejected in favor of the alternative \nhypothesis, as well as the p-value that led to the decision\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a58f11a90bea210c70f12eeac3af53d65\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.summary </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a string containing basic summary information \nof this multi-order model instance.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"aaa9b2f2852c4ae513e5d42a96008c030\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.MultiOrderModel.MultiOrderModel.testNetworkAssumption </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>paths</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>method</em> = <code>'AIC'</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Tests whether the assumption that the observed paths result\nfrom an underlying network topology is justified. Roughly speaking, \nthis test yields true if the explanatory gained by the assumption of \na network topology justifies the additional model complexity\n\nThe decision will be made based on a comparison between the first- \nand zero-order model layers. Different from the comparison of different \nmulti-order models which is the basis of the order detection procedure, \nhere the first- and the zero-order model are fundamentally different:\nFor the zero-order model we do not assume a network topology, while for \nthe first-order model we do assume an underlying network topology. \nFor this reason, the zero-order model cannot be cast as a special case \nof the first-order model, i.e. the models are not nested!\n\nSo we need to use the AIC or BIC rather than a likelihood ratio test.\n</pre> \n</div>\n</div>\n<hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/MultiOrderModel.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Member List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>Paths</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">Paths</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.Paths.Paths Member List</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n\n<p>This is the complete list of members for <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a>, including all inherited members.</p>\n<table class=\"directory\">\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5e853dc424f2142bc53df219e33be29f\">__init__</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afdbe49d3727f7fd3022899cb3130c6db\">__str__</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5af2d99bd84797a960e43bc78c57db5a\">addPath</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ab1c50bfee7d28f3a180fc036000bc146\">addPathTuple</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#add4dc55240d6ca92645f3709152d0545\">BetweennessCentrality</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afc66998bbb85b5fd3e7b8cd049a2bfa1\">BetweennessPreference</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a45a27a4b5d8fe5aa368da221ab68502a\">ClosenessCentrality</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a6e1941415a937fa9aa86b6b442f858e1\">expandSubPaths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a1c27b7c76d16437518f28734de2b86a5\">filterPaths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a02f8ee9b6b4fa8c1e8012c9d21bfb76d\">fromTemporalNetwork</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#af830708eaa318dd450193b8e6d7fb37a\">getContainedPaths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5b6ad214815f9fbd687c457d53367f19\">getDistanceMatrix</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a9bf65d20a97eb629a4d618a4e19160a7\">getEntropyGrowthRateRatio</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a287a429de9a958ccc48658b9fb9f7665\">getNodes</a></td><td class=\"entry\"><a 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href=\"classpathpy_1_1Paths_1_1Paths.html#a160fb269d25c24adc4fbfdc3df71075c\">projectPaths</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#adba5db911e0be900c03908f8b2ca511e\">readEdges</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a349974504cf0ef9fd1ff97a0249e649e\">readFile</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\">pathpy.Paths.Paths</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a0642d710d46cf929c00f22ed53510d92\">separator</a></td><td class=\"entry\"><a class=\"el\" 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1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: pathpy.Paths.Paths Class Reference</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not 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class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5b6ad214815f9fbd687c457d53367f19\">getDistanceMatrix</a></td></tr>\n<tr class=\"separator:a5b6ad214815f9fbd687c457d53367f19\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a99a295d0674ca4eb5ba6906b48893b26\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a99a295d0674ca4eb5ba6906b48893b26\">getShortestPaths</a></td></tr>\n<tr class=\"separator:a99a295d0674ca4eb5ba6906b48893b26\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:add4dc55240d6ca92645f3709152d0545\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#add4dc55240d6ca92645f3709152d0545\">BetweennessCentrality</a></td></tr>\n<tr class=\"separator:add4dc55240d6ca92645f3709152d0545\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a45a27a4b5d8fe5aa368da221ab68502a\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a45a27a4b5d8fe5aa368da221ab68502a\">ClosenessCentrality</a></td></tr>\n<tr class=\"separator:a45a27a4b5d8fe5aa368da221ab68502a\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a5c964af581ac3c48fbd2a3955f711bb2\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5c964af581ac3c48fbd2a3955f711bb2\">VisitationProbabilities</a></td></tr>\n<tr class=\"separator:a5c964af581ac3c48fbd2a3955f711bb2\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table><table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-static-methods\"></a>\nStatic Public Member Functions</h2></td></tr>\n<tr class=\"memitem:a02f8ee9b6b4fa8c1e8012c9d21bfb76d\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a02f8ee9b6b4fa8c1e8012c9d21bfb76d\">fromTemporalNetwork</a></td></tr>\n<tr class=\"separator:a02f8ee9b6b4fa8c1e8012c9d21bfb76d\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table><table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-attribs\"></a>\nPublic Attributes</h2></td></tr>\n<tr class=\"memitem:aacbff90d31fabf41c2413246aafc8275\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#aacbff90d31fabf41c2413246aafc8275\">paths</a></td></tr>\n<tr class=\"memdesc:aacbff90d31fabf41c2413246aafc8275\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary of paths that has the following structure:  <a href=\"#aacbff90d31fabf41c2413246aafc8275\">More...</a><br/></td></tr>\n<tr class=\"separator:aacbff90d31fabf41c2413246aafc8275\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a0642d710d46cf929c00f22ed53510d92\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a0642d710d46cf929c00f22ed53510d92\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a0642d710d46cf929c00f22ed53510d92\">separator</a></td></tr>\n<tr class=\"memdesc:a0642d710d46cf929c00f22ed53510d92\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">The character used to separate nodes on paths. <br/></td></tr>\n<tr class=\"separator:a0642d710d46cf929c00f22ed53510d92\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">Instances of this class represent path statistics which can be analyzed using higher- and multi-order network\nmodels. The origin of the path statistics can be (i) n-gram files which provide us with a list of paths \nin terms of n-grams of varying lengths, or (ii) a temporal network instance which provides us with a set of\ntime-respecting paths based on a given maximum time difference delta.\n</pre> </div><h2 class=\"groupheader\">Constructor &amp; Destructor Documentation</h2>\n<a class=\"anchor\" id=\"a5e853dc424f2142bc53df219e33be29f\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.__init__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Creates an empty Paths object\n</pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Function Documentation</h2>\n<a class=\"anchor\" id=\"afdbe49d3727f7fd3022899cb3130c6db\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.__str__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the default string representation of \nthis Paths instance\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a5af2d99bd84797a960e43bc78c57db5a\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.addPath </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>ngram</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>separator</em> = <code>'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>expandSubPaths</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>pathFrequency</em> = <code>None</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Adds the path(s) of a single n-gram to the path statistics object.\n\n@param ngram: An ngram representing a path between nodes, separated by the separator character, e.g. \n    the 4-gram a;b;c;d represents a path of length three (with separator ';')\n\n@param separator: The character used as separator for the ngrams (';' by default)\n\n@param expandSubPaths: by default all subpaths of the given ngram are generated, i.e. \n    for the trigram a;b;c a path a-&gt;b-&gt;c of length two will be generated \n    as well as two subpaths a-&gt;b and b-&gt;c of length one\n\n@weight weight: the weight (i.e. frequency) of the ngram\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ab1c50bfee7d28f3a180fc036000bc146\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.addPathTuple </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>path</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>expandSubPaths</em> = <code>True</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>frequency</em> = <code>(0,1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Adds a tuple of elements as a path. If the elements are not strings, \na conversion to strings will be made. This function can be used to \nto set custom subpath statistics, via the frequency tuple (see below).\n\n@path: The path tuple to be added, e.g. ('a', 'b', 'c')\n@expandSubPaths: Whether or not to calculate subpath statistics for this path\n@frequency: A tuple (x,y) indicating the frequency of this path as subpath \n    (first component) and longest path (second component). Default is (0,1).\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"add4dc55240d6ca92645f3709152d0545\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.BetweennessCentrality </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the betweenness centrality of nodes based on\nobserved shortest paths between all pairs of nodes\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"afc66998bbb85b5fd3e7b8cd049a2bfa1\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.BetweennessPreference </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>False</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>method</em> = <code>'MLE'</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the k-th order betweenness preferences of \nk-th order nodes based on the mutual information of path \nstatistics of length k+1. The minimum order k for which \nbetweenness preference can be computed is one, in which \ncase for each first-order node v all paths s-&gt;v-&gt;d of length \ntwo will be considered for all nodes s and d. In the general case of \norder k, for a k-th order node v_1-...-v_{k} the statistics \nof all paths s-v_1-...v_{k-1} -&gt; v_1-...-v_{k} -&gt; v_2-...-v_{k}-d\nof length two in the k-th order network (i.e. length k+1) in the first-order\nnetwork will be considered in the calculation.\n\n@order: The order of nodes for which to calculate betweenness preference\n\n@nornalized: whether or not to normalize betweenness preference values\n\n@method: which method to use for the entropy calculation. The default 'MLE' uses \n    the standard Maximum-Likelihood estimation of entropy. Setting method to \n    'Miller' additionally applies a Miller-correction. see e.g. \n    Liam Paninski: Estimation of Entropy and Mutual Information, Neural Computation 5, 2003 or \n    http://www.nowozin.net/sebastian/blog/estimating-discrete-entropy-part-2.html\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a45a27a4b5d8fe5aa368da221ab68502a\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.ClosenessCentrality </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the closeness centrality of nodes based on\nobserved shortest paths between all nodes \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a6e1941415a937fa9aa86b6b442f858e1\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.expandSubPaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">This function implements the sub path expansion, i.e. \nfor a four-gram a,b,c,d, the paths a-&gt;b, b-&gt;c, c-&gt;d of \nlength one and the paths a-&gt;b-&gt;c and b-&gt;c-&gt;d of length \ntwo will be counted.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a1c27b7c76d16437518f28734de2b86a5\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.filterPaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>node_filter</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>minLength</em> = <code>0</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxLength</em> = <code>sys.maxsize</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a new paths object which contains only paths between nodes in a given \nfilter set. For each of the paths in the current Paths object, the set of maximally \ncontained subpaths between nodes in node_filter is extracted. This method is useful \nwhen studying (sub-)paths passing through a subset of nodes.\n\n@param node_filter: the nodes for which paths with be extracted from the current\n    set of paths\n@param minLength: the minimum length of paths to extract (default 0)\n@param maxLength: the maximum length of paths to extract (default sys.maxsize)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a02f8ee9b6b4fa8c1e8012c9d21bfb76d\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n<table class=\"mlabels\">\n  <tr>\n  <td class=\"mlabels-left\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.fromTemporalNetwork </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>tempnet</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>delta</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxLength</em> = <code>_sys.maxsize</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n  </td>\n  <td class=\"mlabels-right\">\n<span class=\"mlabels\"><span class=\"mlabel\">static</span></span>  </td>\n  </tr>\n</table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the frequency of all time-respecting paths up to maximum length of k, assuming \na maximum temporal distance of delta between consecutive time-stamped links on a path. \nThis (static) method returns an instance of the class Paths, which can subsequently be used to \ngenerate higher-order network representations based on the path statistics.\n\n@param delta: Indicates the maximum temporal distance up to which time-stamped links will be \nconsidered to contribute to time-respecting paths. For (u,v;3) and (v,w;7) a time-respecting path (u,v)-&gt;(v,w) \nwill be inferred for all 0 &lt; delta &lt;= 4, while no time-respecting path will be inferred for all delta &gt; 4. \nIf the max time diff is not set specifically, the default value of delta=1 will be used, meaning that a\ntime-respecting path u -&gt; v -&gt; w will only be inferred if there are *directly consecutive* time-stamped \nlinks (u,v;t) (v,w;t+1). Every time-stamped edge is further considered a path of length one, i.e. for maxLength=1 \nthis function will simply return the statistics of time-stamped edges.\n\n@param maxLength: Indicates the maximum length up to which time-respecting paths should be calculated, \n     which can be limited due to computational efficiency. A value of k will generate all time-respecting paths \n     consisting of up to k time-stamped links. Note that generating a multi-order model with a maximum order of k \n     requires to extract time-respecting paths with *at least* length k. If a limitation of the maxLength is not \n     required for computational reasons, this parameter should not be set (as it will change the statistics of \n     paths)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"af830708eaa318dd450193b8e6d7fb37a\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getContainedPaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>p</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>node_filter</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the set of maximum-length sub-paths of the path p, which\nonly contain nodes that appear in the node_filter. As an example, \nfor the path (a,b,c,d,e,f,g) and a node_filter [a,b,d,f,g], the method \nwill return [(a,b), (d,), (f,g)].\n\n@param p: a path tuple to check for contained paths\n@param node_filter: a set of nodes to which the contained paths should be limited\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a5b6ad214815f9fbd687c457d53367f19\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getDistanceMatrix </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates shortest path distances between all pairs of \nnodes based on the observed shortest paths (and subpaths)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a9bf65d20a97eb629a4d618a4e19160a7\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getEntropyGrowthRateRatio </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>method</em> = <code>'MLE'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>2</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>1000</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Computes the ratio between the entropy growth rate ratio between\nthe k-order and first-order model of a temporal network t. Ratios smaller\nthan one indicate that the temporal network exhibits non-Markovian characteristics\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a287a429de9a958ccc48658b9fb9f7665\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getNodes </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the list of nodes for the underlying \nset of paths\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ab457e7e3f439e193410be707a2cd39bd\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getSequence </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>stopchar</em> = <code>'|'</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a single sequence in which all \npaths have been concatenated. Individual \npaths are separated by a stop character.\n\n@stopchar: The character used to separate paths\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a99a295d0674ca4eb5ba6906b48893b26\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getShortestPaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates all observed shortest paths (and subpaths) between \nall pairs of nodes\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a60aa117d37a599f912122263bf9e3eea\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getSlowDownFactor </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>k</em> = <code>2</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>lanczosVecs</em> = <code>15</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxiter</em> = <code>1000</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a factor S that indicates how much slower (S&gt;1) or faster (S&lt;1)\na diffusion process evolves in a k-order model of the path statistics\ncompared to what is expected based on a first-order model. This value captures \nthe effect of order correlations of length k on a diffusion process which evolves \nbased on the observed paths.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"afc152a2783167b289326e528f0077951\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.getUniquePaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>l</em> = <code>-1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the number of unique paths up to a given length l. For the default \nvalue of l=-1 paths of any length will be counted. \n\n@param l: the (inclusive) maximum length up to which path shall be counted. \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a4b4e13eb898dd55b9c1a381eaf22aea9\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.ObservationCount </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the total number of observed pathways of any length \n(includes multiple observations for paths with a frequency weight)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a160fb269d25c24adc4fbfdc3df71075c\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.projectPaths </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>mapping</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a new path object in which nodes have been mapped to different labels\ngiven by an arbitrary mapping function. For instance, for the mapping \n{'a': 'x', 'b': 'x', 'c': 'y', 'd': 'y'} the path (a,b,c,d) is mapped to \n(x,x,y,y). This is useful, e.g., to map page page click streams to topic \nclick streams, using a mapping from pages to topics.\n\n@param mapping: a dictionary that maps nodes to the new labels\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"adba5db911e0be900c03908f8b2ca511e\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.readEdges </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>filename</em> = <code>None</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>separator</em> = <code>'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>weight</em> = <code>False</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>undirected</em> = <code>False</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxlines</em> = <code>_sys.maxsize</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>expandSubPaths</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Reads data from a file containing multiple lines of *edges* of the\nform \"v,w,frequency,X\" (where frequency is optional and X are arbitrary additional columns). The default separating \ncharacter ',' can be changed. In order to calculate the statistics of paths of any length, \nby default all subpaths of length 1 (i.e. single nodes) contained in an edge will be considered.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a349974504cf0ef9fd1ff97a0249e649e\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.readFile </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>filename</em> = <code>None</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>separator</em> = <code>'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>pathFrequency</em> = <code>False</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxlines</em> = <code>_sys.maxsize</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxN</em> = <code>_sys.maxsize</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>expandSubPaths</em> = <code>True</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Reads path data from a file containing multiple lines of n-grams of the \nform \"a,b,c,d,frequency\" (where frequency is optional). The default separating \ncharacter ',' can be changed. Each n-gram will be interpreted as a path of length n-1, \ni.e. bigrams a,b are considered as path of length one, trigrams a,b,c as path of length two, etc.\nIn order to calculate the statistics of paths of any length, by default all subpaths of \nlength k &lt; n-1 contained in an n-gram will be considered. I.e. for n=4 the four-gram a,b,c,d \nwill be considered as a single (longest) path of length n-1 = 3 and three subpaths \na-&gt;b, b-&gt;c, c-&gt;d of length k=1 and two subpaths a-&gt;b-&gt;c amd b-&gt;c-&gt;d of length k=2 will be \nadditionally counted.\n\n@param filename: name of the n-gram file to read data from\n\n@param separator: the character used to separate nodes on the path, i.e. using a \n    separator character of ';' n-grams are represented as a;b;c;...\n\n@param pathFrequency: if set to true, the last entry in each n-gram will be interpreted as \n    weight (i.e. frequency of the path), e.g. a,b,c,d,4 means that four-gram a,b,c,d has weight four.\n    False by default, which means each path occurrence is assigned a default weight of one (adding weights \n    of multiple occurrences).\n\n@param maxlines: The maximum number of lines (i.e. ngrams) to read from the input file\n\n@param maxN: The maximum n for the n-grams to read, i.e. setting maxN to 15 will ignore all n-grams of length \n    16 and longer, which means that only paths up to length n-1 are considered.\n\n@param expandSubPaths: by default all subpaths of the given ngrams are generated, i.e. \n    for an input file with a single trigram a;b;c a path a-&gt;b-&gt;c of length two will be generated\n    as well as two subpaths a-&gt;b and b-&gt;c of length one\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ad519d020de268ea49898e7520be4ffcf\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.summary </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a string containing basic summary info of this Paths instance\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a5c964af581ac3c48fbd2a3955f711bb2\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.Paths.Paths.VisitationProbabilities </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the probabilities that randomly chosen paths\npass through nodes. If 5 out of 100 paths (of any length) contain \nnode v, it will be assigned a value of 0.05. This measure can be \ninterpreted as path-based ground truth for the notion of importance \ncaptured by PageRank applied to a graphical abstraction of the paths.\n</pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Data Documentation</h2>\n<a class=\"anchor\" id=\"aacbff90d31fabf41c2413246aafc8275\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">pathpy.Paths.Paths.paths</td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n\n<p>A dictionary of paths that has the following structure: </p>\n<ul>\n<li>paths[k] is a dictionary containing all paths of length k, indexed by a path tuple p = (u,v,w,...)</li>\n<li>for each tuple p of length k, paths[k][p] contains a tuple (i,j) where i refers to the number of times p occurs as a subpath of a longer path, and j refers to the number of times p occurs as a <em>real</em> or <em>longest</em> path (i.e. not being a subpath of a longer path) </li>\n</ul>\n\n</div>\n</div>\n<hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/Paths.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
  },
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    "path": "docs/classpathpy_1_1TemporalNetwork_1_1TemporalNetwork-members.html",
    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Member List</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>TemporalNetwork</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">TemporalNetwork</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.TemporalNetwork.TemporalNetwork Member List</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n\n<p>This is the complete list of members for <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a>, including all inherited members.</p>\n<table class=\"directory\">\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a778b3b6f649c7057ac1f3f7dc3e43aff\">__init__</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0deb7b84090e2667840606ab0eeae509\">__str__</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae9ea417aa751edb7e58bc1c23602b9b8\">activities</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a65a5f24cb5e6cf06960f8c93c6c8aa84\">activities_sets</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aac7da90422a66a8123776b3d698b4615\">addEdge</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a040f4101e98fb4507709976c2ef7452b\">ecount</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa1b765e6f119b214d78ba16de3b36cf4\">filterEdges</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a3e43ba8b22586d6f55b2ab28a65bf35b\">getInterEventTimes</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a7a385a1b17104011975aa9c0c3f2912a\">getInterPathTimes</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0ac006ac7818ef042fde95ea90deee80\">getObservationLength</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a948503bbe0104c9678778cdecfe8ceb7\">GetTemporalBetweenness</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa6a91612301e802d287f1fb58fe31961\">GetTemporalBetweennessInstantaneous</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a9dd08ea1ab218cb69b2fdd56cbcab61f\">GetTemporalCloseness</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ac73ea9fa5c13c1df5ac101d45473243d\">GetTemporalClosenessInstantaneous</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a86e49baf6e63c58a3f993e2768097001\">nodes</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a051e7d352da3d194387762134bd70992\">ordered_times</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a54bf4b554d7ca2a45a49cc76a0080cea\">readFile</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"><span class=\"mlabel\">static</span></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a50ea38c4325e1035966c4d2173599a1d\">ShuffleEdges</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0716456e27f19af5522af30e071df0ff\">sources</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a39d2b3a872f2a19554ee505393a6f708\">summary</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae82b4df377620a626771cdcdcbb43b42\">targets</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a2ed001aa13e863e9ff6ab9ffc9ab998d\">tedges</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr class=\"even\"><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a1f61dbedd4a4edde5176fec56fc1f0b3\">time</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n  <tr><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a727ebc2cc2eb2ad8c1ee23cdd5a25b6e\">vcount</a></td><td class=\"entry\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"entry\"></td></tr>\n</table></div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: pathpy.TemporalNetwork.TemporalNetwork Class Reference</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" 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<li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div id=\"nav-path\" class=\"navpath\">\n  <ul>\n<li class=\"navelem\"><b>pathpy</b></li><li class=\"navelem\"><b>TemporalNetwork</b></li><li class=\"navelem\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\">TemporalNetwork</a></li>  </ul>\n</div>\n</div><!-- top -->\n<div class=\"header\">\n  <div class=\"summary\">\n<a href=\"#pub-methods\">Public Member Functions</a> &#124;\n<a href=\"#pub-static-methods\">Static Public Member Functions</a> &#124;\n<a href=\"#pub-attribs\">Public Attributes</a> &#124;\n<a href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork-members.html\">List of all members</a>  </div>\n  <div class=\"headertitle\">\n<div class=\"title\">pathpy.TemporalNetwork.TemporalNetwork Class Reference</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<table class=\"memberdecls\">\n<tr class=\"heading\"><td colspan=\"2\"><h2 class=\"groupheader\"><a name=\"pub-methods\"></a>\nPublic Member Functions</h2></td></tr>\n<tr class=\"memitem:a778b3b6f649c7057ac1f3f7dc3e43aff\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a778b3b6f649c7057ac1f3f7dc3e43aff\">__init__</a></td></tr>\n<tr class=\"separator:a778b3b6f649c7057ac1f3f7dc3e43aff\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:aa1b765e6f119b214d78ba16de3b36cf4\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa1b765e6f119b214d78ba16de3b36cf4\">filterEdges</a></td></tr>\n<tr class=\"separator:aa1b765e6f119b214d78ba16de3b36cf4\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:aac7da90422a66a8123776b3d698b4615\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aac7da90422a66a8123776b3d698b4615\">addEdge</a></td></tr>\n<tr class=\"separator:aac7da90422a66a8123776b3d698b4615\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a727ebc2cc2eb2ad8c1ee23cdd5a25b6e\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\">def&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a727ebc2cc2eb2ad8c1ee23cdd5a25b6e\">vcount</a></td></tr>\n<tr 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name=\"pub-attribs\"></a>\nPublic Attributes</h2></td></tr>\n<tr class=\"memitem:a2ed001aa13e863e9ff6ab9ffc9ab998d\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a2ed001aa13e863e9ff6ab9ffc9ab998d\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a2ed001aa13e863e9ff6ab9ffc9ab998d\">tedges</a></td></tr>\n<tr class=\"memdesc:a2ed001aa13e863e9ff6ab9ffc9ab998d\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A list of time-stamped edges of this temporal network. <br/></td></tr>\n<tr class=\"separator:a2ed001aa13e863e9ff6ab9ffc9ab998d\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a86e49baf6e63c58a3f993e2768097001\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a86e49baf6e63c58a3f993e2768097001\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a86e49baf6e63c58a3f993e2768097001\">nodes</a></td></tr>\n<tr class=\"memdesc:a86e49baf6e63c58a3f993e2768097001\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A list of nodes of this temporal network. <br/></td></tr>\n<tr class=\"separator:a86e49baf6e63c58a3f993e2768097001\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a1f61dbedd4a4edde5176fec56fc1f0b3\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a1f61dbedd4a4edde5176fec56fc1f0b3\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a1f61dbedd4a4edde5176fec56fc1f0b3\">time</a></td></tr>\n<tr class=\"memdesc:a1f61dbedd4a4edde5176fec56fc1f0b3\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary storing all time-stamped links, indexed by time-stamps. <br/></td></tr>\n<tr class=\"separator:a1f61dbedd4a4edde5176fec56fc1f0b3\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ae82b4df377620a626771cdcdcbb43b42\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ae82b4df377620a626771cdcdcbb43b42\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae82b4df377620a626771cdcdcbb43b42\">targets</a></td></tr>\n<tr class=\"memdesc:ae82b4df377620a626771cdcdcbb43b42\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary storing all time-stamped links, indexed by time and target node. <br/></td></tr>\n<tr class=\"separator:ae82b4df377620a626771cdcdcbb43b42\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a0716456e27f19af5522af30e071df0ff\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a0716456e27f19af5522af30e071df0ff\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0716456e27f19af5522af30e071df0ff\">sources</a></td></tr>\n<tr class=\"memdesc:a0716456e27f19af5522af30e071df0ff\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary storing all time-stamped links, indexed by time and source node. <br/></td></tr>\n<tr class=\"separator:a0716456e27f19af5522af30e071df0ff\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:ae9ea417aa751edb7e58bc1c23602b9b8\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"ae9ea417aa751edb7e58bc1c23602b9b8\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae9ea417aa751edb7e58bc1c23602b9b8\">activities</a></td></tr>\n<tr class=\"memdesc:ae9ea417aa751edb7e58bc1c23602b9b8\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary storing time stamps at which links (v,*;t) originate from node v. <br/></td></tr>\n<tr class=\"separator:ae9ea417aa751edb7e58bc1c23602b9b8\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a65a5f24cb5e6cf06960f8c93c6c8aa84\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a65a5f24cb5e6cf06960f8c93c6c8aa84\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a65a5f24cb5e6cf06960f8c93c6c8aa84\">activities_sets</a></td></tr>\n<tr class=\"memdesc:a65a5f24cb5e6cf06960f8c93c6c8aa84\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">A dictionary storing sets of time stamps at which links (v,*;t) originate from node v Note that the insertion into a set is much faster than repeatedly checking whether an element already exists in a list! <br/></td></tr>\n<tr class=\"separator:a65a5f24cb5e6cf06960f8c93c6c8aa84\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n<tr class=\"memitem:a051e7d352da3d194387762134bd70992\"><td class=\"memItemLeft\" align=\"right\" valign=\"top\"><a class=\"anchor\" id=\"a051e7d352da3d194387762134bd70992\"></a>\n&#160;</td><td class=\"memItemRight\" valign=\"bottom\"><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a051e7d352da3d194387762134bd70992\">ordered_times</a></td></tr>\n<tr class=\"memdesc:a051e7d352da3d194387762134bd70992\"><td class=\"mdescLeft\">&#160;</td><td class=\"mdescRight\">An ordered list of time-stamps. <br/></td></tr>\n<tr class=\"separator:a051e7d352da3d194387762134bd70992\"><td class=\"memSeparator\" colspan=\"2\">&#160;</td></tr>\n</table>\n<a name=\"details\" id=\"details\"></a><h2 class=\"groupheader\">Detailed Description</h2>\n<div class=\"textblock\"><pre class=\"fragment\">This class represents a sequence of time-stamped edges.\n   Instances of this class can be used to generate path statistics \n   based on the time-respecting paths resulting from a given maximum\n   time difference between consecutive time-stamped edges.\n</pre> </div><h2 class=\"groupheader\">Constructor &amp; Destructor Documentation</h2>\n<a class=\"anchor\" id=\"a778b3b6f649c7057ac1f3f7dc3e43aff\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.__init__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>tedges</em> = <code>None</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Constructor that generates a temporal network instance. \n\n@param tedges: an optional list of (possibly unordered time-stamped) links \n    from which to construct a temporal network instance. For the default value None        \n    an empty temporal network will be created.\n</pre> \n</div>\n</div>\n<h2 class=\"groupheader\">Member Function Documentation</h2>\n<a class=\"anchor\" id=\"a0deb7b84090e2667840606ab0eeae509\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.__str__ </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the default string representation of \nthis temporal network instance.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"aac7da90422a66a8123776b3d698b4615\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.addEdge </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>source</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>target</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>ts</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Adds a directed time-stamped edge (source,target;time) to the temporal network. To add an undirected \n    time-stamped link (u,v;t) at time t, please call addEdge(u,v;t) and addEdge(v,u;t).\n\n@param source: name of the source node of a directed, time-stamped link\n@param target: name of the target node of a directed, time-stamped link\n@param ts: (integer) time-stamp of the time-stamped link\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a040f4101e98fb4507709976c2ef7452b\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.ecount </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the number of time-stamped edges (u,v;t) in the temporal network.\nThis number corresponds to the sum of link weights in the (first-order)\ntime-aggregated network.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"aa1b765e6f119b214d78ba16de3b36cf4\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.filterEdges </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>edge_filter</em>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Filter time-stamped edges according to a given filter expression. \n\n@param edge_filter: an arbitrary filter function of the form filter_func(v, w, time) that \n    returns True for time-stamped edges that shall pass the filter, and False for all edges that \n    shall be filtered out.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a3e43ba8b22586d6f55b2ab28a65bf35b\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.getInterEventTimes </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns an array containing all time differences between any \ntwo consecutive time-stamped links (involving any node)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a7a385a1b17104011975aa9c0c3f2912a\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.getInterPathTimes </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a dictionary which, for each node v, contains all time differences \nbetween any time-stamped link (*,v;t) and the next link (v,*;t') (t'&gt;t)\nin the temporal network\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a0ac006ac7818ef042fde95ea90deee80\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.getObservationLength </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the length of the observation time in time units.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a948503bbe0104c9678778cdecfe8ceb7\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.GetTemporalBetweenness </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>t</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>delta</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the temporal betweenness centralities of all nodes based on the shortest \ntime-respecting paths with a maximum waiting time of delta. This function returns a \nnumpy array of temporal betweenness centrality values of nodes.\n    \n@param t: the temporal network for which temporal closeness centralities will be computed    \n@param delta: the maximum time difference used in the time-respecting path definition (default 1).\n    Note that this parameter is independent from the delta used internally for the extraction \n    of two-paths by the class TemporalNetwork\n@param normalized: whether or not to normalize centralities by dividing each value by the total \n    number of shortest time-respecting paths.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"aa6a91612301e802d287f1fb58fe31961\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.GetTemporalBetweennessInstantaneous </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>t</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>start_t</em> = <code>0</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>delta</em> = <code>1</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>normalized</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the temporal betweennness values of \nall nodes fir a given start time start_t in an empirical temporal network t.\nThis function returns a numpy array of (temporal) betweenness centrality values. \nThe ordering of these values corresponds to the ordering of nodes in the vertex \nsequence of the igraph first order time-aggregated network. A mapping between node names\nand array indices can be found in Utilities.firstOrderNameMap().\n    \n@param t: the temporal network for which temporal betweenness centralities will be computed\n@param start_t: the start time for which to consider time-respecting paths (default 0). This is \n    important, since any unambigious definition of a shortest time-respecting path between\n    two nodes must include the time range to be considered (c.f. Holme and Saramäki, Phys. Rep., 2012)\n@param delta: the maximum waiting time used in the time-respecting path definition (default 1)\n    Note that this parameter is independent from the delta used internally for the extraction of two-paths\n    by the class TemporalNetwork\n@param normalized: whether or not to normalize the temporal betweenness centrality values by\n    dividing by the number of all shortest time-respecting paths in the temporal network.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a9dd08ea1ab218cb69b2fdd56cbcab61f\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.GetTemporalCloseness </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>t</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>delta</em> = <code>1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the temporal closeness centralities of all nodes based on the minimal \nshortest time-respecting paths with a maximum time difference of delta. This function \nreturns a numpy array of average (temporal) closeness centrality values of nodes.\n    \n@param t: the temporal network for which temporal closeness centralities will be computed   \n@param delta: the maximum waiting time used in the time-respecting path definition (default 1)            \n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"ac73ea9fa5c13c1df5ac101d45473243d\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.GetTemporalClosenessInstantaneous </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>t</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>start_t</em> = <code>0</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>delta</em> = <code>1</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Calculates the temporal closeness values of all nodes for a given start time start_t.\nThis function returns a numpy array of (temporal) closeness centrality values.        \n    \n@param t: the temporal network for which temporal closeness centralities will be computed\n@param start_t: the start time for which to consider time-respecting paths (default 0). This is \n    important, since any unambigious definition of a shortest time-respecting path between\n    two nodes must include the time range to be considered (c.f. Holme and Saramäki, Phys. Rep., 2012)\n@param delta: the maximum time difference time used in the time-respecting path definition (default 1)\n    Note that this parameter is independent from the delta used internally for the extraction of two-paths\n    by the class TemporalNetwork.\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a54bf4b554d7ca2a45a49cc76a0080cea\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n<table class=\"mlabels\">\n  <tr>\n  <td class=\"mlabels-left\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.readFile </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>filename</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>sep</em> = <code>'</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>timestampformat</em> = <code>&quot;%Y-%m-%d&#160;%H:%M&quot;</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>maxlines</em> = <code>_sys.maxsize</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n  </td>\n  <td class=\"mlabels-right\">\n<span class=\"mlabels\"><span class=\"mlabel\">static</span></span>  </td>\n  </tr>\n</table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Reads time-stamped links from a file and returns a new instance \n    of the class TemporalNetwork. The file is assumed to have a header \n\nsource target time \n\n    where columns can be in arbitrary order and separated by arbitrary characters. \n    Each time-stamped link must occur in a separate line and links are assumed to be\n    directed.\n     \n    The time column can be omitted and in this case all links are assumed to occur \n    in consecutive time stamps (that have a distance of one). Time stamps can be simple \n    integers, or strings to be converted to UNIX time stamps via a custom timestamp format. \n    For this, the python function datetime.strptime will be used. \n\n    @param sep: the character that separates columns \n    @param filename: path of the file to read from\n    @param timestampformat: used to convert string timestamps to UNIX timestamps. This parameter is \nignored, if the timestamps are digit types (like a simple int).\n    @param maxlines: limit reading of file to certain number of lines, default sys.maxsize</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a50ea38c4325e1035966c4d2173599a1d\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.ShuffleEdges </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>l</em> = <code>0</code>, </td>\n        </tr>\n        <tr>\n          <td class=\"paramkey\"></td>\n          <td></td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>with_replacement</em> = <code>False</code>&#160;</td>\n        </tr>\n        <tr>\n          <td></td>\n          <td>)</td>\n          <td></td><td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Generates a shuffled version of the temporal network in which edge statistics (i.e.\nthe frequencies of time-stamped edges) are preserved, while all order correlations are \ndestroyed. The shuffling procedure randomly reshuffles the time-stamps of links.\n\n@param l: the length of the sequence to be generated (i.e. the number of time-stamped links.\n    For the default value l=0, the length of the generated shuffled temporal network will be \n    equal to that of the original temporal network. \n@param with_replacement: Whether or not the sampling should be with replacement (default False)\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a39d2b3a872f2a19554ee505393a6f708\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.summary </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns a string containing basic summary statistics of this temporal network\n</pre> \n</div>\n</div>\n<a class=\"anchor\" id=\"a727ebc2cc2eb2ad8c1ee23cdd5a25b6e\"></a>\n<div class=\"memitem\">\n<div class=\"memproto\">\n      <table class=\"memname\">\n        <tr>\n          <td class=\"memname\">def pathpy.TemporalNetwork.TemporalNetwork.vcount </td>\n          <td>(</td>\n          <td class=\"paramtype\">&#160;</td>\n          <td class=\"paramname\"><em>self</em></td><td>)</td>\n          <td></td>\n        </tr>\n      </table>\n</div><div class=\"memdoc\">\n<pre class=\"fragment\">Returns the number of vertices in the temporal network. \nThis number corresponds to the number of nodes in the (first-order) \ntime-aggregated network.\n</pre> \n</div>\n</div>\n<hr/>The documentation for this class was generated from the following file:<ul>\n<li>/mnt/c/Users/ingos/Desktop/pathpy/pathpy/TemporalNetwork.py</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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border-top-left-radius: 4px;\n        /* firefox specific markup */\n        -moz-box-shadow: rgba(0, 0, 0, 0.15) 5px 5px 5px;\n        -moz-border-radius-topright: 4px;\n        -moz-border-radius-topleft: 4px;\n        /* webkit specific markup */\n        -webkit-box-shadow: 5px 5px 5px rgba(0, 0, 0, 0.15);\n        -webkit-border-top-right-radius: 4px;\n        -webkit-border-top-left-radius: 4px;\n\n}\n\n.memdoc, dl.reflist dd {\n        border-bottom: 1px solid #A8B8D9;      \n        border-left: 1px solid #A8B8D9;      \n        border-right: 1px solid #A8B8D9; \n        padding: 6px 10px 2px 10px;\n        background-color: #FBFCFD;\n        border-top-width: 0;\n        background-image:url('nav_g.png');\n        background-repeat:repeat-x;\n        background-color: #FFFFFF;\n        /* opera specific markup */\n        border-bottom-left-radius: 4px;\n        border-bottom-right-radius: 4px;\n        box-shadow: 5px 5px 5px rgba(0, 0, 0, 0.15);\n        /* firefox specific markup */\n        -moz-border-radius-bottomleft: 4px;\n        -moz-border-radius-bottomright: 4px;\n        -moz-box-shadow: rgba(0, 0, 0, 0.15) 5px 5px 5px;\n        /* webkit specific markup */\n        -webkit-border-bottom-left-radius: 4px;\n        -webkit-border-bottom-right-radius: 4px;\n        -webkit-box-shadow: 5px 5px 5px rgba(0, 0, 0, 0.15);\n}\n\ndl.reflist dt {\n        padding: 5px;\n}\n\ndl.reflist dd {\n        margin: 0px 0px 10px 0px;\n        padding: 5px;\n}\n\n.paramkey {\n\ttext-align: right;\n}\n\n.paramtype {\n\twhite-space: nowrap;\n}\n\n.paramname {\n\tcolor: #602020;\n\twhite-space: nowrap;\n}\n.paramname em {\n\tfont-style: normal;\n}\n.paramname code {\n        line-height: 14px;\n}\n\n.params, .retval, .exception, .tparams {\n        margin-left: 0px;\n        padding-left: 0px;\n}       \n\n.params .paramname, .retval .paramname {\n        font-weight: bold;\n        vertical-align: top;\n}\n        \n.params .paramtype {\n        font-style: italic;\n        vertical-align: top;\n}       \n        \n.params .paramdir {\n        font-family: \"courier new\",courier,monospace;\n        vertical-align: top;\n}\n\ntable.mlabels {\n\tborder-spacing: 0px;\n}\n\ntd.mlabels-left {\n\twidth: 100%;\n\tpadding: 0px;\n}\n\ntd.mlabels-right {\n\tvertical-align: bottom;\n\tpadding: 0px;\n\twhite-space: nowrap;\n}\n\nspan.mlabels {\n        margin-left: 8px;\n}\n\nspan.mlabel {\n        background-color: #728DC1;\n        border-top:1px solid #5373B4;\n        border-left:1px solid #5373B4;\n        border-right:1px solid #C4CFE5;\n        border-bottom:1px solid #C4CFE5;\n\ttext-shadow: none;\n\tcolor: white;\n\tmargin-right: 4px;\n\tpadding: 2px 3px;\n\tborder-radius: 3px;\n\tfont-size: 7pt;\n\twhite-space: nowrap;\n\tvertical-align: middle;\n}\n\n\n\n/* @end */\n\n/* these are for tree view when not used as main index */\n\ndiv.directory {\n        margin: 10px 0px;\n        border-top: 1px solid #A8B8D9;\n        border-bottom: 1px solid #A8B8D9;\n        width: 100%;\n}\n\n.directory table {\n        border-collapse:collapse;\n}\n\n.directory td {\n        margin: 0px;\n        padding: 0px;\n\tvertical-align: top;\n}\n\n.directory td.entry {\n        white-space: nowrap;\n        padding-right: 6px;\n\tpadding-top: 3px;\n}\n\n.directory td.entry a {\n        outline:none;\n}\n\n.directory td.entry a img {\n        border: none;\n}\n\n.directory td.desc {\n        width: 100%;\n        padding-left: 6px;\n\tpadding-right: 6px;\n\tpadding-top: 3px;\n\tborder-left: 1px solid rgba(0,0,0,0.05);\n}\n\n.directory tr.even {\n\tpadding-left: 6px;\n\tbackground-color: #F7F8FB;\n}\n\n.directory img {\n\tvertical-align: -30%;\n}\n\n.directory .levels {\n        white-space: nowrap;\n        width: 100%;\n        text-align: right;\n        font-size: 9pt;\n}\n\n.directory .levels span {\n        cursor: pointer;\n        padding-left: 2px;\n        padding-right: 2px;\n\tcolor: #3D578C;\n}\n\ndiv.dynheader {\n        margin-top: 8px;\n\t-webkit-touch-callout: none;\n\t-webkit-user-select: none;\n\t-khtml-user-select: none;\n\t-moz-user-select: none;\n\t-ms-user-select: none;\n\tuser-select: none;\n}\n\naddress {\n\tfont-style: normal;\n\tcolor: #2A3D61;\n}\n\ntable.doxtable {\n\tborder-collapse:collapse;\n        margin-top: 4px;\n        margin-bottom: 4px;\n}\n\ntable.doxtable td, table.doxtable th {\n\tborder: 1px solid #2D4068;\n\tpadding: 3px 7px 2px;\n}\n\ntable.doxtable th {\n\tbackground-color: #374F7F;\n\tcolor: #FFFFFF;\n\tfont-size: 110%;\n\tpadding-bottom: 4px;\n\tpadding-top: 5px;\n}\n\ntable.fieldtable {\n        /*width: 100%;*/\n        margin-bottom: 10px;\n        border: 1px solid #A8B8D9;\n        border-spacing: 0px;\n        -moz-border-radius: 4px;\n        -webkit-border-radius: 4px;\n        border-radius: 4px;\n        -moz-box-shadow: rgba(0, 0, 0, 0.15) 2px 2px 2px;\n        -webkit-box-shadow: 2px 2px 2px rgba(0, 0, 0, 0.15);\n        box-shadow: 2px 2px 2px rgba(0, 0, 0, 0.15);\n}\n\n.fieldtable td, .fieldtable th {\n        padding: 3px 7px 2px;\n}\n\n.fieldtable td.fieldtype, .fieldtable td.fieldname {\n        white-space: nowrap;\n        border-right: 1px solid #A8B8D9;\n        border-bottom: 1px solid #A8B8D9;\n        vertical-align: top;\n}\n\n.fieldtable td.fieldname {\n        padding-top: 3px;\n}\n\n.fieldtable td.fielddoc {\n        border-bottom: 1px solid #A8B8D9;\n        /*width: 100%;*/\n}\n\n.fieldtable td.fielddoc p:first-child {\n        margin-top: 0px;\n}       \n        \n.fieldtable td.fielddoc p:last-child {\n        margin-bottom: 2px;\n}\n\n.fieldtable tr:last-child td {\n        border-bottom: none;\n}\n\n.fieldtable th {\n        background-image:url('nav_f.png');\n        background-repeat:repeat-x;\n        background-color: #E2E8F2;\n        font-size: 90%;\n        color: #253555;\n        padding-bottom: 4px;\n        padding-top: 5px;\n        text-align:left;\n        -moz-border-radius-topleft: 4px;\n        -moz-border-radius-topright: 4px;\n        -webkit-border-top-left-radius: 4px;\n        -webkit-border-top-right-radius: 4px;\n        border-top-left-radius: 4px;\n        border-top-right-radius: 4px;\n        border-bottom: 1px solid #A8B8D9;\n}\n\n\n.tabsearch {\n\ttop: 0px;\n\tleft: 10px;\n\theight: 36px;\n\tbackground-image: url('tab_b.png');\n\tz-index: 101;\n\toverflow: hidden;\n\tfont-size: 13px;\n}\n\n.navpath ul\n{\n\tfont-size: 11px;\n\tbackground-image:url('tab_b.png');\n\tbackground-repeat:repeat-x;\n\tbackground-position: 0 -5px;\n\theight:30px;\n\tline-height:30px;\n\tcolor:#8AA0CC;\n\tborder:solid 1px #C2CDE4;\n\toverflow:hidden;\n\tmargin:0px;\n\tpadding:0px;\n}\n\n.navpath li\n{\n\tlist-style-type:none;\n\tfloat:left;\n\tpadding-left:10px;\n\tpadding-right:15px;\n\tbackground-image:url('bc_s.png');\n\tbackground-repeat:no-repeat;\n\tbackground-position:right;\n\tcolor:#364D7C;\n}\n\n.navpath li.navelem a\n{\n\theight:32px;\n\tdisplay:block;\n\ttext-decoration: none;\n\toutline: none;\n\tcolor: #283A5D;\n\tfont-family: 'Lucida Grande',Geneva,Helvetica,Arial,sans-serif;\n\ttext-shadow: 0px 1px 1px rgba(255, 255, 255, 0.9);\n\ttext-decoration: none;        \n}\n\n.navpath li.navelem a:hover\n{\n\tcolor:#6884BD;\n}\n\n.navpath li.footer\n{\n        list-style-type:none;\n        float:right;\n        padding-left:10px;\n        padding-right:15px;\n        background-image:none;\n        background-repeat:no-repeat;\n        background-position:right;\n        color:#364D7C;\n        font-size: 8pt;\n}\n\n\ndiv.summary\n{\n\tfloat: right;\n\tfont-size: 8pt;\n\tpadding-right: 5px;\n\twidth: 50%;\n\ttext-align: right;\n}       \n\ndiv.summary a\n{\n\twhite-space: nowrap;\n}\n\ndiv.ingroups\n{\n\tfont-size: 8pt;\n\twidth: 50%;\n\ttext-align: left;\n}\n\ndiv.ingroups a\n{\n\twhite-space: nowrap;\n}\n\ndiv.header\n{\n        background-image:url('nav_h.png');\n        background-repeat:repeat-x;\n\tbackground-color: #F9FAFC;\n\tmargin:  0px;\n\tborder-bottom: 1px solid #C4CFE5;\n}\n\ndiv.headertitle\n{\n\tpadding: 5px 5px 5px 10px;\n}\n\ndl\n{\n        padding: 0 0 0 10px;\n}\n\n/* dl.note, dl.warning, dl.attention, dl.pre, dl.post, dl.invariant, dl.deprecated, dl.todo, dl.test, dl.bug */\ndl.section\n{\n\tmargin-left: 0px;\n\tpadding-left: 0px;\n}\n\ndl.note\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #D0C000;\n}\n\ndl.warning, dl.attention\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #FF0000;\n}\n\ndl.pre, dl.post, dl.invariant\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #00D000;\n}\n\ndl.deprecated\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #505050;\n}\n\ndl.todo\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #00C0E0;\n}\n\ndl.test\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #3030E0;\n}\n\ndl.bug\n{\n        margin-left:-7px;\n        padding-left: 3px;\n        border-left:4px solid;\n        border-color: #C08050;\n}\n\ndl.section dd {\n\tmargin-bottom: 6px;\n}\n\n\n#projectlogo\n{\n\ttext-align: center;\n\tvertical-align: bottom;\n\tborder-collapse: separate;\n}\n \n#projectlogo img\n{ \n\tborder: 0px none;\n}\n \n#projectname\n{\n\tfont: 300% Tahoma, Arial,sans-serif;\n\tmargin: 0px;\n\tpadding: 2px 0px;\n}\n    \n#projectbrief\n{\n\tfont: 120% Tahoma, Arial,sans-serif;\n\tmargin: 0px;\n\tpadding: 0px;\n}\n\n#projectnumber\n{\n\tfont: 50% Tahoma, Arial,sans-serif;\n\tmargin: 0px;\n\tpadding: 0px;\n}\n\n#titlearea\n{\n\tpadding: 0px;\n\tmargin: 0px;\n\twidth: 100%;\n\tborder-bottom: 1px solid #5373B4;\n}\n\n.image\n{\n        text-align: center;\n}\n\n.dotgraph\n{\n        text-align: center;\n}\n\n.mscgraph\n{\n        text-align: center;\n}\n\n.diagraph\n{\n        text-align: center;\n}\n\n.caption\n{\n\tfont-weight: bold;\n}\n\ndiv.zoom\n{\n\tborder: 1px solid #90A5CE;\n}\n\ndl.citelist {\n        margin-bottom:50px;\n}\n\ndl.citelist dt {\n        color:#334975;\n        float:left;\n        font-weight:bold;\n        margin-right:10px;\n        padding:5px;\n}\n\ndl.citelist dd {\n        margin:2px 0;\n        padding:5px 0;\n}\n\ndiv.toc {\n        padding: 14px 25px;\n        background-color: #F4F6FA;\n        border: 1px solid #D8DFEE;\n        border-radius: 7px 7px 7px 7px;\n        float: right;\n        height: auto;\n        margin: 0 20px 10px 10px;\n        width: 200px;\n}\n\ndiv.toc li {\n        background: url(\"bdwn.png\") no-repeat scroll 0 5px transparent;\n        font: 10px/1.2 Verdana,DejaVu Sans,Geneva,sans-serif;\n        margin-top: 5px;\n        padding-left: 10px;\n        padding-top: 2px;\n}\n\ndiv.toc h3 {\n        font: bold 12px/1.2 Arial,FreeSans,sans-serif;\n\tcolor: #4665A2;\n        border-bottom: 0 none;\n        margin: 0;\n}\n\ndiv.toc ul {\n        list-style: none outside none;\n        border: medium none;\n        padding: 0px;\n}       \n\ndiv.toc li.level1 {\n        margin-left: 0px;\n}\n\ndiv.toc li.level2 {\n        margin-left: 15px;\n}\n\ndiv.toc li.level3 {\n        margin-left: 30px;\n}\n\ndiv.toc li.level4 {\n        margin-left: 45px;\n}\n\n.inherit_header {\n        font-weight: bold;\n        color: gray;\n        cursor: pointer;\n\t-webkit-touch-callout: none;\n\t-webkit-user-select: none;\n\t-khtml-user-select: none;\n\t-moz-user-select: none;\n\t-ms-user-select: none;\n\tuser-select: none;\n}\n\n.inherit_header td {\n        padding: 6px 0px 2px 5px;\n}\n\n.inherit {\n        display: none;\n}\n\ntr.heading h2 {\n        margin-top: 12px;\n        margin-bottom: 4px;\n}\n\n/* tooltip related style info */\n\n.ttc {\n        position: absolute;\n        display: none;\n}\n\n#powerTip {\n\tcursor: default;\n\twhite-space: nowrap;\n\tbackground-color: white;\n\tborder: 1px solid gray;\n\tborder-radius: 4px 4px 4px 4px;\n\tbox-shadow: 1px 1px 7px gray;\n\tdisplay: none;\n\tfont-size: smaller;\n\tmax-width: 80%;\n\topacity: 0.9;\n\tpadding: 1ex 1em 1em;\n\tposition: absolute;\n\tz-index: 2147483647;\n}\n\n#powerTip div.ttdoc {\n        color: grey;\n\tfont-style: italic;\n}\n\n#powerTip div.ttname a {\n        font-weight: bold;\n}\n\n#powerTip div.ttname {\n        font-weight: bold;\n}\n\n#powerTip div.ttdeci {\n        color: #006318;\n}\n\n#powerTip div {\n        margin: 0px;\n        padding: 0px;\n        font: 12px/16px Roboto,sans-serif;\n}\n\n#powerTip:before, #powerTip:after {\n\tcontent: \"\";\n\tposition: absolute;\n\tmargin: 0px;\n}\n\n#powerTip.n:after,  #powerTip.n:before,\n#powerTip.s:after,  #powerTip.s:before,\n#powerTip.w:after,  #powerTip.w:before,\n#powerTip.e:after,  #powerTip.e:before,\n#powerTip.ne:after, #powerTip.ne:before,\n#powerTip.se:after, #powerTip.se:before,\n#powerTip.nw:after, #powerTip.nw:before,\n#powerTip.sw:after, #powerTip.sw:before {\n\tborder: solid transparent;\n\tcontent: \" \";\n\theight: 0;\n\twidth: 0;\n\tposition: absolute;\n}\n\n#powerTip.n:after,  #powerTip.s:after,\n#powerTip.w:after,  #powerTip.e:after,\n#powerTip.nw:after, #powerTip.ne:after,\n#powerTip.sw:after, #powerTip.se:after {\n\tborder-color: rgba(255, 255, 255, 0);\n}\n\n#powerTip.n:before,  #powerTip.s:before,\n#powerTip.w:before,  #powerTip.e:before,\n#powerTip.nw:before, #powerTip.ne:before,\n#powerTip.sw:before, #powerTip.se:before {\n\tborder-color: rgba(128, 128, 128, 0);\n}\n\n#powerTip.n:after,  #powerTip.n:before,\n#powerTip.ne:after, #powerTip.ne:before,\n#powerTip.nw:after, #powerTip.nw:before {\n\ttop: 100%;\n}\n\n#powerTip.n:after, #powerTip.ne:after, #powerTip.nw:after {\n\tborder-top-color: #ffffff;\n\tborder-width: 10px;\n\tmargin: 0px -10px;\n}\n#powerTip.n:before {\n\tborder-top-color: #808080;\n\tborder-width: 11px;\n\tmargin: 0px -11px;\n}\n#powerTip.n:after, #powerTip.n:before {\n\tleft: 50%;\n}\n\n#powerTip.nw:after, #powerTip.nw:before {\n\tright: 14px;\n}\n\n#powerTip.ne:after, #powerTip.ne:before {\n\tleft: 14px;\n}\n\n#powerTip.s:after,  #powerTip.s:before,\n#powerTip.se:after, #powerTip.se:before,\n#powerTip.sw:after, #powerTip.sw:before {\n\tbottom: 100%;\n}\n\n#powerTip.s:after, #powerTip.se:after, #powerTip.sw:after {\n\tborder-bottom-color: #ffffff;\n\tborder-width: 10px;\n\tmargin: 0px -10px;\n}\n\n#powerTip.s:before, #powerTip.se:before, #powerTip.sw:before {\n\tborder-bottom-color: #808080;\n\tborder-width: 11px;\n\tmargin: 0px -11px;\n}\n\n#powerTip.s:after, #powerTip.s:before {\n\tleft: 50%;\n}\n\n#powerTip.sw:after, #powerTip.sw:before {\n\tright: 14px;\n}\n\n#powerTip.se:after, #powerTip.se:before {\n\tleft: 14px;\n}\n\n#powerTip.e:after, #powerTip.e:before {\n\tleft: 100%;\n}\n#powerTip.e:after {\n\tborder-left-color: #ffffff;\n\tborder-width: 10px;\n\ttop: 50%;\n\tmargin-top: -10px;\n}\n#powerTip.e:before {\n\tborder-left-color: #808080;\n\tborder-width: 11px;\n\ttop: 50%;\n\tmargin-top: -11px;\n}\n\n#powerTip.w:after, #powerTip.w:before {\n\tright: 100%;\n}\n#powerTip.w:after {\n\tborder-right-color: #ffffff;\n\tborder-width: 10px;\n\ttop: 50%;\n\tmargin-top: -10px;\n}\n#powerTip.w:before {\n\tborder-right-color: #808080;\n\tborder-width: 11px;\n\ttop: 50%;\n\tmargin-top: -11px;\n}\n\n@media print\n{\n  #top { display: none; }\n  #side-nav { display: none; }\n  #nav-path { display: none; }\n  body { overflow:visible; }\n  h1, h2, h3, h4, h5, h6 { page-break-after: avoid; }\n  .summary { display: none; }\n  .memitem { page-break-inside: avoid; }\n  #doc-content\n  {\n    margin-left:0 !important;\n    height:auto !important;\n    width:auto !important;\n    overflow:inherit;\n    display:inline;\n  }\n}\n\n"
  },
  {
    "path": "docs/dynsections.js",
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    "path": "docs/functions.html",
    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Class Members</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li class=\"current\"><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n  <div id=\"navrow3\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li class=\"current\"><a href=\"functions.html\"><span>All</span></a></li>\n      <li><a href=\"functions_func.html\"><span>Functions</span></a></li>\n      <li><a href=\"functions_vars.html\"><span>Variables</span></a></li>\n    </ul>\n  </div>\n  <div id=\"navrow4\" class=\"tabs3\">\n    <ul class=\"tablist\">\n      <li><a href=\"#index__\"><span>_</span></a></li>\n      <li><a href=\"#index_a\"><span>a</span></a></li>\n      <li><a href=\"#index_b\"><span>b</span></a></li>\n      <li><a href=\"#index_c\"><span>c</span></a></li>\n      <li><a href=\"#index_d\"><span>d</span></a></li>\n      <li><a href=\"#index_e\"><span>e</span></a></li>\n      <li><a href=\"#index_f\"><span>f</span></a></li>\n      <li><a href=\"#index_g\"><span>g</span></a></li>\n      <li><a href=\"#index_h\"><span>h</span></a></li>\n      <li><a href=\"#index_i\"><span>i</span></a></li>\n      <li><a href=\"#index_l\"><span>l</span></a></li>\n      <li><a href=\"#index_m\"><span>m</span></a></li>\n      <li><a href=\"#index_n\"><span>n</span></a></li>\n      <li><a href=\"#index_o\"><span>o</span></a></li>\n      <li><a href=\"#index_p\"><span>p</span></a></li>\n      <li><a href=\"#index_r\"><span>r</span></a></li>\n      <li><a href=\"#index_s\"><span>s</span></a></li>\n      <li><a href=\"#index_t\"><span>t</span></a></li>\n      <li><a href=\"#index_v\"><span>v</span></a></li>\n      <li class=\"current\"><a href=\"#index_w\"><span>w</span></a></li>\n    </ul>\n  </div>\n</div><!-- top -->\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div class=\"contents\">\n<div class=\"textblock\">Here is a list of all documented class members with links to the class documentation for each member:</div>\n\n<h3><a class=\"anchor\" id=\"index__\"></a>- _ -</h3><ul>\n<li>__init__()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a63d36720423ee8d6d88c5a06f4655c84\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a670778608688e4926328cf2d851b7d6d\">pathpy.MarkovSequence.MarkovSequence</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ac8dc89b42d8e51906ce15124da699409\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5e853dc424f2142bc53df219e33be29f\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a778b3b6f649c7057ac1f3f7dc3e43aff\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>__str__()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a4aec883869195967a9209655905ace52\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#af69183cc68e6b8aae85cce91341dbf44\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afdbe49d3727f7fd3022899cb3130c6db\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0deb7b84090e2667840606ab0eeae509\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_a\"></a>- a -</h3><ul>\n<li>activities\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae9ea417aa751edb7e58bc1c23602b9b8\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>activities_sets\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a65a5f24cb5e6cf06960f8c93c6c8aa84\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>add()\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a7b948e9bbdcd1ab31bf6ee1a425195f7\">pathpy.Log.Log</a>\n</li>\n<li>addEdge()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aac7da90422a66a8123776b3d698b4615\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>addPath()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5af2d99bd84797a960e43bc78c57db5a\">pathpy.Paths.Paths</a>\n</li>\n<li>addPathTuple()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ab1c50bfee7d28f3a180fc036000bc146\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_b\"></a>- b -</h3><ul>\n<li>BetweennessCentrality()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#afcfdcfacef4f3beb7594a58600e833e4\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#add4dc55240d6ca92645f3709152d0545\">pathpy.Paths.Paths</a>\n</li>\n<li>BetweennessPreference()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afc66998bbb85b5fd3e7b8cd049a2bfa1\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_c\"></a>- c -</h3><ul>\n<li>ClosenessCentrality()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a2f94ebbc8141be6f5194ec922e3b01a0\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a45a27a4b5d8fe5aa368da221ab68502a\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_d\"></a>- d -</h3><ul>\n<li>DEBUG\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#ad6d71133083c6972a8400a1e4b355381\">pathpy.Log.Severity</a>\n</li>\n<li>degrees()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#acd9ab003f80216ed6beff9c513a7e876\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>dof_ngrams\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a25d69b9cc9b7b328fbd201244e68ca95\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>dof_paths\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a8a271893d9fb656f805e36335afca257\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_e\"></a>- e -</h3><ul>\n<li>ecount()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ae425665357e88b0adf493854143a3f72\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a040f4101e98fb4507709976c2ef7452b\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>edges\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6eec968e72178ab2930f83928b3ca842\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>ERROR\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#aa2e434a659bbb9f2e2c25e47fab1dc37\">pathpy.Log.Severity</a>\n</li>\n<li>estimateOrder()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a0e794225267c8195f091f5ed452d34e6\">pathpy.MarkovSequence.MarkovSequence</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a866b2b0b96f4bafa594b2a8d5a64efbf\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>EvCent()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a57e8494220dd4a5b2a9a45de17f9d26a\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>expandSubPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a6e1941415a937fa9aa86b6b442f858e1\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_f\"></a>- f -</h3><ul>\n<li>factorial()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a7882e4dcbe9ec932e863e20d6b49a4ed\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>filterEdges()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa1b765e6f119b214d78ba16de3b36cf4\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>filterPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a1c27b7c76d16437518f28734de2b86a5\">pathpy.Paths.Paths</a>\n</li>\n<li>fitMarkovModel()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a422f6b70f888eaab6a6f942158355072\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>fromTemporalNetwork()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a02f8ee9b6b4fa8c1e8012c9d21bfb76d\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_g\"></a>- g -</h3><ul>\n<li>getAdjacencyMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a8e10f45369dff5f7ccff3bcf7e6c5b33\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getAIC()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#aba5377e966e3bfb9c4c736e782863484\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>getAlgebraicConnectivity()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ae48d8ad635f7cf263897016d876c6fa2\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getBIC()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a975922931ec471f436c4d340830a7ca3\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>getContainedPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#af830708eaa318dd450193b8e6d7fb37a\">pathpy.Paths.Paths</a>\n</li>\n<li>getDegreesOfFreedom()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a219e12dca2b474515d74c65c4ad15c69\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>getDistanceMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20092d5a4a182df408af6063a0887630\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5b6ad214815f9fbd687c457d53367f19\">pathpy.Paths.Paths</a>\n</li>\n<li>getDistanceMatrixFirstOrder()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abed4839be0864210c0b5aff9376fe307\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getDoF()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a3a03ec6087add6dd0b54f7264b2e21c5\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getEigenValueGap()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a556545310735cba27128afd37c59ed35\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getEntropyGrowthRateRatio()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a9bf65d20a97eb629a4d618a4e19160a7\">pathpy.Paths.Paths</a>\n</li>\n<li>getFiedlerVectorDense()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#adea7800343373793dbd9688c77fb6191\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getFiedlerVectorSparse()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#aa8f3ed627c16c15c877fc0316c88bdb3\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getInterEventTimes()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a3e43ba8b22586d6f55b2ab28a65bf35b\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getInterPathTimes()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a7a385a1b17104011975aa9c0c3f2912a\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getLaplacianMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a69ea9c565b0d8bf7f1a2d0cb409f0e15\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getLayerLikelihood()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ab56ae5b47770b09178ae5aa49f695d17\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>getLeadingEigenvector()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a1b757112293f9093efc437ffb113df83\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getLikelihood()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ab820bbcfb5569ef05e361aa478c521f6\">pathpy.MarkovSequence.MarkovSequence</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#acceb5eabd7d1cb8a8856a485a29fc5f8\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>getNodeNameMap()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abd69fc6003eb13d11390466182a63357\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getNodes()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a287a429de9a958ccc48658b9fb9f7665\">pathpy.Paths.Paths</a>\n</li>\n<li>getObservationLength()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0ac006ac7818ef042fde95ea90deee80\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getSequence()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ab457e7e3f439e193410be707a2cd39bd\">pathpy.Paths.Paths</a>\n</li>\n<li>getShortestPaths()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a672cdad613e84eb0f528bbc02e7c6163\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a99a295d0674ca4eb5ba6906b48893b26\">pathpy.Paths.Paths</a>\n</li>\n<li>getSlowDownFactor()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a60aa117d37a599f912122263bf9e3eea\">pathpy.Paths.Paths</a>\n</li>\n<li>GetTemporalBetweenness()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a948503bbe0104c9678778cdecfe8ceb7\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>GetTemporalBetweennessInstantaneous()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa6a91612301e802d287f1fb58fe31961\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>GetTemporalCloseness()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a9dd08ea1ab218cb69b2fdd56cbcab61f\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>GetTemporalClosenessInstantaneous()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ac73ea9fa5c13c1df5ac101d45473243d\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getTransitionMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20c4a62ca4706bdab81534332e3843fe\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getUniquePaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afc152a2783167b289326e528f0077951\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_h\"></a>- h -</h3><ul>\n<li>HigherOrderNodeToPath()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a1963d0e4370e3818de3cf6886bba8594\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>HigherOrderPathToFirstOrder()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab71b78f1c9ffe7a06364841572f1fee2\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_i\"></a>- i -</h3><ul>\n<li>INFO\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a12fb98e4cc0d9e68114e29e8f6758c22\">pathpy.Log.Severity</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_l\"></a>- l -</h3><ul>\n<li>layers\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#abc678904e6dd23fc36bace35f8c8b651\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>likeliHoodRatioTest()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a0518e905c00b8c3a2df5cce509084fb8\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_m\"></a>- m -</h3><ul>\n<li>maxOrder\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ad77293d316dbc4264e07d33f15c43f55\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>min_severity\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a327ac21443db1980997ddb0c8ef65313\">pathpy.Log.Log</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_n\"></a>- n -</h3><ul>\n<li>nodes\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6626777ff215fde5f7d92368a407c683\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a86e49baf6e63c58a3f993e2768097001\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_o\"></a>- o -</h3><ul>\n<li>ObservationCount()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a4b4e13eb898dd55b9c1a381eaf22aea9\">pathpy.Paths.Paths</a>\n</li>\n<li>order\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a6dea6fe6e34178adb395ad8e79403d5c\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>ordered_times\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a051e7d352da3d194387762134bd70992\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>output_stream\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a8f7664b2a5379f9e94402117e22ed058\">pathpy.Log.Log</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_p\"></a>- p -</h3><ul>\n<li>P\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ac0d2ff028f2c2c88349555527e44a898\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>PageRank()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abc76633f53a0747353e7ab0e15744d94\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>paths\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ac4c8ee9f7775478793d88680d6f99fc8\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#adf751f249355e9a26e8062050567cf54\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#aacbff90d31fabf41c2413246aafc8275\">pathpy.Paths.Paths</a>\n</li>\n<li>pathToHigherOrderNodes()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a117fe621fb02d356f6591620f9340eaa\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>projectPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a160fb269d25c24adc4fbfdc3df71075c\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_r\"></a>- r -</h3><ul>\n<li>readEdges()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#adba5db911e0be900c03908f8b2ca511e\">pathpy.Paths.Paths</a>\n</li>\n<li>readFile()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a349974504cf0ef9fd1ff97a0249e649e\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a54bf4b554d7ca2a45a49cc76a0080cea\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>reduceToGCC()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a4b71eed8268df33814725ae7832729e6\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_s\"></a>- s -</h3><ul>\n<li>separator\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#af3e51491a2417e471eeb1404b44df204\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a0642d710d46cf929c00f22ed53510d92\">pathpy.Paths.Paths</a>\n</li>\n<li>sequence\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ac0cbbe436a3938f2ff94d313c72c4e67\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>setMinSeverity()\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a9e0171144845116732949b87d348d27d\">pathpy.Log.Log</a>\n</li>\n<li>setOutputStream()\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a0e7ec3decada72adee6edcda4951e720\">pathpy.Log.Log</a>\n</li>\n<li>ShuffleEdges()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a50ea38c4325e1035966c4d2173599a1d\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>sources\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0716456e27f19af5522af30e071df0ff\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>states\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a77ca53bfcfb5458a8834b8b6392422ce\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>successors\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a522350b2e4a401732b64bb0acf1634ea\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>summary()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab35dd7f65e3bfeb280fcd38c1e7448f7\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a58f11a90bea210c70f12eeac3af53d65\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ad519d020de268ea49898e7520be4ffcf\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a39d2b3a872f2a19554ee505393a6f708\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_t\"></a>- t -</h3><ul>\n<li>T\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a93118aa6719067efdbe8b38ef85a578a\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>targets\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae82b4df377620a626771cdcdcbb43b42\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>tedges\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a2ed001aa13e863e9ff6ab9ffc9ab998d\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>testNetworkAssumption()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#aaa9b2f2852c4ae513e5d42a96008c030\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>time\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a1f61dbedd4a4edde5176fec56fc1f0b3\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>TIMING\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#a0c184f8e48c1f5cb7f795d00c9205ed5\">pathpy.Log.Severity</a>\n</li>\n<li>totalEdgeWeight()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a91626f933af603a73f8bb39249ab6c51\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_v\"></a>- v -</h3><ul>\n<li>vcount()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a89ba7bbeb54449a87ba78eecc591fca4\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a727ebc2cc2eb2ad8c1ee23cdd5a25b6e\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>VisitationProbabilities()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5c964af581ac3c48fbd2a3955f711bb2\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_w\"></a>- w -</h3><ul>\n<li>WARNING\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html#ae4a13b6d9d1d2485bceb5843df6e3f40\">pathpy.Log.Severity</a>\n</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:10 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
  },
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    "path": "docs/functions_func.html",
    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Class Members - Functions</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li class=\"current\"><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n  <div id=\"navrow3\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"functions.html\"><span>All</span></a></li>\n      <li class=\"current\"><a href=\"functions_func.html\"><span>Functions</span></a></li>\n      <li><a href=\"functions_vars.html\"><span>Variables</span></a></li>\n    </ul>\n  </div>\n  <div id=\"navrow4\" class=\"tabs3\">\n    <ul class=\"tablist\">\n      <li><a href=\"#index__\"><span>_</span></a></li>\n      <li><a href=\"#index_a\"><span>a</span></a></li>\n      <li><a href=\"#index_b\"><span>b</span></a></li>\n      <li><a href=\"#index_c\"><span>c</span></a></li>\n      <li><a href=\"#index_d\"><span>d</span></a></li>\n      <li><a href=\"#index_e\"><span>e</span></a></li>\n      <li><a href=\"#index_f\"><span>f</span></a></li>\n      <li><a href=\"#index_g\"><span>g</span></a></li>\n      <li><a href=\"#index_h\"><span>h</span></a></li>\n      <li><a href=\"#index_l\"><span>l</span></a></li>\n      <li><a href=\"#index_o\"><span>o</span></a></li>\n      <li><a href=\"#index_p\"><span>p</span></a></li>\n      <li><a href=\"#index_r\"><span>r</span></a></li>\n      <li><a href=\"#index_s\"><span>s</span></a></li>\n      <li><a href=\"#index_t\"><span>t</span></a></li>\n      <li class=\"current\"><a href=\"#index_v\"><span>v</span></a></li>\n    </ul>\n  </div>\n</div><!-- top -->\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div class=\"contents\">\n&#160;\n\n<h3><a class=\"anchor\" id=\"index__\"></a>- _ -</h3><ul>\n<li>__init__()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a63d36720423ee8d6d88c5a06f4655c84\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a670778608688e4926328cf2d851b7d6d\">pathpy.MarkovSequence.MarkovSequence</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ac8dc89b42d8e51906ce15124da699409\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5e853dc424f2142bc53df219e33be29f\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a778b3b6f649c7057ac1f3f7dc3e43aff\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>__str__()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a4aec883869195967a9209655905ace52\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#af69183cc68e6b8aae85cce91341dbf44\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afdbe49d3727f7fd3022899cb3130c6db\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0deb7b84090e2667840606ab0eeae509\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_a\"></a>- a -</h3><ul>\n<li>add()\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a7b948e9bbdcd1ab31bf6ee1a425195f7\">pathpy.Log.Log</a>\n</li>\n<li>addEdge()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aac7da90422a66a8123776b3d698b4615\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>addPath()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5af2d99bd84797a960e43bc78c57db5a\">pathpy.Paths.Paths</a>\n</li>\n<li>addPathTuple()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ab1c50bfee7d28f3a180fc036000bc146\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_b\"></a>- b -</h3><ul>\n<li>BetweennessCentrality()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#afcfdcfacef4f3beb7594a58600e833e4\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#add4dc55240d6ca92645f3709152d0545\">pathpy.Paths.Paths</a>\n</li>\n<li>BetweennessPreference()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afc66998bbb85b5fd3e7b8cd049a2bfa1\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_c\"></a>- c -</h3><ul>\n<li>ClosenessCentrality()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a2f94ebbc8141be6f5194ec922e3b01a0\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a45a27a4b5d8fe5aa368da221ab68502a\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_d\"></a>- d -</h3><ul>\n<li>degrees()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#acd9ab003f80216ed6beff9c513a7e876\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_e\"></a>- e -</h3><ul>\n<li>ecount()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ae425665357e88b0adf493854143a3f72\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a040f4101e98fb4507709976c2ef7452b\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>estimateOrder()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a0e794225267c8195f091f5ed452d34e6\">pathpy.MarkovSequence.MarkovSequence</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a866b2b0b96f4bafa594b2a8d5a64efbf\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>EvCent()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a57e8494220dd4a5b2a9a45de17f9d26a\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>expandSubPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a6e1941415a937fa9aa86b6b442f858e1\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_f\"></a>- f -</h3><ul>\n<li>factorial()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a7882e4dcbe9ec932e863e20d6b49a4ed\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>filterEdges()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa1b765e6f119b214d78ba16de3b36cf4\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>filterPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a1c27b7c76d16437518f28734de2b86a5\">pathpy.Paths.Paths</a>\n</li>\n<li>fitMarkovModel()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a422f6b70f888eaab6a6f942158355072\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>fromTemporalNetwork()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a02f8ee9b6b4fa8c1e8012c9d21bfb76d\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_g\"></a>- g -</h3><ul>\n<li>getAdjacencyMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a8e10f45369dff5f7ccff3bcf7e6c5b33\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getAIC()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#aba5377e966e3bfb9c4c736e782863484\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>getAlgebraicConnectivity()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ae48d8ad635f7cf263897016d876c6fa2\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getBIC()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#a975922931ec471f436c4d340830a7ca3\">pathpy.MarkovSequence.MarkovSequence</a>\n</li>\n<li>getContainedPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#af830708eaa318dd450193b8e6d7fb37a\">pathpy.Paths.Paths</a>\n</li>\n<li>getDegreesOfFreedom()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a219e12dca2b474515d74c65c4ad15c69\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>getDistanceMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20092d5a4a182df408af6063a0887630\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5b6ad214815f9fbd687c457d53367f19\">pathpy.Paths.Paths</a>\n</li>\n<li>getDistanceMatrixFirstOrder()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abed4839be0864210c0b5aff9376fe307\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getDoF()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a3a03ec6087add6dd0b54f7264b2e21c5\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getEigenValueGap()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a556545310735cba27128afd37c59ed35\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getEntropyGrowthRateRatio()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a9bf65d20a97eb629a4d618a4e19160a7\">pathpy.Paths.Paths</a>\n</li>\n<li>getFiedlerVectorDense()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#adea7800343373793dbd9688c77fb6191\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getFiedlerVectorSparse()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#aa8f3ed627c16c15c877fc0316c88bdb3\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getInterEventTimes()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a3e43ba8b22586d6f55b2ab28a65bf35b\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getInterPathTimes()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a7a385a1b17104011975aa9c0c3f2912a\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getLaplacianMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a69ea9c565b0d8bf7f1a2d0cb409f0e15\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getLayerLikelihood()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#ab56ae5b47770b09178ae5aa49f695d17\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>getLeadingEigenvector()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a1b757112293f9093efc437ffb113df83\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getLikelihood()\n: <a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html#ab820bbcfb5569ef05e361aa478c521f6\">pathpy.MarkovSequence.MarkovSequence</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#acceb5eabd7d1cb8a8856a485a29fc5f8\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>getNodeNameMap()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abd69fc6003eb13d11390466182a63357\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getNodes()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a287a429de9a958ccc48658b9fb9f7665\">pathpy.Paths.Paths</a>\n</li>\n<li>getObservationLength()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a0ac006ac7818ef042fde95ea90deee80\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getSequence()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ab457e7e3f439e193410be707a2cd39bd\">pathpy.Paths.Paths</a>\n</li>\n<li>getShortestPaths()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a672cdad613e84eb0f528bbc02e7c6163\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a99a295d0674ca4eb5ba6906b48893b26\">pathpy.Paths.Paths</a>\n</li>\n<li>getSlowDownFactor()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a60aa117d37a599f912122263bf9e3eea\">pathpy.Paths.Paths</a>\n</li>\n<li>GetTemporalBetweenness()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a948503bbe0104c9678778cdecfe8ceb7\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>GetTemporalBetweennessInstantaneous()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#aa6a91612301e802d287f1fb58fe31961\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>GetTemporalCloseness()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a9dd08ea1ab218cb69b2fdd56cbcab61f\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>GetTemporalClosenessInstantaneous()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ac73ea9fa5c13c1df5ac101d45473243d\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>getTransitionMatrix()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a20c4a62ca4706bdab81534332e3843fe\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>getUniquePaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#afc152a2783167b289326e528f0077951\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_h\"></a>- h -</h3><ul>\n<li>HigherOrderNodeToPath()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a1963d0e4370e3818de3cf6886bba8594\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>HigherOrderPathToFirstOrder()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab71b78f1c9ffe7a06364841572f1fee2\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_l\"></a>- l -</h3><ul>\n<li>likeliHoodRatioTest()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a0518e905c00b8c3a2df5cce509084fb8\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_o\"></a>- o -</h3><ul>\n<li>ObservationCount()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a4b4e13eb898dd55b9c1a381eaf22aea9\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_p\"></a>- p -</h3><ul>\n<li>PageRank()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#abc76633f53a0747353e7ab0e15744d94\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>pathToHigherOrderNodes()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a117fe621fb02d356f6591620f9340eaa\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n<li>projectPaths()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a160fb269d25c24adc4fbfdc3df71075c\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_r\"></a>- r -</h3><ul>\n<li>readEdges()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#adba5db911e0be900c03908f8b2ca511e\">pathpy.Paths.Paths</a>\n</li>\n<li>readFile()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a349974504cf0ef9fd1ff97a0249e649e\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a54bf4b554d7ca2a45a49cc76a0080cea\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>reduceToGCC()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a4b71eed8268df33814725ae7832729e6\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_s\"></a>- s -</h3><ul>\n<li>setMinSeverity()\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a9e0171144845116732949b87d348d27d\">pathpy.Log.Log</a>\n</li>\n<li>setOutputStream()\n: <a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html#a0e7ec3decada72adee6edcda4951e720\">pathpy.Log.Log</a>\n</li>\n<li>ShuffleEdges()\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a50ea38c4325e1035966c4d2173599a1d\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>summary()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#ab35dd7f65e3bfeb280fcd38c1e7448f7\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#a58f11a90bea210c70f12eeac3af53d65\">pathpy.MultiOrderModel.MultiOrderModel</a>\n, <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#ad519d020de268ea49898e7520be4ffcf\">pathpy.Paths.Paths</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a39d2b3a872f2a19554ee505393a6f708\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_t\"></a>- t -</h3><ul>\n<li>testNetworkAssumption()\n: <a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html#aaa9b2f2852c4ae513e5d42a96008c030\">pathpy.MultiOrderModel.MultiOrderModel</a>\n</li>\n<li>totalEdgeWeight()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a91626f933af603a73f8bb39249ab6c51\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" id=\"index_v\"></a>- v -</h3><ul>\n<li>vcount()\n: <a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html#a89ba7bbeb54449a87ba78eecc591fca4\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a>\n, <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a727ebc2cc2eb2ad8c1ee23cdd5a25b6e\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>VisitationProbabilities()\n: <a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html#a5c964af581ac3c48fbd2a3955f711bb2\">pathpy.Paths.Paths</a>\n</li>\n</ul>\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:10 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
  },
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    "path": "docs/functions_vars.html",
    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Class Members - Variables</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n 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href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li class=\"current\"><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n  <div id=\"navrow3\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"functions.html\"><span>All</span></a></li>\n      <li><a href=\"functions_func.html\"><span>Functions</span></a></li>\n      <li class=\"current\"><a href=\"functions_vars.html\"><span>Variables</span></a></li>\n    </ul>\n  </div>\n  <div id=\"navrow4\" class=\"tabs3\">\n    <ul class=\"tablist\">\n      <li><a href=\"#index_a\"><span>a</span></a></li>\n      <li><a href=\"#index_d\"><span>d</span></a></li>\n      <li><a href=\"#index_e\"><span>e</span></a></li>\n      <li><a href=\"#index_i\"><span>i</span></a></li>\n      <li><a href=\"#index_l\"><span>l</span></a></li>\n      <li><a href=\"#index_m\"><span>m</span></a></li>\n      <li><a href=\"#index_n\"><span>n</span></a></li>\n      <li><a href=\"#index_o\"><span>o</span></a></li>\n      <li><a href=\"#index_p\"><span>p</span></a></li>\n      <li><a href=\"#index_s\"><span>s</span></a></li>\n      <li><a href=\"#index_t\"><span>t</span></a></li>\n      <li class=\"current\"><a href=\"#index_w\"><span>w</span></a></li>\n    </ul>\n  </div>\n</div><!-- top -->\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div class=\"contents\">\n&#160;\n\n<h3><a class=\"anchor\" id=\"index_a\"></a>- a -</h3><ul>\n<li>activities\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#ae9ea417aa751edb7e58bc1c23602b9b8\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n<li>activities_sets\n: <a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html#a65a5f24cb5e6cf06960f8c93c6c8aa84\">pathpy.TemporalNetwork.TemporalNetwork</a>\n</li>\n</ul>\n\n\n<h3><a class=\"anchor\" 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    "content": "<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Transitional//EN\" \"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd\">\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\n<head>\n<meta http-equiv=\"Content-Type\" content=\"text/xhtml;charset=UTF-8\"/>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=9\"/>\n<meta name=\"generator\" content=\"Doxygen 1.8.6\"/>\n<title>pathpy: Class Hierarchy</title>\n<link href=\"tabs.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"jquery.js\"></script>\n<script type=\"text/javascript\" src=\"dynsections.js\"></script>\n<link href=\"search/search.css\" rel=\"stylesheet\" type=\"text/css\"/>\n<script type=\"text/javascript\" src=\"search/search.js\"></script>\n<script type=\"text/javascript\">\n  $(document).ready(function() { searchBox.OnSelectItem(0); });\n</script>\n<link href=\"doxygen.css\" rel=\"stylesheet\" type=\"text/css\" />\n</head>\n<body>\n<div id=\"top\"><!-- do not remove this div, it is closed by doxygen! -->\n<div id=\"titlearea\">\n<table cellspacing=\"0\" cellpadding=\"0\">\n <tbody>\n <tr style=\"height: 56px;\">\n  <td style=\"padding-left: 0.5em;\">\n   <div id=\"projectname\">pathpy\n   &#160;<span id=\"projectnumber\">1.0</span>\n   </div>\n   <div id=\"projectbrief\">pathpy is an OpenSource python package for the modeling and analysis of pathways and temporal networks using higher-order and multi-order graphical models</div>\n  </td>\n </tr>\n </tbody>\n</table>\n</div>\n<!-- end header part -->\n<!-- Generated by Doxygen 1.8.6 -->\n<script type=\"text/javascript\">\nvar searchBox = new SearchBox(\"searchBox\", \"search\",false,'Search');\n</script>\n  <div id=\"navrow1\" class=\"tabs\">\n    <ul class=\"tablist\">\n      <li><a href=\"index.html\"><span>Main&#160;Page</span></a></li>\n      <li class=\"current\"><a href=\"annotated.html\"><span>Classes</span></a></li>\n      <li>\n        <div id=\"MSearchBox\" class=\"MSearchBoxInactive\">\n        <span class=\"left\">\n          <img id=\"MSearchSelect\" src=\"search/mag_sel.png\"\n               onmouseover=\"return searchBox.OnSearchSelectShow()\"\n               onmouseout=\"return searchBox.OnSearchSelectHide()\"\n               alt=\"\"/>\n          <input type=\"text\" id=\"MSearchField\" value=\"Search\" accesskey=\"S\"\n               onfocus=\"searchBox.OnSearchFieldFocus(true)\" \n               onblur=\"searchBox.OnSearchFieldFocus(false)\" \n               onkeyup=\"searchBox.OnSearchFieldChange(event)\"/>\n          </span><span class=\"right\">\n            <a id=\"MSearchClose\" href=\"javascript:searchBox.CloseResultsWindow()\"><img id=\"MSearchCloseImg\" border=\"0\" src=\"search/close.png\" alt=\"\"/></a>\n          </span>\n        </div>\n      </li>\n    </ul>\n  </div>\n  <div id=\"navrow2\" class=\"tabs2\">\n    <ul class=\"tablist\">\n      <li><a href=\"annotated.html\"><span>Class&#160;List</span></a></li>\n      <li><a href=\"classes.html\"><span>Class&#160;Index</span></a></li>\n      <li class=\"current\"><a href=\"hierarchy.html\"><span>Class&#160;Hierarchy</span></a></li>\n      <li><a href=\"functions.html\"><span>Class&#160;Members</span></a></li>\n    </ul>\n  </div>\n</div><!-- top -->\n<!-- window showing the filter options -->\n<div id=\"MSearchSelectWindow\"\n     onmouseover=\"return searchBox.OnSearchSelectShow()\"\n     onmouseout=\"return searchBox.OnSearchSelectHide()\"\n     onkeydown=\"return searchBox.OnSearchSelectKey(event)\">\n<a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(0)\"><span class=\"SelectionMark\">&#160;</span>All</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(1)\"><span class=\"SelectionMark\">&#160;</span>Classes</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(2)\"><span class=\"SelectionMark\">&#160;</span>Functions</a><a class=\"SelectItem\" href=\"javascript:void(0)\" onclick=\"searchBox.OnSelectItem(3)\"><span class=\"SelectionMark\">&#160;</span>Variables</a></div>\n\n<!-- iframe showing the search results (closed by default) -->\n<div id=\"MSearchResultsWindow\">\n<iframe src=\"javascript:void(0)\" frameborder=\"0\" \n        name=\"MSearchResults\" id=\"MSearchResults\">\n</iframe>\n</div>\n\n<div class=\"header\">\n  <div class=\"headertitle\">\n<div class=\"title\">Class Hierarchy</div>  </div>\n</div><!--header-->\n<div class=\"contents\">\n<div class=\"textblock\">This inheritance list is sorted roughly, but not completely, alphabetically:</div><div class=\"directory\">\n<div class=\"levels\">[detail level <span onclick=\"javascript:toggleLevel(1);\">1</span><span onclick=\"javascript:toggleLevel(2);\">2</span>]</div><table class=\"directory\">\n<tr id=\"row_0_\" class=\"even\"><td class=\"entry\"><img id=\"arr_0_\" src=\"ftv2mnode.png\" alt=\"o\" width=\"16\" height=\"22\" onclick=\"toggleFolder('0_')\"/><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><b>Exception</b></td><td class=\"desc\"></td></tr>\n<tr id=\"row_0_0_\"><td class=\"entry\"><img src=\"ftv2vertline.png\" alt=\"|\" width=\"16\" height=\"22\" /><img src=\"ftv2lastnode.png\" alt=\"\\\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1EmptySCCError.html\" target=\"_self\">pathpy.HigherOrderNetwork.EmptySCCError</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_1_\" class=\"even\"><td class=\"entry\"><img src=\"ftv2node.png\" alt=\"o\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1HigherOrderNetwork_1_1HigherOrderNetwork.html\" target=\"_self\">pathpy.HigherOrderNetwork.HigherOrderNetwork</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_2_\"><td class=\"entry\"><img id=\"arr_2_\" src=\"ftv2mnode.png\" alt=\"o\" width=\"16\" height=\"22\" onclick=\"toggleFolder('2_')\"/><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><b>IntEnum</b></td><td class=\"desc\"></td></tr>\n<tr id=\"row_2_0_\" class=\"even\"><td class=\"entry\"><img src=\"ftv2vertline.png\" alt=\"|\" width=\"16\" height=\"22\" /><img src=\"ftv2lastnode.png\" alt=\"\\\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1Log_1_1Severity.html\" target=\"_self\">pathpy.Log.Severity</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_3_\"><td class=\"entry\"><img src=\"ftv2node.png\" alt=\"o\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1Log_1_1Log.html\" target=\"_self\">pathpy.Log.Log</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_4_\" class=\"even\"><td class=\"entry\"><img src=\"ftv2node.png\" alt=\"o\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1MarkovSequence_1_1MarkovSequence.html\" target=\"_self\">pathpy.MarkovSequence.MarkovSequence</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_5_\"><td class=\"entry\"><img src=\"ftv2node.png\" alt=\"o\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1MultiOrderModel_1_1MultiOrderModel.html\" target=\"_self\">pathpy.MultiOrderModel.MultiOrderModel</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_6_\" class=\"even\"><td class=\"entry\"><img src=\"ftv2node.png\" alt=\"o\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1Paths_1_1Paths.html\" target=\"_self\">pathpy.Paths.Paths</a></td><td class=\"desc\"></td></tr>\n<tr id=\"row_7_\"><td class=\"entry\"><img src=\"ftv2lastnode.png\" alt=\"\\\" width=\"16\" height=\"22\" /><img src=\"ftv2cl.png\" alt=\"C\" width=\"24\" height=\"22\" /><a class=\"el\" href=\"classpathpy_1_1TemporalNetwork_1_1TemporalNetwork.html\" target=\"_self\">pathpy.TemporalNetwork.TemporalNetwork</a></td><td class=\"desc\"></td></tr>\n</table>\n</div><!-- directory -->\n</div><!-- contents -->\n<!-- start footer part -->\n<hr class=\"footer\"/><address class=\"footer\"><small>\nGenerated on Wed Mar 1 2017 10:39:09 for pathpy by &#160;<a href=\"http://www.doxygen.org/index.html\">\n<img class=\"footer\" src=\"doxygen.png\" alt=\"doxygen\"/>\n</a> 1.8.6\n</small></address>\n</body>\n</html>\n"
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.fa-rotate-90{filter:none}.fa-stack{position:relative;display:inline-block;width:2em;height:2em;line-height:2em;vertical-align:middle}.fa-stack-1x,.fa-stack-2x{position:absolute;left:0;width:100%;text-align:center}.fa-stack-1x{line-height:inherit}.fa-stack-2x{font-size:2em}.fa-inverse{color:#fff}.fa-glass:before{content:\"\\f000\"}.fa-music:before{content:\"\\f001\"}.fa-search:before{content:\"\\f002\"}.fa-envelope-o:before{content:\"\\f003\"}.fa-heart:before{content:\"\\f004\"}.fa-star:before{content:\"\\f005\"}.fa-star-o:before{content:\"\\f006\"}.fa-user:before{content:\"\\f007\"}.fa-film:before{content:\"\\f008\"}.fa-th-large:before{content:\"\\f009\"}.fa-th:before{content:\"\\f00a\"}.fa-th-list:before{content:\"\\f00b\"}.fa-check:before{content:\"\\f00c\"}.fa-close:before,.fa-remove:before,.fa-times:before{content:\"\\f00d\"}.fa-search-plus:before{content:\"\\f00e\"}.fa-search-minus:before{content:\"\\f010\"}.fa-power-off:before{content:\"\\f011\"}.fa-signal:before{content:\"\\f012\"}.fa-cog:before,.fa-gear:before{content:\"\\f013\"}.fa-trash-o:before{content:\"\\f014\"}.fa-home:before{content:\"\\f015\"}.fa-file-o:before{content:\"\\f016\"}.fa-clock-o:before{content:\"\\f017\"}.fa-road:before{content:\"\\f018\"}.fa-download:before{content:\"\\f019\"}.fa-arrow-circle-o-down:before{content:\"\\f01a\"}.fa-arrow-circle-o-up:before{content:\"\\f01b\"}.fa-inbox:before{content:\"\\f01c\"}.fa-play-circle-o:before{content:\"\\f01d\"}.fa-repeat:before,.fa-rotate-right:before{content:\"\\f01e\"}.fa-refresh:before{content:\"\\f021\"}.fa-list-alt:before{content:\"\\f022\"}.fa-lock:before{content:\"\\f023\"}.fa-flag:before{content:\"\\f024\"}.fa-headphones:before{content:\"\\f025\"}.fa-volume-off:before{content:\"\\f026\"}.fa-volume-down:before{content:\"\\f027\"}.fa-volume-up:before{content:\"\\f028\"}.fa-qrcode:before{content:\"\\f029\"}.fa-barcode:before{content:\"\\f02a\"}.fa-tag:before{content:\"\\f02b\"}.fa-tags:before{content:\"\\f02c\"}.fa-book:before{content:\"\\f02d\"}.fa-bookmark:before{content:\"\\f02e\"}.fa-print:before{content:\"\\f02f\"}.fa-camera:before{content:\"\\f030\"}.fa-font:before{content:\"\\f031\"}.fa-bold:before{content:\"\\f032\"}.fa-italic:before{content:\"\\f033\"}.fa-text-height:before{content:\"\\f034\"}.fa-text-width:before{content:\"\\f035\"}.fa-align-left:before{content:\"\\f036\"}.fa-align-center:before{content:\"\\f037\"}.fa-align-right:before{content:\"\\f038\"}.fa-align-justify:before{content:\"\\f039\"}.fa-list:before{content:\"\\f03a\"}.fa-dedent:before,.fa-outdent:before{content:\"\\f03b\"}.fa-indent:before{content:\"\\f03c\"}.fa-video-camera:before{content:\"\\f03d\"}.fa-image:before,.fa-photo:before,.fa-picture-o:before{content:\"\\f03e\"}.fa-pencil:before{content:\"\\f040\"}.fa-map-marker:before{content:\"\\f041\"}.fa-adjust:before{content:\"\\f042\"}.fa-tint:before{content:\"\\f043\"}.fa-edit:before,.fa-pencil-square-o:before{content:\"\\f044\"}.fa-share-square-o:before{content:\"\\f045\"}.fa-check-square-o:before{content:\"\\f046\"}.fa-arrows:before{content:\"\\f047\"}.fa-step-backward:before{content:\"\\f048\"}.fa-fast-backward:before{content:\"\\f049\"}.fa-backward:before{content:\"\\f04a\"}.fa-play:before{content:\"\\f04b\"}.fa-pause:before{content:\"\\f04c\"}.fa-stop:before{content:\"\\f04d\"}.fa-forward:before{content:\"\\f04e\"}.fa-fast-forward:before{content:\"\\f050\"}.fa-step-forward:before{content:\"\\f051\"}.fa-eject:before{content:\"\\f052\"}.fa-chevron-left:before{content:\"\\f053\"}.fa-chevron-right:before{content:\"\\f054\"}.fa-plus-circle:before{content:\"\\f055\"}.fa-minus-circle:before{content:\"\\f056\"}.fa-times-circle:before{content:\"\\f057\"}.fa-check-circle:before{content:\"\\f058\"}.fa-question-circle:before{content:\"\\f059\"}.fa-info-circle:before{content:\"\\f05a\"}.fa-crosshairs:before{content:\"\\f05b\"}.fa-times-circle-o:before{content:\"\\f05c\"}.fa-check-circle-o:before{content:\"\\f05d\"}.fa-ban:before{content:\"\\f05e\"}.fa-arrow-left:before{content:\"\\f060\"}.fa-arrow-right:before{content:\"\\f061\"}.fa-arrow-up:before{content:\"\\f062\"}.fa-arrow-down:before{content:\"\\f063\"}.fa-mail-forward:before,.fa-share:before{content:\"\\f064\"}.fa-expand:before{content:\"\\f065\"}.fa-compress:before{content:\"\\f066\"}.fa-plus:before{content:\"\\f067\"}.fa-minus:before{content:\"\\f068\"}.fa-asterisk:before{content:\"\\f069\"}.fa-exclamation-circle:before{content:\"\\f06a\"}.fa-gift:before{content:\"\\f06b\"}.fa-leaf:before{content:\"\\f06c\"}.fa-fire:before{content:\"\\f06d\"}.fa-eye:before{content:\"\\f06e\"}.fa-eye-slash:before{content:\"\\f070\"}.fa-exclamation-triangle:before,.fa-warning:before{content:\"\\f071\"}.fa-plane:before{content:\"\\f072\"}.fa-calendar:before{content:\"\\f073\"}.fa-random:before{content:\"\\f074\"}.fa-comment:before{content:\"\\f075\"}.fa-magnet:before{content:\"\\f076\"}.fa-chevron-up:before{content:\"\\f077\"}.fa-chevron-down:before{content:\"\\f078\"}.fa-retweet:before{content:\"\\f079\"}.fa-shopping-cart:b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print {\n  div.cell {\n    display: block;\n    page-break-inside: avoid;\n  } \n  div.output_wrapper { \n    display: block;\n    page-break-inside: avoid; \n  }\n  div.output { \n    display: block;\n    page-break-inside: avoid; \n  }\n}\n</style>\n\n<!-- Custom stylesheet, it must be in the same directory as the html file -->\n<link rel=\"stylesheet\" href=\"custom.css\">\n\n<!-- Loading mathjax macro -->\n<!-- Load mathjax -->\n    <script src=\"https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML\"></script>\n    <!-- MathJax configuration -->\n    <script type=\"text/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {\n            inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n            displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ],\n            processEscapes: true,\n            processEnvironments: true\n        },\n        // Center justify equations in code and markdown cells. Elsewhere\n        // we use CSS to left justify single line equations in code cells.\n        displayAlign: 'center',\n        \"HTML-CSS\": {\n            styles: {'.MathJax_Display': {\"margin\": 0}},\n            linebreaks: { automatic: true }\n        }\n    });\n    </script>\n    <!-- End of mathjax configuration --></head>\n<body>\n  <div tabindex=\"-1\" id=\"notebook\" class=\"border-box-sizing\">\n    <div class=\"container\" id=\"notebook-container\">\n\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h1 id=\"When-is-a-network-a-network?-Multi-Order-Model-Selection-in-Pathways-and-Temporal-Networks\">When is a network a network? Multi-Order Model Selection in Pathways and Temporal Networks<a class=\"anchor-link\" href=\"#When-is-a-network-a-network?-Multi-Order-Model-Selection-in-Pathways-and-Temporal-Networks\">&#182;</a></h1><h3 id=\"An-educational-tutorial-introducing-the-OpenSource-python-package-pathpy\">An educational tutorial introducing the OpenSource <code>python</code> package <a href=\"https://github.com/IngoScholtes/pathpy\"><code>pathpy</code></a><a class=\"anchor-link\" href=\"#An-educational-tutorial-introducing-the-OpenSource-python-package-pathpy\">&#182;</a></h3><p>Ingo Scholtes<br>\n<a href=\"http://www.sg.ethz.ch\">Chair of Systems Design</a><br>\nETH Zürich</p>\n<p><em>February 23 2017</em></p>\n<h2 id=\"Summary\">Summary<a class=\"anchor-link\" href=\"#Summary\">&#182;</a></h2><p>This educational tutorial introduces the <strong>analysis of sequential data using multi-order graphical models</strong>, based on the python package <a href=\"https://github.com/IngoScholtes/pathpy\"><code>pathpy</code></a>.</p>\n<p><a name=\"references\"></a></p>\n<p>The theoretical foundation of this package has been outlined in the recent paper:</p>\n<ul>\n<li>I Scholtes: <a href=\"https://arxiv.org/abs/1702.05499\">When is a network a network? Multi-Order Graphical Model Selection in Pathways and Temporal Networks</a>, In <a href=\"http://www.kdd.org/kdd2017/\">KDD'17 - Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining</a>, Halifax, Nova Scotia, Canada, August 13-17, 2017</li>\n</ul>\n<p>Moreover, it builds on <strong>higher-order network abstractions</strong> of time-stamped data, as well as <strong>high-order centrality measures</strong> developed in:</p>\n<ul>\n<li>I Scholtes, N Wider, A Garas: <a href=\"http://link.springer.com/article/10.1140%2Fepjb%2Fe2016-60663-0\">Higher-Order Aggregate Networks in the Analysis of Temporal Networks: Path structures and centralities</a>, The European Physical Journal B, 89:61, March 2016  </li>\n<li>I Scholtes, N Wider, R Pfitzner, A Garas, CJ Tessone, F Schweitzer: <a href=\"http://dx.doi.org/10.1140/epjb/e2016-60663-0\">Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks</a>, Nature Communications, 5, September 2014  </li>\n<li>R Pfitzner, I Scholtes, A Garas, CJ Tessone, F Schweitzer: <a href=\"http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.198701\">Betweenness preference: Quantifying correlations in the topological dynamics of temporal networks</a>, Phys Rev Lett, 110(19), 198701, May 2013  </li>\n</ul>\n<p>A key feature of <code>pathpy</code> is that it provides a unified approach to the analysis of pathways and temporal networks. Let us first define what we mean by pathway data. We consider sequential data of the form ...</p>\n<p>$(a \\rightarrow b)$<br>\n$(b \\rightarrow c)$<br>\n$(a \\rightarrow b \\rightarrow c)$<br>\n$(b \\rightarrow c \\rightarrow a)$<br>\n$(a \\rightarrow b \\rightarrow c \\rightarrow d)$<br>\n$(c)$</p>\n<p>... which capture multiple (typically short) paths with varying lengths, observed in a network topology. Such data are relevant in a number of data mining scenarios: Consider, for instance, click streams of multiple users in the Web. Each line above could be the navigation path of a user in a Web graph. Considering biological pathways, each line could be one activation sequence of genes observed in a cell. In social media, paths could be traces of information propagating through a social network. Finally, we will show that pathway data also naturally emerge in time-stamped interaction data, which makes <code>pathpy</code> <strong>particularly useful for those studying temporal networks</strong>.</p>\n<p>In this tutorial we show that such sequential data allow us to provide a principled answer to the crucial question: <strong>Is it justified to model a system as a network</strong>, i.e. can we apply graph-theoretic or network-analytic methods to a relational data set?</p>\n<p>Apart from answering this important question, <code>pathpy</code> allows to infer <strong>optimal higher-order graphical models</strong> which generalize the commonly used network abstraction and facilitate the analysis of sequential data.</p>\n<p>The outline of this tutorial is as follows:</p>\n<p><a name=\"outline\"></a></p>\n<ol>\n<li><a href=\"#setup\">Setting up pathpy</a></li>\n<li><a href=\"#paths\">Getting started: the Paths object</a></li>\n<li><a href=\"#temporal\">Analyzing temporal networks: the TemporalNetwork class</a></li>\n<li><a href=\"#data\">Importing data on pathways and temporal networks</a></li>\n<li><a href=\"#multiorder\">Multi-Order graphical models of pathways and temporal networks</a></li>\n<li><a href=\"#network\">When is a network a network?</a></li>\n<li><a href=\"#conclusion\">Conclusion</a></li>\n</ol>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"setup\"></a></p>\n<h2 id=\"1.-Setting-up-pathpy\">1. Setting up <code>pathpy</code><a class=\"anchor-link\" href=\"#1.-Setting-up-pathpy\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n<p>Before diving into the theoretical foundation of our framework, let us first install and setup <code>pathpy</code>. The package consist of pure <code>python</code> code which means that there are no platform-specific dependencies that complicate the setup. The only requirement is that you have a <code>python</code> interpreter (version 3 and above) as well as the packages <code>numpy</code> and <code>scipy</code>, which are used for mathematical calculations.</p>\n<p>The code of <code>pathpy</code> is available at <a href=\"https://github.com/IngoScholtes/pathpy\">gitHub</a>. Downloading and installing the latest version of <code>pathpy</code> is simple. Just fire up a console and type:</p>\n<p><code>&gt; pip install git+git://github.com/IngoScholtes/pathpy.git</code></p>\n<p>This will download and install the latest development version of <code>pathpy</code> and its dependencies. If you want to install a specific <a href=\"https://github.com/IngoScholtes/pathpy/releases\">release version</a>, you can type:</p>\n<p><code>&gt; pip install https://github.com/IngoScholtes/pathpy/archive/VERSIONTAG.zip</code></p>\n<p>where <code>VERSIONTAG</code> is the tag of the release. In this tutorial, we use the first beta release <code>v1.0-beta.1</code>, so we run</p>\n<p><code>&gt; pip install https://github.com/IngoScholtes/pathpy/archive/v1.0-beta.1.zip</code></p>\n<p>While <code>pathpy</code> does not depend on any specific graph library, for illustration purposes this tutorial will use network visualizations generated by <code>python-igraph</code>. We can set this up by running:</p>\n<p><code>&gt; pip install python-igraph</code></p>\n<p>However, <code>igraph</code> is not needed to use <code>pathpy</code> unless you wan to visualize higher-order graphical models. Now that everything is installed, we can import <code>pathpy</code>, <code>numpy</code> and <code>igraph</code> in our script. We will also use the <code>IPython.display</code> function to plot some figures later, so let us import that as well.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[1]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"kn\">import</span> <span class=\"nn\">igraph</span>\n<span class=\"kn\">import</span> <span class=\"nn\">numpy</span> <span class=\"k\">as</span> <span class=\"nn\">np</span>\n<span class=\"kn\">import</span> <span class=\"nn\">pathpy</span> <span class=\"k\">as</span> <span class=\"nn\">pp</span>\n\n<span class=\"kn\">from</span> <span class=\"nn\">IPython.display</span> <span class=\"k\">import</span> <span class=\"o\">*</span>\n<span class=\"kn\">from</span> <span class=\"nn\">IPython.display</span> <span class=\"k\">import</span> <span class=\"n\">HTML</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"paths\"></a></p>\n<h2 id=\"2.-Getting-started:-the-Paths-object\">2. Getting started: the <code>Paths</code> object<a class=\"anchor-link\" href=\"#2.-Getting-started:-the-Paths-object\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>One of the key objects in <code>pathpy</code> is the <code>Paths</code> class. It can be used to import, manipulate and analyze pathways like in the example above. As we will see later, we can also use it to generate pathways from temporal networks. For now, let us create an empty <code>Paths</code> instance as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[2]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"p\">()</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We can manually add paths to this object using the function <code>addPath</code>. Since all classes and functions in <code>pathpy</code> are documented using so-called <code>python docstrings</code> we can use <code>python</code>'s interactive help system to print the documentation of this function.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[3]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">help</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Help on method addPath in module pathpy.Paths:\n\naddPath(ngram, separator=&apos;,&apos;, expandSubPaths=True, pathFrequency=None) method of pathpy.Paths.Paths instance\n    Adds the path(s) of a single n-gram to the path statistics object.\n    \n    @param ngram: An ngram representing a path between nodes, separated by the separator character, e.g. \n        the 4-gram a;b;c;d represents a path of length three (with separator &apos;;&apos;)\n    \n    @param separator: The character used as separator for the ngrams (&apos;;&apos; by default)\n    \n    @param expandSubPaths: by default all subpaths of the given ngram are generated, i.e. \n        for the trigram a;b;c a path a-&gt;b-&gt;c of length two will be generated \n        as well as two subpaths a-&gt;b and b-&gt;c of length one\n    \n    @weight weight: the weight (i.e. frequency) of the ngram\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>The function takes a string argument, as well as a separator character. The string is an <em>n-gram</em> which consists of $n$ symbols separated by the separator character (default: \",\"). Each symbol in this n-gram represents a node or vertex traversed by a path of length $n-1$ (we define the length of a path as the number of links it traverses). So the pathways from the example above can be generated as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[4]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b,c&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,a&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b,c,d&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c&#39;</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We can print the <code>Paths</code> instance to get a human-readable summary:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[5]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Number of paths (unique/sub paths/total):\t6 (6/23/29)\nNodes:\t\t\t\t4\nEdges:\t\t\t\t4\nMax. path length:\t\t3\nAvg path length:\t\t1.5\nPaths of length k = 0\t\t1 (1/14/15)\nPaths of length k = 1\t\t2 (2/7/9)\nPaths of length k = 2\t\t2 (2/2/4)\nPaths of length k = 3\t\t1 (1/0/1)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>This instance contains six paths: One path consists of a single node and has length zero (remember the path length is the number of traverse links). Two paths have length one and two respectively, and a single path has length three. The network in which these paths occur has four nodes $a$, $b$, $c$, and $d$ connected by four directed links/edges $(a,b)$, $(b,c)$, $(c,a)$, and $(c,d)$.</p>\n<p>You will notice a group of three numbers after each path length. The first number counts unique ocurrences of paths. This becomes clear if we add a second occurrence of path $(a \\rightarrow b)$:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[6]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b&#39;</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Number of paths (unique/sub paths/total):\t7 (6/25/32)\nNodes:\t\t\t\t4\nEdges:\t\t\t\t4\nMax. path length:\t\t3\nAvg path length:\t\t1.42857142857\nPaths of length k = 0\t\t1 (1/16/17)\nPaths of length k = 1\t\t3 (2/7/10)\nPaths of length k = 2\t\t2 (2/2/4)\nPaths of length k = 3\t\t1 (1/0/1)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We now have a total of seven observations of six unique paths (one path occuring twice). Rather than adding multiple observations by hand, we can actually set frequencies (or weights) of paths. So, rather than adding $(a \\rightarrow b)$ twice we could have written the following to get the same result:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[7]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"p\">()</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b,c&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,a&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,b,c,d&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c&#39;</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Number of paths (unique/sub paths/total):\t7 (6/25/32)\nNodes:\t\t\t\t4\nEdges:\t\t\t\t4\nMax. path length:\t\t3\nAvg path length:\t\t1.42857142857\nPaths of length k = 0\t\t1 (1/16/17)\nPaths of length k = 1\t\t3 (2/7/10)\nPaths of length k = 2\t\t2 (2/2/4)\nPaths of length k = 3\t\t1 (1/0/1)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>What about the second number in the group of three numbers? It counts the number of so-called sub-paths. For a path $p=(a \\rightarrow b \\rightarrow c \\rightarrow d)$ we call any sequence of nodes $q$ which is contained in $p$ a sub-path of $p$. I.e. $q_1=(a \\rightarrow b)$, $q_2=(b \\rightarrow c \\rightarrow d)$ or the single node $q_3=(b)$ are sub-paths of path $p$.</p>\n<p>Correctly accounting for sub-paths of any length is crucial for our graphical modeling framework. Whenever we add a path (using the default parameter <code>expandSubPaths=True</code>) all sub-paths of the added path will be automatically calculated. In the example above, this means that there are a total of $16$ sub-paths of length zero (single nodes). Note that $(a\\rightarrow b)$ is occurring twice and $(c)$ is an actual path with length zero for which no additional sub-path is counted. Similarly, there are seven sub-paths of length one, etc.</p>\n<p>We can access paths via a public dictionary <code>paths</code>, which contains the list (and frequencies) of paths of any length. For the example above, we can, e.g., access paths of length two as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[8]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">paths</span><span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">]</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt output_prompt\">Out[8]:</div>\n\n\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>defaultdict(&lt;function pathpy.Paths.Paths.__init__.&lt;locals&gt;.&lt;lambda&gt;.&lt;locals&gt;.&lt;lambda&gt;&gt;,\n            {(&apos;a&apos;, &apos;b&apos;, &apos;c&apos;): array([1, 1]),\n             (&apos;b&apos;, &apos;c&apos;, &apos;a&apos;): array([0, 1]),\n             (&apos;b&apos;, &apos;c&apos;, &apos;d&apos;): array([1, 0])})</pre>\n</div>\n\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>This returns a dictionary which contains three different paths of length two, each of which is associated with a tuple consisting of two numbers. The first counts the <strong>actual path observations</strong>, the second number counts the <strong>occurrences of a path as a sub-path</strong> of a longer path observation. In the example above, path $(a \\rightarrow b \\rightarrow c)$ has been observed one time as an actual path, and one time as a sub-path of the longer path $(a\\rightarrow b \\rightarrow c \\rightarrow d)$. The path $(b \\rightarrow c \\rightarrow a)$ never occurs as sub-path, while $(b \\rightarrow c \\rightarrow d)$ only occurs as a sub path of $(a \\rightarrow b \\rightarrow c \\rightarrow d)$.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"temporal\"></a></p>\n<h2 id=\"3.-Analyzing-temporal-networks:-the-TemporalNetwork-class\">3. Analyzing temporal networks: the <code>TemporalNetwork</code> class<a class=\"anchor-link\" href=\"#3.-Analyzing-temporal-networks:-the-TemporalNetwork-class\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>A key design principle behind <code>pathpy</code> is that it unifies the analysis of pathway data (introduced above) and <strong>time-stamped data on temporal networks</strong>. We consider a temporal network as a collection of triplets of the form $(a,b;t)$ which capture that a node $a$ was connected to node $b$ (via a directed link) at a discrete time $t$. Such time-stamped data are of increasing importance, for instance when studying time-stamped interactions in a social network.</p>\n<p>We can represent such time-stamped network data using <code>pathpy</code>'s <code>TemporalNetwork</code> class. Let us create an empty instance:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[9]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">t</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">TemporalNetwork</span><span class=\"p\">()</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We can add time-stamped edges to this temporal network in any order:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[10]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">addEdge</span><span class=\"p\">(</span><span class=\"n\">source</span><span class=\"o\">=</span><span class=\"s1\">&#39;a&#39;</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"o\">=</span><span class=\"s1\">&#39;b&#39;</span><span class=\"p\">,</span> <span class=\"n\">ts</span><span class=\"o\">=</span><span class=\"mi\">42</span><span class=\"p\">)</span>\n<span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">addEdge</span><span class=\"p\">(</span><span class=\"n\">source</span><span class=\"o\">=</span><span class=\"s1\">&#39;b&#39;</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"o\">=</span><span class=\"s1\">&#39;c&#39;</span><span class=\"p\">,</span> <span class=\"n\">ts</span><span class=\"o\">=</span><span class=\"mi\">21</span><span class=\"p\">)</span>\n<span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">addEdge</span><span class=\"p\">(</span><span class=\"n\">source</span><span class=\"o\">=</span><span class=\"s1\">&#39;c&#39;</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"o\">=</span><span class=\"s1\">&#39;d&#39;</span><span class=\"p\">,</span> <span class=\"n\">ts</span><span class=\"o\">=</span><span class=\"mi\">51</span><span class=\"p\">)</span>\n<span class=\"n\">t</span><span class=\"o\">.</span><span class=\"n\">addEdge</span><span class=\"p\">(</span><span class=\"n\">source</span><span class=\"o\">=</span><span class=\"s1\">&#39;b&#39;</span><span class=\"p\">,</span> <span class=\"n\">target</span><span class=\"o\">=</span><span class=\"s1\">&#39;c&#39;</span><span class=\"p\">,</span> <span class=\"n\">ts</span><span class=\"o\">=</span><span class=\"mi\">44</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Again, printing the instance will return a human-readable summary:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[11]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Nodes:\t\t\t4\nTime-stamped links:\t4\nLinks/Nodes:\t\t1.0\nObservation period:\t[21, 51]\nObservation length:\t30\nTime stamps:\t\t4\nAvg. inter-event dt:\t10.0\nMin/Max inter-event dt:\t2/21\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>This temporal network consists of four nodes connected by four time-stamped links. The observation period covers 30 time units and contains four different time stamps with observed edges. The average inter-event time between \"events\" is 10 time units, the minimum and the maximum difference between any two consecutive events are two and 21 time units respectively.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Now you may ask: How are temporal networks related to the pathways above and how can we study both from the same perspective?</p>\n<p>Well, the point is that temporal networks naturally give rise to <strong>time-respecting paths</strong>, i.e. paths consisting of sequences of time-stamped links which (minimally) satisfy <strong>causality</strong>. Specifically, two time-stamped links $(a,b;t_1)$ and $(b,c;t_2)$ contribute to a time-respecting path $a \\rightarrow b \\rightarrow c$ if $t_1<t_2$, i.e. if the link $(a,b)$ occurs <strong>before</strong> $(b,c)$. Apart from the condition that links have to occur in the correct order, it is common to impose a <strong>maximum time difference</strong> between consecutive links. I.e. we define a maximum time difference $\\delta$ and consider two time-stamped edges $(a,b;t)$ and $(b,c;t')$ to contribute to a time-respecting path if $0 \\leq t'-t \\leq \\delta$. Imposing this additional condition is natural, since we are typically interested in paths which occur at short time scales. Especially, when considering time-stamped data collected over a period of several days, weeks or even years, it is usually not reasonable to consider a path definition where links can be weeks or years apart.</p>\n<p>With this definition of time-respecting paths at hand, we can extract pathways from a sequence of time-stamped edges based on a given value of $\\delta$. We can directly do this in <code>pathpy</code> using a built-in method to extract time-respecting paths for arbitrary $\\delta$. Let us try this for $\\delta=\\inf$, i.e. we don't impose a constraint for the maximum time difference, but still require that links occur in the right order:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[12]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">trp</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"o\">.</span><span class=\"n\">fromTemporalNetwork</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">,</span> <span class=\"n\">delta</span><span class=\"o\">=</span><span class=\"n\">np</span><span class=\"o\">.</span><span class=\"n\">inf</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">trp</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:16:53 [Severity.INFO]\tExtracting time-respecting paths for delta = inf ...\n2017-03-01 17:16:53 [Severity.INFO]\tCalculating sub path statistics ... \n2017-03-01 17:16:53 [Severity.INFO]\tfinished.\nNumber of paths (unique/sub paths/total):\t3 (2/19/22)\nNodes:\t\t\t\t4\nEdges:\t\t\t\t3\nMax. path length:\t\t3\nAvg path length:\t\t2.33333333333\nPaths of length k = 0\t\t0 (0/10/10)\nPaths of length k = 1\t\t0 (0/7/7)\nPaths of length k = 2\t\t2 (1/2/4)\nPaths of length k = 3\t\t1 (1/0/1)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Here we have two time-respecting paths $(b,c;21) \\rightarrow (c,d;51)$ and $(b,c;44) \\rightarrow (c,d;51)$ of length two and a path $(a,b;41) \\rightarrow (b,c;44) \\rightarrow (c,d;51)$ of length three. In addition, all shorter time-respecting paths which are sub paths of the three detected paths are automatically accounted for in the sub-path counts given in the second number in the brackets. If we instead set $\\delta$ to a smaller value like $\\delta=5$, we get different (time-respecting) paths:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[13]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">trp</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"o\">.</span><span class=\"n\">fromTemporalNetwork</span><span class=\"p\">(</span><span class=\"n\">t</span><span class=\"p\">,</span> <span class=\"n\">delta</span><span class=\"o\">=</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">trp</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:16:53 [Severity.INFO]\tExtracting time-respecting paths for delta = 5 ...\n2017-03-01 17:16:53 [Severity.INFO]\tCalculating sub path statistics ... \n2017-03-01 17:16:53 [Severity.INFO]\tfinished.\nNumber of paths (unique/sub paths/total):\t3 (3/9/12)\nNodes:\t\t\t\t4\nEdges:\t\t\t\t3\nMax. path length:\t\t2\nAvg path length:\t\t1.33333333333\nPaths of length k = 0\t\t0 (0/7/7)\nPaths of length k = 1\t\t2 (2/2/4)\nPaths of length k = 2\t\t1 (1/0/1)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Now we only have two paths of length one (which are just time-stamped edges without a continuation towards a longer time-respecting path) and a single path $(a,b;42) \\rightarrow (b,c;44)$ of length two for which the time difference between edges is less than five.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>In summary, <code>pathpy</code> makes it very easy to extract (time-respecting) paths from time-stamped data on temporal networks. Thanks to this, <strong>both pathway and temporal network data can be studied from the same perspective</strong>.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"data\"></a></p>\n<h2 id=\"4.-Importing-data-on-pathways-and-temporal-networks\">4. Importing data on pathways and temporal networks<a class=\"anchor-link\" href=\"#4.-Importing-data-on-pathways-and-temporal-networks\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Rather than manually adding paths or time-stamped edges, we can directly read <code>Paths</code> and <code>TemporalNetwork</code> instances from data files.</p>\n<p>For pathway data, the <code>Paths</code> class can be used to read <strong>n-gram files</strong>, i.e. text files where each line contains a path with variable length. Vertices can be arbitrary strings, separated by a special character that can be set when reading the file. <code>pathpy</code> supports data where each path is observed multiple times. This either works by reading files where identical paths occur in multiple lines of the file, or by including a special last column in each line which contains the number of observations of a path.</p>\n<p>Such an n-gram file can look like the following excerpt from a file that captures <a href=\"https://tfl.gov.uk/info-for/open-data-users/\">travel itineraries of passengers of the London Tube</a>, recorded via smartcard readers:</p>\n<p><code>344,314,445,440,513,346,305,312,289,367,356,299,376,9</code><br>\n<code>339,303,323,376,299,356,367,289,312,305,346,400,465,382,325,7</code><br>\n<code>296,430,474,271,332,331,441,341,280,294,362,528,344,493,29</code><br>\n<code>...</code></p>\n<p>Numbers indicate metro stations passed by on an itinerary, except for the last column which indicates the number of times the given path was observed. If we save such data in a textfile called <code>tube_paths.ngram</code>, we can read it as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[14]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">tube_paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"o\">.</span><span class=\"n\">readFile</span><span class=\"p\">(</span><span class=\"s1\">&#39;pathpy_tutorial/tube_paths.ngram&#39;</span><span class=\"p\">,</span> <span class=\"n\">separator</span><span class=\"o\">=</span><span class=\"s1\">&#39;,&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">tube_paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:16:53 [Severity.INFO]\tReading ngram data ... \n2017-03-01 17:16:54 [Severity.INFO]\tfinished. Read 32918 paths with maximum length 35\n2017-03-01 17:16:54 [Severity.INFO]\tCalculating sub path statistics ... \n2017-03-01 17:17:15 [Severity.INFO]\tfinished.\nNumber of paths (unique/sub paths/total):\t4295731 (32313/182041358/186337089)\nNodes:\t\t\t\t276\nEdges:\t\t\t\t663\nMax. path length:\t\t35\nAvg path length:\t\t6.86497129359\nPaths of length k = 0\t\t0 (0/33785801/33785801)\nPaths of length k = 1\t\t186435 (594/29303635/29490070)\nPaths of length k = 2\t\t337975 (912/24856364/25194339)\nPaths of length k = 3\t\t446941 (1288/20638102/21085043)\nPaths of length k = 4\t\t453255 (1696/16860467/17313722)\nPaths of length k = 5\t\t467697 (2040/13521645/13989342)\nPaths of length k = 6\t\t438374 (2316/10679843/11118217)\nPaths of length k = 7\t\t369090 (2460/8345699/8714789)\nPaths of length k = 8\t\t317013 (2529/6432722/6749735)\nPaths of length k = 9\t\t277742 (2572/4876029/5153771)\nPaths of length k = 10\t\t229761 (2481/3645059/3874820)\nPaths of length k = 11\t\t181738 (2308/2691873/2873611)\nPaths of length k = 12\t\t147092 (2080/1955071/2102163)\nPaths of length k = 13\t\t123089 (1855/1389364/1512453)\nPaths of length k = 14\t\t85503 (1612/984332/1069835)\nPaths of length k = 15\t\t70859 (1329/679447/750306)\nPaths of length k = 16\t\t48461 (1073/467819/516280)\nPaths of length k = 17\t\t37015 (841/316098/353113)\nPaths of length k = 18\t\t24104 (655/214303/238407)\nPaths of length k = 19\t\t16498 (470/144218/160716)\nPaths of length k = 20\t\t10852 (335/96277/107129)\nPaths of length k = 21\t\t10416 (294/59624/70040)\nPaths of length k = 22\t\t5155 (179/38648/43803)\nPaths of length k = 23\t\t3274 (118/24708/27982)\nPaths of length k = 24\t\t3136 (98/14180/17316)\nPaths of length k = 25\t\t2253 (76/7671/9924)\nPaths of length k = 26\t\t506 (34/5162/5668)\nPaths of length k = 27\t\t567 (26/3098/3665)\nPaths of length k = 28\t\t400 (13/1768/2168)\nPaths of length k = 29\t\t225 (13/1013/1238)\nPaths of length k = 30\t\t161 (8/547/708)\nPaths of length k = 31\t\t50 (2/353/403)\nPaths of length k = 32\t\t12 (1/247/259)\nPaths of length k = 33\t\t4 (1/161/165)\nPaths of length k = 34\t\t73 (3/10/83)\nPaths of length k = 35\t\t5 (1/0/5)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>The large numbers result from the calculation of sub-path statistics outlined above. Here, the fact that this data set contains many long paths results in more than 180 million shorter sub-paths which are contained within longer pathways.</p>\n<p>We finally show how we can read temporal networks from data files. Here we assume that we are given a data file that has the following format</p>\n<p><code>time source target</code><br>\n<code>28820 492 938</code><br>\n<code>28860 267 272</code><br>\n<code>29300 181 826</code><br>\n<code>...</code></p>\n<p>Each line captures a directed time-stamped edge from source to target, happening instantaneously at the indicated time stamp. The header column tells which column is which. The ordering of columns can be arbitrary and the character that separates column can be specified.</p>\n<p>The above lines are actually an excerpt from a time-stamped data set released by the <a href=\"http://www.sociopatterns.org\">SocioPatterns collaboration</a>. It captures <a href=\"http://www.sociopatterns.org/datasets/contacts-in-a-workplace/\">face-to-face encounters of workers in a company</a>, recorded via sensor badges. We can read this data file to a <code>TemporalNetwork</code> instance as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[15]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">work_t</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">TemporalNetwork</span><span class=\"o\">.</span><span class=\"n\">readFile</span><span class=\"p\">(</span><span class=\"s1\">&#39;pathpy_tutorial/WorkplaceContacts.tedges&#39;</span><span class=\"p\">,</span> <span class=\"n\">sep</span><span class=\"o\">=</span><span class=\"s1\">&#39; &#39;</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">work_t</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:16 [Severity.INFO]\tReading time-stamped links ...\n2017-03-01 17:17:16 [Severity.INFO]\tBuilding index data structures ...\n2017-03-01 17:17:16 [Severity.INFO]\tSorting time stamps ...\n2017-03-01 17:17:16 [Severity.INFO]\tfinished.\nNodes:\t\t\t92\nTime-stamped links:\t9827\nLinks/Nodes:\t\t106.81521739130434\nObservation period:\t[28820, 1016440]\nObservation length:\t987620\nTime stamps:\t\t7104\nAvg. inter-event dt:\t139.042658032\nMin/Max inter-event dt:\t20/222680\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Now we can pick a maximum time difference $\\delta$ and extract all time-respecting paths as explained above. Here we choose a maximum time difference of three minutes, which gives the following time-respecting path statistics:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[16]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">work_paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"o\">.</span><span class=\"n\">fromTemporalNetwork</span><span class=\"p\">(</span><span class=\"n\">work_t</span><span class=\"p\">,</span> <span class=\"n\">delta</span><span class=\"o\">=</span><span class=\"mi\">180</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">work_paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:16 [Severity.INFO]\tExtracting time-respecting paths for delta = 180 ...\n2017-03-01 17:17:17 [Severity.INFO]\tCalculating sub path statistics ... \n2017-03-01 17:17:17 [Severity.INFO]\tfinished.\nNumber of paths (unique/sub paths/total):\t10939 (968/31879/42818)\nNodes:\t\t\t\t92\nEdges:\t\t\t\t755\nMax. path length:\t\t4\nAvg path length:\t\t1.28393820276\nPaths of length k = 0\t\t0 (0/24984/24984)\nPaths of length k = 1\t\t8467 (669/5578/14045)\nPaths of length k = 2\t\t1887 (252/1219/3106)\nPaths of length k = 3\t\t536 (41/98/634)\nPaths of length k = 4\t\t49 (6/0/49)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"multiorder\"></a></p>\n<h2 id=\"5.-Multi-Order-Graphical-Models-of-Pathways-and-Temporal-Networks\">5. Multi-Order Graphical Models of Pathways and Temporal Networks<a class=\"anchor-link\" href=\"#5.-Multi-Order-Graphical-Models-of-Pathways-and-Temporal-Networks\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Now that we can import data on pathways and temporal networks, we turn our attention to the multi-order graphical modeling framework which is the theoretical foundation of our data mining techniques. The mathematical details have been developed in <a href=\"https://arxiv.org/abs/1702.05499\">this recent article</a>. Here we provide a short (and rather high-level) introduction.</p>\n<p>Consider a toy network which consists of five nodes $a,b,c,d,e$ connected by four links $(a,c), (b,c), (c,d), (c,e)$. We further assume that we have a total of 18 observations of the following eight unique paths:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[17]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"p\">()</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">5</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">6</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Number of paths (unique/sub paths/total):\t19 (6/71/90)\nNodes:\t\t\t\t5\nEdges:\t\t\t\t4\nMax. path length:\t\t2\nAvg path length:\t\t1.57894736842\nPaths of length k = 0\t\t0 (0/49/49)\nPaths of length k = 1\t\t8 (4/22/30)\nPaths of length k = 2\t\t11 (2/0/11)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>The question we are going to address is: Considering the statistics of observed paths, is it justified to model the underlying system as a network? And if not, what would be an optimal graphical abstraction for the data set?</p>\n<p>We address these important questions based on <a href=\"https://www.sg.ethz.ch/team/people/ischoltes/research-insights/temporal-networks-demo/\">\"higher-order\" networks</a> introduced in <a href=\"http://www.nature.com/ncomms/2014/140924/ncomms6024/full/ncomms6024.html\">this Nature Communications article</a>. Here, we generalize this approach to (i) multi-order models which combine multiple layers of higher-order networks, and (ii) pathway data.</p>\n<p>So what is the problem if we model the system above as a graph or network? The problem is that graph- and network-analytic methods like centrality measures, community detection, etc. are implicitly based on the <strong>assumption that paths in a network are transitive</strong>, i.e. if we observe a path from node $u$ to node $v$ and a path from node $v$ to node $w$, we implicitly assume that there is a transitive path from $u$ via $v$ to $w$. This fundamental assumption is due to the way how graph algorithms (as well as algebraic methods which rely on matrix multiplication or spectral analysis) work. However, what is important to see is that <strong>correlations in the sequence of nodes traversed by paths can invalidate the assumption of transitivity</strong> (see <a href=\"http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.198701\">discussion here</a>).</p>\n<p>Consider the example above: Since the network has two edges $(a,c)$ and $(c,e)$ we would expect that a path of length two from $a$ via $c$ to $e$ exists. But this is actually not the case. Instead, whenever a path traverses from $a$ to $c$, it continues to $d$. Likewise, whenever a path traverses from $b$ to $c$, it continues to $e$. The transitive path $(b \\rightarrow c \\rightarrow d)$, which we expect based on the link topology of the graph, never occurs.</p>\n<p>So the question really is: Is the topology of the underlying graph enough to explain the statistics of observed paths? Note that the example above is an extreme example, where two transitive paths are completely absent. Rather than being completely absent, we could also have cases where paths are just less (or more) frequent than what we expect. Clearly, such an under- or overrepresentation of paths violates the transitivity assumption of a network abstraction as well (though possibly to a lesser degree).</p>\n<p>While we refer to <a href=\"#references\">the publications above</a> for a detailed mathematical description of our approach, the key idea is to consider <a href=\"https://en.wikipedia.org/wiki/Markov_chain\">Markov chain models of different orders</a> which are tailored to reproduce the statistics of paths observed in a given graph topology. We specifically use a graphical construction that resembles <a href=\"https://en.wikipedia.org/wiki/De_Bruijn_graph\">De Bruijn graphs</a> known from sequence modeling.</p>\n<p>This is how it works: We first consider a \"first-order\" network abstraction which simply counts the frequencies at which edges are traversed by paths. We can generate such a first-order abstractions of our observed paths using the <code>HigherOrderNetwork</code> class provided by <code>pathpy</code> as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[18]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">network</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">HigherOrderNetwork</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>To visualize this network model, we provide the following helper function, which takes an instance of the <code>HigherOrderNetwork</code> class and returns an <code>igraph</code> <code>Graph</code> object that we can use to plot the network topology. The current version of <code>pathpy</code> specifically does not include plotting and visualization tools since (i) we want the package to have minimal dependencies, and (ii) the function below shows that it is very easy to construct <code>igraph</code> (and similarly <a href=\"https://graph-tool.skewed.de/\"><code>graph-tool</code></a>) instances from a <code>HigherOrderNetwork</code> object.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[19]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">Network2igraph</span><span class=\"p\">(</span><span class=\"n\">network</span><span class=\"p\">):</span>\n    <span class=\"sd\">&quot;&quot;&quot; </span>\n<span class=\"sd\">    Returns an igraph Graph object which represents </span>\n<span class=\"sd\">    the k-th layer of a multi-order graphical model.</span>\n<span class=\"sd\">    &quot;&quot;&quot;</span>\n    <span class=\"n\">g</span> <span class=\"o\">=</span> <span class=\"n\">igraph</span><span class=\"o\">.</span><span class=\"n\">Graph</span><span class=\"p\">(</span><span class=\"n\">directed</span><span class=\"o\">=</span><span class=\"kc\">True</span><span class=\"p\">)</span>\n\n    <span class=\"k\">for</span> <span class=\"n\">e</span> <span class=\"ow\">in</span> <span class=\"n\">network</span><span class=\"o\">.</span><span class=\"n\">edges</span><span class=\"p\">:</span>\n        <span class=\"k\">if</span> <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">vcount</span><span class=\"p\">()</span><span class=\"o\">==</span> <span class=\"mi\">0</span> <span class=\"ow\">or</span> <span class=\"n\">e</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">vs</span><span class=\"p\">()[</span><span class=\"s2\">&quot;name&quot;</span><span class=\"p\">]:</span>\n            <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">add_vertex</span><span class=\"p\">(</span><span class=\"n\">e</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span>\n        <span class=\"k\">if</span> <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">vcount</span><span class=\"p\">()</span><span class=\"o\">==</span> <span class=\"mi\">0</span> <span class=\"ow\">or</span> <span class=\"n\">e</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">]</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">vs</span><span class=\"p\">()[</span><span class=\"s2\">&quot;name&quot;</span><span class=\"p\">]:</span>\n            <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">add_vertex</span><span class=\"p\">(</span><span class=\"n\">e</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">])</span>\n        <span class=\"n\">g</span><span class=\"o\">.</span><span class=\"n\">add_edge</span><span class=\"p\">(</span><span class=\"n\">e</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">],</span> <span class=\"n\">e</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">],</span> <span class=\"n\">weight</span><span class=\"o\">=</span><span class=\"n\">network</span><span class=\"o\">.</span><span class=\"n\">edges</span><span class=\"p\">[</span><span class=\"n\">e</span><span class=\"p\">]</span><span class=\"o\">.</span><span class=\"n\">sum</span><span class=\"p\">())</span>\n    <span class=\"k\">return</span> <span class=\"n\">g</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We can now use <code>igraph</code>'s visual styling and plotting features to plot the network representation.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[20]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">g1</span> <span class=\"o\">=</span> <span class=\"n\">Network2igraph</span><span class=\"p\">(</span><span class=\"n\">network</span><span class=\"p\">)</span>\n<span class=\"n\">igraph</span><span class=\"o\">.</span><span class=\"n\">plot</span><span class=\"p\">(</span><span class=\"n\">g1</span><span class=\"p\">)</span>\n\n<span class=\"n\">visual_style</span> <span class=\"o\">=</span> <span class=\"p\">{}</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;bbox&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"p\">(</span><span class=\"mi\">600</span><span class=\"p\">,</span> <span class=\"mi\">400</span><span class=\"p\">)</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;margin&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"mi\">60</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;vertex_size&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"mi\">80</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;vertex_label_size&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"mi\">24</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;vertex_color&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"s2\">&quot;lightblue&quot;</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;edge_curved&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"mf\">0.2</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;edge_width&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"mi\">1</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;edge_arrow_size&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"mi\">2</span>\n\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;layout&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g1</span><span class=\"o\">.</span><span class=\"n\">layout_auto</span><span class=\"p\">()</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;vertex_label&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g1</span><span class=\"o\">.</span><span class=\"n\">vs</span><span class=\"p\">[</span><span class=\"s2\">&quot;name&quot;</span><span class=\"p\">]</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;edge_label&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g1</span><span class=\"o\">.</span><span class=\"n\">es</span><span class=\"p\">[</span><span class=\"s2\">&quot;weight&quot;</span><span class=\"p\">]</span>\n\n<span class=\"n\">igraph</span><span class=\"o\">.</span><span class=\"n\">plot</span><span class=\"p\">(</span><span class=\"n\">g1</span><span class=\"p\">,</span> <span class=\"s1\">&#39;pathpy_tutorial/g1.png&#39;</span><span class=\"p\">,</span> <span class=\"o\">**</span><span class=\"n\">visual_style</span><span class=\"p\">)</span>\n<span class=\"n\">display</span><span class=\"p\">(</span><span class=\"n\">Image</span><span class=\"p\">(</span><span class=\"n\">filename</span><span class=\"o\">=</span><span class=\"s1\">&#39;pathpy_tutorial/g1.png&#39;</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n\n\n<div class=\"output_png output_subarea \">\n<img 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DBg2/dujV5\n8mTUWQRBZWXlvXv3Hj9+XFVV5ejo6OPjEx4eLikpiToXTzMyNq4oKSKgCDEMy0pO/OqRwUv7d2EY\n5uw5koDZK0qKDI2MOBkBihAArhg9evSyZcsmTJggJgZ/yrrp7du39+/fT05ObmxsHDJkyPTp083M\nzHhkDTPeN3jAgOyMxzZD3AmYK2Jm4JePz48xUWd/Qcwto3lPH7sM5mg5J/gjCgBX6OjoaGpqpqSk\nuLi4oM7CZ2g0WnJy8t27d9++fevo6Dh9+nRra2tOTnwJJzc319Oz5xAwEfv8Z8TMr+8d3Xs3jYDZ\nMQx79SR1+eypnIwARQgAt/j4+Ny4cQOKsJNYLFZWVtbdu3efPHliY2MTEBBgY2MjbI894Mje3r6q\n/A218ZOMXA9uzxV5NvbLtUY9/AO/vWrIJY31de8r39nY2HAyCInFYuEVCADwJSaTGRAQsGPHjt69\ne6POwtMKCwtv3ryZnJxsaGjo5eXl4OAA1/9w4THc08prjONQL9RBuOjhzWv592/dvM7RGt9wRAgA\nt4iJibm7u8fHxwcHB6POwovq6+vv3r0bHx/f1tY2YsSIgwcPKikpoQ4lUIImBe45fkqwizD1+qXF\nITM5HASOCAHgojdv3vzxxx9nzpwRERFBnYWH5Ofnx8TEPHr0yNbWdujQoba2tvDvhxvodLqahsbO\nG0nyArpbff2HmiW+blUVFRwulQ5HhABwkY6OjpKSUkZGhr29PfuVysrKgoKCDx8+NDU10Wg0Op2O\nYRh773JZWVllZWUTExMVFRWkqbmFyWQmJyfHxMQ0NDT4+PjMmTOHk/UhwS+RyWQvL69Ht294Bgrm\nOYm0uFgfbx/ONwyBIgSAu+zs7LZs2UKWkX327FlJcZG4hKSmnoGCsoqkNFlSmiwhLY1hWCuD0UKn\ntTQz6qqr3pYUdbQz9Q0NrSythrq7ubi4qKqqov6HwDAMYzKZGzdunDhxop6eXlc/W1NTc/ny5YSE\nBEtLy5kzZ5qamnIjIfjWzOnTp0yfMXxCEEngjrk72tvjz548feI450PBqVEAuOL58+cnT0Vfi439\nWPfRwMzKYsAQfVMLDV19mR7yv/xsY31dZWlx0ctn+ZmPXz1NV1FRGT1q1KTAif369SMg+XcxmczI\nyMiUlBRnZ+c1a9Z08lMsFis7O/vq1av5+fne3t5eXl49e/bkZkzwHU4DBjr6TRjgNQp1EJw9uHYp\n+9aV5KT7nA8FRQgAnurq6g4eOnTi1KkmKn3gyDH9h3tr6htxsjchq6OjrOup3/EAACAASURBVCDv\n0Z3rKbGXe/VUDJo0acaMGT16cP2G+C99bkEMw0gk0l9//WVsbPzzj7S3tyclJcXExDAYDF9fX3d3\nd072TQWcuHPnzuzQsG3XEgTpoJDV0bF4pMvRA/vd3Nw4Hw2KEAB8fPjwYdv2HQcPHbR18XAZM8HE\n2g7fvXlZHR2vMx4nXjrzPPXBvLlzF4SFKSoq4jj+j3zZgmwODg4bNmz40fvb2tru3Llz4cIFbW3t\nMWPGWFhYwB7FaLFYLHNLK6+ZYfbuw1Fnwc2jOzfunTiYnYnPNk9QhABwikajbdgQceDQwQEjfL2D\nQ5TUNbk6Xc3bstgje9PvxYXOm/fHsmVcPdL6tgXZoqKi+vbt+9WLVCr1ypUrN2/e7N+//7hx44R8\nIySe8vDhw7HjA7Zfvy9FFoRNielNjYt8XG7GXsNr108oQgA4cvHixQULF/WxdwpYtFxBibi7PWur\nKqO3rn/z6vme3VEjR3JlaeMftSCGYdbW1ps3b/78y6ampvPnz9++fdvV1XXMmDGCetcrX5s8JYgu\nJTthkSBsDRa9dYM8q+XY0SN4DQhFCEA3VVVVTZoypbi0LPjPjf3s+iPJ8Oxh0vENqyzM+h0/ehTf\np9F/0oJs27Zts7Cw+Pjx4/nz5xMTE11dXceOHQtPxPOs6upqUzPzNaeuqOvqo87CkXfFheuDxr7K\neamsjNvDkYJz7RQAIsXHx1tYWino9dl8NR5VC2IYZjlgyLZrCSKKqhaWVj8pra76ZQtiGHbs2LEj\nR47MmjWro6Nj3759c+bMgRbkZaqqqps3b4paHNLW0oI6S/e1NDN2Lw7ZunULji2IwREhAF3FYrFW\nh4cfOHRoTuQuywFDUMf5x5OEO4fCf/99yeLlf/zB4VCdaUE2X1/fCRMmEHPPDsDFlKlTyz5+mr91\nL+og3RS1OERftdexI7idFGWDB+oB6AImkxkUHJydk7vp0h1FZR66EmbvNky3T7/todPKysr27d3b\n7RXL2E/Nd/LgMjc3V0FBoXsTAST27N5taW39IObi4FFjUWfpssQr56qK8m6ew+dO0S/BqVEAOqup\nqcndY+jbusbwE5d4qgXZlNQ1156Oyc4v9h7py2AwujECuwWTk5M7+f6CgoL09PRuTARQkZWVvXr5\ncvS29TnpqaizdM3LRw/P7oi8cumSjIwM7oNDEQLQKc3NzT6+o8R7Ks/b8pc4r24SKyklvXj3ERom\nOspvTFtbW5c+29UWZDtx4gRcXuEv5ubmt27c+Ov3eSWvX6LO0lkFzzN3L5kTd/MmlxbngyIE4Nfa\n29v9xweIKSjN3rBDVJSnLyiIiYvP37qXISoxaUpQR0dHJz/V3t7ejRbEMKyoqCg1lc+OLYCDg8PB\n/fu2h06rLn+DOsuvVZe/iVoUcvTI4c8r1+MOihCAX+jo6Jg4aXJDa0fIhu18sUgKSUQkdPPukqr3\ns2aHdOb97e3tkZGR3WhBNjgo5Ed+fn5bN21cF/Rb6esc1Fl+pvjVi3VBv23fsnnUKC6ulSra+fVz\nARBOkRs3JqY+Wrz7iJi4OOosnSUiImrj4nFyz86OtlZHB4efvJPDFsQwrKGhoXfv3rq6ut0eASBh\nbm5ua2OzYGqgmq6+uk6XdxQhQEZS/N6loRfOnvXx8eHqRPD4BAA/c+fOnSnB0zZdui2nyH/bJjR8\n/LDiN8+L588NHDjwu2/gvAXZNDQ0jh49KioqyuE4gHjp6em+o/18gkOGBwbzzgkPFot18+SRuBOH\nrl+LsbW15fZ0cGoUgB+qqKiYHBQ0b/Nf/NiCGIbJ91SasW7rb/7+1dXV3/4ui8Xavn07hy2orKxs\nY2Njb2/f0NDAyTgAFQcHh5fPn5U+SYmYNr6h9j3qOBiGYfUfajZMG/82M+3Fs2wCWhCDI0IAfmLo\n8OHyuibjFyxDHYQjJzb+KdL08erly1++2I1jQWlpaT09PR0dHTU1NW1tbW1tbSUlJTExnr51CHQS\nk8lcsXLl6XPng1dHWg10QZgkO+X+0XXLgyZNWrd2DWH/d0ERAvB958+fX7VufeTFOB6/TfSXmG1t\nv49y27trp7e3N/uVzrSgoqKijo4Ou/DYzQd76gq8xMTEmbNnq+oaBC79U1lTi+DZ62qqT21eU1Wc\nf/jgwcGDBxM5NRQhAN9BpVKNTfrM3brXxBqffV7Qep6WfHL9itzXr6SkpL5tQRKJpKqqqv0v9jEf\nNx5bRisqKmrBggVfvnLr1i1PT09UeXhTS0tL5MaNO3ft8hg3yWfqbDkFItbPozc13jnz961TR8Pm\nh/6xbJkE4c/pQhEC8B2rVoenvngVumUP6iC42Tpv6igP1yWLF2/durW8vJx9kKejo6OlpaWhoSHO\nPzfE4iIuLm7Dhg3wBOSPFBUVRURuvBpzddi4SSOCZsnKc2shvaaG+tunjsRfiPYd6btyxXI9PTQ3\nr0IRAvC1uro6fUPDLVfie6oKztayVWWlf04cVViQLy0tLSUlhToOYiQSqbCw0MDAAHUQnvbu3btt\n23dER0ebOjo7evpaDXTBa02lttbW7OTEx3HXctLTJk2atGTxIg0NDVxG7h4oQgC+tmLlqudvKqaF\nR6IOgrO9f4S52lmuXCEIW7NyIioq6vXr1wcPHkQdhD9QqdRLly4dOnrsVU6OzRB3U8cB5v0HKqqo\ndmOoj9VVLx+nvExLyXqQYGZhMSN46pgxY3jhJDwUIQD/p6GhQc/AYNOl273UUP6Iyg0VJUVrp4x5\nU1LCC996EILDwe4pLi6OjY29ey/hYUpyL1U1Q3MrZW09dR09DT0DBWVVaTJZVOy/E+ztzDYGnV5X\nU1VZWlz1pqT6TXHRi+yP72sGDBw0zMN95MiRPLUCAxQhAP9nV1RUzL2k+dv2oQ7CFVvnTp02/rfp\n06ejDoIMXB3kHJPJzMzMfP78eX5BQW5efn5+/sfaD1QqVURERFqajGEYg0Hv6OiQkZHppaRsbGzc\nx8TY2MjIwsLCxsaGN1ddgCIE4P+Y9O03cfm6fvZOqINwRUZS/L3j+588foQ6CDKzZs3q27dvWFgY\n6iACqKWlhU6nYxhGoVCIv/OTE7CyDAD/yc7OptLpfe36ow7CLVYDXUpKS/Pz81EHQSYnJ8fLywt1\nCsEkKSmpoKCgoKDAXy2IQREC8KWTp6L7e/ryzoqLuBMVFes/zPvMmTOogyCTlpaGOgLgOVCEAPwn\n9vp1x2HeqFNwl+Nwn6ux11GnQIbFYsFtMuArUIQA/KOsrOzTp086Jn1RB+EuI0vr0pLi2tpa1EEA\n4BVQhAD848GDB/3s+wvweVE2UVGxfjb2nG+9BIDAgCIE4B/3kx4Y2/xsD1uBYWzrmJiUhDoFALwC\nihCAf2Q9yzYws0SdgggGZpZZ2dmoUwDAK6AIAcAwDGOxWEUFBRp6QnEbhYaeQX5eHuoUAPAK/t5o\nDQAMw1gsFo1GwzCsvb2d/Tzvt6Slpb/a5POrZcYqKirIMrLSFKFYe0y+l3JbW1tdXZ2iIhGb7ADA\n46AIAa9gsVj19fX19fWfPn2iUqm0f9Hp9M9ff369vb0dwzAajcZisUgkEoVCwTBMVFSUTCZ/d3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t5PHjri6un5uQUlJSfbRno6ODvtUp5KSkmB/7yCRSIWFhQYGBlFRURcuXEhNTUWdiBe1trbu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\n2SEAAAAASUVORK5CYII=\n\"\n>\n</div>\n\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>As you can see, this first-order model corresponds to a weighted network, where link weights  count the number of times an edge has been traversed by paths. Considering that (i) each node in this network is actually a path of length zero, and (ii) each link provides the frequency of paths of length one, we can generalize this to higher-order graphical models. For $k=2$ we get a second-order model, where second-order nodes are paths of length one, and links provide the frequencies of paths of length two.</p>\n<p>We can easily generate this using the <code>HigherOrderNetwork</code> class:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[21]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">network2</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">HigherOrderNetwork</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">g2</span> <span class=\"o\">=</span> <span class=\"n\">Network2igraph</span><span class=\"p\">(</span><span class=\"n\">network2</span><span class=\"p\">)</span>\n\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;layout&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g2</span><span class=\"o\">.</span><span class=\"n\">layout_auto</span><span class=\"p\">()</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;vertex_label&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g2</span><span class=\"o\">.</span><span class=\"n\">vs</span><span class=\"p\">[</span><span class=\"s2\">&quot;name&quot;</span><span class=\"p\">]</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;edge_label&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g2</span><span class=\"o\">.</span><span class=\"n\">es</span><span class=\"p\">[</span><span class=\"s2\">&quot;weight&quot;</span><span class=\"p\">]</span>\n\n<span class=\"n\">igraph</span><span class=\"o\">.</span><span class=\"n\">plot</span><span class=\"p\">(</span><span class=\"n\">g2</span><span class=\"p\">,</span> <span class=\"s1\">&#39;pathpy_tutorial/g2.png&#39;</span><span class=\"p\">,</span> <span class=\"o\">**</span><span class=\"n\">visual_style</span><span class=\"p\">)</span>\n<span class=\"n\">display</span><span class=\"p\">(</span><span class=\"n\">Image</span><span class=\"p\">(</span><span class=\"n\">filename</span><span class=\"o\">=</span><span class=\"s1\">&#39;pathpy_tutorial/g2.png&#39;</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n\n\n<div class=\"output_png output_subarea \">\n<img 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Wsa2Akz6EJ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\nIQgBAIDV/g8E97SdfgQbywAAAABJRU5ErkJggg==\n\"\n>\n</div>\n\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>In this representation only two paths of length two actually exist, while - under the assumption that paths in the first-order network are transitive - we would expect four possible paths. If we would have longer paths, we could continue this approach and generate network models with order 3, 4, etc. In a third-order model nodes represent paths of length two, while links capture paths of length three. In general, the link weights of a $k$-th order model capture the statistics of paths of length $k$, thus generalizing the commonly used first-order network view.</p>\n<p>As shown <a href=\"http://dx.doi.org/10.1140/epjb/e2016-60663-0\">in our previous work</a> and <a href=\"http://link.springer.com/article/10.1140%2Fepjb%2Fe2016-60663-0\">EPJ B</a> such <strong>higher-order network abstractions</strong> are interesting, since they <strong>capture the temporal-topological topology of sequential data on networks</strong>. Moreover, just like the commonly used first-order abstractions, we can interpret them as network topologies which can be analyzed using network-analytic and algebraic methods. <a href=\"http://dx.doi.org/10.1140/epjb/e2016-60663-0\">We have further shown</a> that these higher-order graphs can be interpreted as Markov models which capture correlations of a given length $k$ that are hidden in the statistics of pathways.</p>\n<p>Building on this idea, here we go one step further: We combine several layers of higher-order models up to a maximum order of $k$ to a single <strong>multi-order model</strong>. We can fit such a multi-order model to a given set of pathways using the <code>MultiOrderModel</code> class of <code>pathpy</code>. We can set the maximum order $maxOrder$ up to which higher-order models should be generated. If we don't specify a maximum order, the model will contain all possible higher-order models up to the maximum path length in the data.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[22]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tfinished.\nMulti-order model (max. order = 2, DoF (paths/ngrams) = 7/124)\n===========================================================================\nLayer k = 0\t6 nodes, 5 links, 49 paths, DoF (paths/ngrams) = 4/4\nLayer k = 1\t5 nodes, 4 links, 30 paths, DoF (paths/ngrams) = 1/20\nLayer k = 2\t4 nodes, 2 links, 11 paths, DoF (paths/ngrams) = 2/100\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Again, printing the instance prints a human-readable summary of the underlying model. Here it actually combines three layers from order zero (which simply captures \"activation frequencies\" of nodes) up to the maximum order of two. Each layer $k$ is simply the $k$-th order model introduced above. We can verify this by plotting the corresponding <code>HigherOrderNetwork</code> instances which are stored in the dictionary <code>layers</code> of the <code>MultiOrderNetwork</code> instance.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[23]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">g2</span> <span class=\"o\">=</span> <span class=\"n\">Network2igraph</span><span class=\"p\">(</span><span class=\"n\">m</span><span class=\"o\">.</span><span class=\"n\">layers</span><span class=\"p\">[</span><span class=\"mi\">2</span><span class=\"p\">])</span>\n\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;layout&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g2</span><span class=\"o\">.</span><span class=\"n\">layout_auto</span><span class=\"p\">()</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;vertex_label&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g2</span><span class=\"o\">.</span><span class=\"n\">vs</span><span class=\"p\">[</span><span class=\"s2\">&quot;name&quot;</span><span class=\"p\">]</span>\n<span class=\"n\">visual_style</span><span class=\"p\">[</span><span class=\"s2\">&quot;edge_label&quot;</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"n\">g2</span><span class=\"o\">.</span><span class=\"n\">es</span><span class=\"p\">[</span><span class=\"s2\">&quot;weight&quot;</span><span class=\"p\">]</span>\n\n<span class=\"n\">igraph</span><span class=\"o\">.</span><span class=\"n\">plot</span><span class=\"p\">(</span><span class=\"n\">g2</span><span class=\"p\">,</span> <span class=\"s1\">&#39;pathpy_tutorial/g2.png&#39;</span><span class=\"p\">,</span> <span class=\"o\">**</span><span class=\"n\">visual_style</span><span class=\"p\">)</span>\n<span class=\"n\">display</span><span class=\"p\">(</span><span class=\"n\">Image</span><span class=\"p\">(</span><span class=\"n\">filename</span><span class=\"o\">=</span><span class=\"s1\">&#39;pathpy_tutorial/g2.png&#39;</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n\n\n<div class=\"output_png output_subarea \">\n<img 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SQq7PT09vJ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\nJHt7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class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"network\"></a></p>\n<h2 id=\"6.-When-is-a-network-a-network?\">6. When is a network a network?<a class=\"anchor-link\" href=\"#6.-When-is-a-network-a-network?\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>What can we do with such a multi-order graphical model of a given set of pathways? We can take a <strong>model selection perspective</strong>, and ask how many layers of higher-order models are needed to model a given set of pathways. In other words, we are interested in the optimal maximum order $K_{opt}$ of a multi-order graphical model needed to model a given data set. By \"optimal\" we refer to the number of layers minimally needed to best explain the observed pathway statistics, however considering the increase in <strong>model complexity</strong> when adding additional layers.</p>\n<p>This optimal maximum order $K_{opt}$ has an interesting interpretation: If for a data set we infer $K_{opt}=1$, this means that there are no significant deviations from the transitivity assumption made by a network representation that would justify the inclusion of higher-order graphical models. In other words: It is - from a model selection perspective - justified to study the underlying system as a network. However, if we find $K_{opt}>1$ this means that the application of a network abstraction (and likewise the use of network-analytic or algebraic methods) is misleading. <strong>Calculating $K_{opt}$ thus allows to answer the crucial question whether a data set should be modeled as a network or not!</strong></p>\n<p>But how can we calculate what is the \"optimal\" order? And what do we mean when we say that this optimal order balances \"model complexity\" and \"explanatory power\". For this, we take a statistical inference view and calculate the likelihoods of multi-order models with different maximum orders under the observed data. Again, if you you are interested in mathematical details please refer to <a href=\"https://arxiv.org/abs/1702.05499\">this recent research paper</a>.</p>\n<p>Here, it is enough to say that we can conveniently calculate the (log-)likelihood of a multi-order model using the likelihood function of the class <code>MultiOrderModel</code>. Let us try this with a multi-order model for our toy example that has a maximum order of one.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[24]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m1</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Likelihood = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m1</span><span class=\"o\">.</span><span class=\"n\">getLikelihood</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tfinished.\nLikelihood =  1.97212806634e-19\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We get a likelihood of the model of $\\approx 1.97 \\cdot 10^{-19}$. Let us compare this to the likelihood of a multi-order model that adds a layer with a second-order model:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[25]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m2</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Likelihood = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m2</span><span class=\"o\">.</span><span class=\"n\">getLikelihood</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tfinished.\nLikelihood =  4.03891827985e-16\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Quite naturally, adding a second layer increases the likelihood, i.e. we have increased the explanatory power of the model for our data data. This may trick us into thinking that the second model is the better one. However, we should also take into account that, by adding an additional layer, we make the model more complex. Applying <a href=\"https://en.wikipedia.org/wiki/Occam's_razor\">Occam's razor</a> we should instead search for the <strong>simplest model</strong> which still has reasonable explanatory power, i.e. we should not make the model more complex than neccessary.</p>\n<p>The key to a principled decision about the optimal maximum order is to correctly account for the complexity of the model in terms of its <strong>degrees of freedom</strong>, i.e. the number of free parameters that we have fitted to the data. The correct calculation of this number for any graph topology and any order $k$ is one of the main contributions of <a href=\"https://arxiv.org/abs/1702.05499\">this work</a>.</p>\n<p>Luckily, you won't have to deal with this because <code>pathpy</code> automatically takes care of it for you. First of all, we can  check the degrees of freedom of our two candidate models simply by printing the model instances.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[26]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">m1</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">m2</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Multi-order model (max. order = 1, DoF (paths/ngrams) = 5/24)\n===========================================================================\nLayer k = 0\t6 nodes, 5 links, 49 paths, DoF (paths/ngrams) = 4/4\nLayer k = 1\t5 nodes, 4 links, 30 paths, DoF (paths/ngrams) = 1/20\n\nMulti-order model (max. order = 2, DoF (paths/ngrams) = 7/124)\n===========================================================================\nLayer k = 0\t6 nodes, 5 links, 49 paths, DoF (paths/ngrams) = 4/4\nLayer k = 1\t5 nodes, 4 links, 30 paths, DoF (paths/ngrams) = 1/20\nLayer k = 2\t4 nodes, 2 links, 11 paths, DoF (paths/ngrams) = 2/100\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>The number that matters here is the first number indicated after Dof (corresponding to paths). The model with maximum order one has five degrees of freedom, while the model of maximum order two has seven degrees of freedom. So the model with maximum order two is more complex than the one with maximum order one because we fit two additional parameters (for the layer of order two).</p>\n<p>Again, omitting mathematical details and referring to <a href=\"https://arxiv.org/abs/1702.05499\">this research paper</a> it turns out that we can apply <a href=\"https://en.wikipedia.org/wiki/Likelihood-ratio_test#Wilks.27_theorem\">Wilk's theorem</a> to perform a series of <a href=\"https://en.wikipedia.org/wiki/Likelihood-ratio_test\">likelihood ratio tests</a> in order to determine which maximum order is optimal, while considering the added complexity of higher-order models. And <code>pathpy</code> does all of this for you! We can simply call the function <code>estimateOrder</code>, which returns the optimal maximum order $K_{opt}$ that we are looking for:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[27]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Optimal maximum order = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m2</span><span class=\"o\">.</span><span class=\"n\">estimateOrder</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tLikelihood ratio test for K_opt = 2, x = 22.0\n2017-03-01 17:17:17 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 2\n2017-03-01 17:17:17 [Severity.INFO]\tLikelihood ratio test, p = 1.67017007903e-05\nOptimal maximum order =  2\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>For our toy example, we see that a (first-order) network abstraction would be misleading. We actually need the second-order graphical model layer, i.e. the added complexity is justified considered the increase in explanatory power for the observed pathways.</p>\n<p>Let us consider another toy example, where pathway statistics are actually <em>exactly</em> as we expect it to be under the assumption that paths in the first-order network are transitive:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[28]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"p\">()</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Number of paths (unique/sub paths/total):\t20 (8/76/96)\nNodes:\t\t\t\t5\nEdges:\t\t\t\t4\nMax. path length:\t\t2\nAvg path length:\t\t1.6\nPaths of length k = 0\t\t0 (0/52/52)\nPaths of length k = 1\t\t8 (4/24/32)\nPaths of length k = 2\t\t12 (4/0/12)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>In this special (and quite artificial case) the model with maximum order one and the model with maximum order two have exactly the same likelihoods. I.e., adding the second-order model layer provides no benefit in terms of explanatory power. The reason for this is that there are no correlations in the data that violate the transitivity assumption. We can confirm this as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[29]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m1</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Likelihood = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m1</span><span class=\"o\">.</span><span class=\"n\">getLikelihood</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tfinished.\nLikelihood =  3.28861452936e-20\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[30]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m2</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Likelihood = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m2</span><span class=\"o\">.</span><span class=\"n\">getLikelihood</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:17:17 [Severity.INFO]\tfinished.\nLikelihood =  3.28861452936e-20\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Since the model <code>m2</code> is more complex, the <code>estimateOrder</code> function will actually reject the more complex model, correctly determining that <strong>a network abstraction of this set of pathways is actually justified</strong>.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[31]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Optimal maximum order = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m2</span><span class=\"o\">.</span><span class=\"n\">estimateOrder</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tLikelihood ratio test for K_opt = 2, x = -0.0\n2017-03-01 17:17:17 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 2\n2017-03-01 17:17:17 [Severity.INFO]\tLikelihood ratio test, p = 1.0\nOptimal maximum order =  1\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>We can vary this example in a way that we slightly distort the statistics of paths, violating the transitivity assumption, but just by a little bit. Precisely, we overrepresent one of the four paths of length two by one occurrence, compared to what we would expect based on the relative frequencies of links:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[32]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"p\">()</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,d&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;b,c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"n\">paths</span><span class=\"o\">.</span><span class=\"n\">addPath</span><span class=\"p\">(</span><span class=\"s1\">&#39;a,c,e&#39;</span><span class=\"p\">,</span> <span class=\"n\">pathFrequency</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">)</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Number of paths (unique/sub paths/total):\t21 (8/81/102)\nNodes:\t\t\t\t5\nEdges:\t\t\t\t4\nMax. path length:\t\t2\nAvg path length:\t\t1.61904761905\nPaths of length k = 0\t\t0 (0/55/55)\nPaths of length k = 1\t\t8 (4/26/34)\nPaths of length k = 2\t\t13 (4/0/13)\n\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Let us again calculate the likelihoods for the models with maximum order one and two respectively:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[33]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m1</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">1</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Likelihood = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m1</span><span class=\"o\">.</span><span class=\"n\">getLikelihood</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:17 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tfinished.\nLikelihood =  2.8147122155e-21\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[34]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">m2</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Likelihood = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m2</span><span class=\"o\">.</span><span class=\"n\">getLikelihood</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">log</span><span class=\"o\">=</span><span class=\"kc\">False</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:18 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tfinished.\nLikelihood =  2.91588212235e-21\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Here, the likelihood of the more complex model is slightly larger, because the additional second-order layer captures the overrepresentation of the path $(a\\rightarrow c \\rightarrow e)$ which is not expected based on the first-order layer. However, since the model is also more complex, our method correctly determines that the small gain in likelihood does not justify the associated increase in model complexity, thus determining that a first-order network abstraction of this data set is optimal.</p>\n<p>In fact, from the output below you can see that we can calculate a $p$-value, which allows us to reject the (alternative) hypothesis that includes a second-order model layer in favor of the (null) hypothesis of the simpler model. We can actually set the significance threshold for the underlying likelihood ratio test as follows:</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[35]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Optimal maximum order = &#39;</span><span class=\"p\">,</span> <span class=\"n\">m2</span><span class=\"o\">.</span><span class=\"n\">estimateOrder</span><span class=\"p\">(</span><span class=\"n\">paths</span><span class=\"p\">,</span> <span class=\"n\">significanceThreshold</span><span class=\"o\">=</span><span class=\"mf\">0.001</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:18 [Severity.INFO]\tLikelihood ratio test for K_opt = 2, x = 0.101889947837\n2017-03-01 17:17:18 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 2\n2017-03-01 17:17:18 [Severity.INFO]\tLikelihood ratio test, p = 0.950330962083\nOptimal maximum order =  1\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Let us finally go beyond synthetic toy examples and test our method in the data sets which we have imported above. We start with the pathway data capturing travel patterns in the London Tube system. Here our method finds that a <strong>network abstraction of the London Tube is misleading</strong>. It actually finds that we need to consider higher-order model layers up to order six, while the layer of order seven is not significant.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[36]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">tube_model</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">tube_paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">7</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Optimal maximum order = &#39;</span><span class=\"p\">,</span> <span class=\"n\">tube_model</span><span class=\"o\">.</span><span class=\"n\">estimateOrder</span><span class=\"p\">(</span><span class=\"n\">tube_paths</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:17:18 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:17:18 [Severity.INFO]\tGenerating 3-th order network layer ...\n2017-03-01 17:17:19 [Severity.INFO]\tGenerating 4-th order network layer ...\n2017-03-01 17:17:20 [Severity.INFO]\tGenerating 5-th order network layer ...\n2017-03-01 17:17:22 [Severity.INFO]\tGenerating 6-th order network layer ...\n2017-03-01 17:17:25 [Severity.INFO]\tGenerating 7-th order network layer ...\n2017-03-01 17:17:29 [Severity.INFO]\tfinished.\n2017-03-01 17:18:17 [Severity.INFO]\tLikelihood ratio test for K_opt = 2, x = 46432008.9276\n2017-03-01 17:18:17 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 1659\n2017-03-01 17:18:17 [Severity.INFO]\tLikelihood ratio test, p = 0.0\n2017-03-01 17:19:19 [Severity.INFO]\tLikelihood ratio test for K_opt = 3, x = 1484643.01227\n2017-03-01 17:19:19 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 5619\n2017-03-01 17:19:19 [Severity.INFO]\tLikelihood ratio test, p = 0.0\n2017-03-01 17:20:21 [Severity.INFO]\tLikelihood ratio test for K_opt = 4, x = 679447.231163\n2017-03-01 17:20:21 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 19527\n2017-03-01 17:20:21 [Severity.INFO]\tLikelihood ratio test, p = 0.0\n2017-03-01 17:21:30 [Severity.INFO]\tLikelihood ratio test for K_opt = 5, x = 365515.096382\n2017-03-01 17:21:30 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 67834\n2017-03-01 17:21:30 [Severity.INFO]\tLikelihood ratio test, p = 0.0\n2017-03-01 17:22:44 [Severity.INFO]\tLikelihood ratio test for K_opt = 6, x = 406258.608459\n2017-03-01 17:22:44 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 246738\n2017-03-01 17:22:44 [Severity.INFO]\tLikelihood ratio test, p = 0.0\n2017-03-01 17:24:02 [Severity.INFO]\tLikelihood ratio test for K_opt = 7, x = 160119.246725\n2017-03-01 17:24:02 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 903486\n2017-03-01 17:24:02 [Severity.INFO]\tLikelihood ratio test, p = 1.0\nOptimal maximum order =  6\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Despite using a data set with more than 4 million paths and generating higher-order graphical models up to order seven, the whole testing procedure only takes a few minutes (on a six year old laptop computer).</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>What is great about the unified approach to pathway and temporal network analysis, is that we can directly apply this method to temporal network data. Here we can test whether the time-respecting path statistics which we extracted from the workplace data set above justifies a network abstraction. This specifically allows us to <strong>test whether there are temporal correlations in the sequence of time-stamped interactions which invalidate a (static, first-order) network abstraction</strong>.</p>\n<p>We can do this with two (!) lines of python code, testing for the significance of mode layers up to order three.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[37]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">work_model</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">work_paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">3</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Optimal maximum order = &#39;</span><span class=\"p\">,</span> <span class=\"n\">work_model</span><span class=\"o\">.</span><span class=\"n\">estimateOrder</span><span class=\"p\">(</span><span class=\"n\">work_paths</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:24:02 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:24:02 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:24:02 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:24:02 [Severity.INFO]\tGenerating 3-th order network layer ...\n2017-03-01 17:24:02 [Severity.INFO]\tfinished.\n2017-03-01 17:24:03 [Severity.INFO]\tLikelihood ratio test for K_opt = 2, x = 6538.14557914\n2017-03-01 17:24:03 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 4521\n2017-03-01 17:24:03 [Severity.INFO]\tLikelihood ratio test, p = 0.0\n2017-03-01 17:24:03 [Severity.INFO]\tLikelihood ratio test for K_opt = 3, x = 123.324081918\n2017-03-01 17:24:03 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 20439\n2017-03-01 17:24:03 [Severity.INFO]\tLikelihood ratio test, p = 1.0\nOptimal maximum order =  2\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Again, we find that a <strong>network abstraction of the time-stamped interactions in this data set is misleading</strong>. Here we actually need to add a second-order model to explain the observed time-respecting paths, while including an additional third-order model layer is not justified!</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>Let us conclude our analysis by studying a third data set, which captures <a href=\"http://link.springer.com/chapter/10.1007/978-3-642-21863-7_17\">time-stamped E-Mail exchanges in a Polish manufacturing company</a>. We again extract time-respecting paths for a given maximum time difference $\\delta$, create a multi-order model and detect the optimal maximum order, all in just <strong>four lines of python code which only require six seconds of computation!</strong></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[38]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span><span class=\"n\">email_t</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">TemporalNetwork</span><span class=\"o\">.</span><span class=\"n\">readFile</span><span class=\"p\">(</span><span class=\"s1\">&#39;pathpy_tutorial/CompanyEmails.tedges&#39;</span><span class=\"p\">,</span> <span class=\"n\">sep</span><span class=\"o\">=</span><span class=\"s1\">&#39;</span><span class=\"se\">\\t</span><span class=\"s1\">&#39;</span><span class=\"p\">)</span>\n<span class=\"n\">email_paths</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">Paths</span><span class=\"o\">.</span><span class=\"n\">fromTemporalNetwork</span><span class=\"p\">(</span><span class=\"n\">email_t</span><span class=\"p\">,</span> <span class=\"n\">delta</span><span class=\"o\">=</span><span class=\"mi\">30</span><span class=\"p\">)</span>\n\n<span class=\"n\">email_model</span> <span class=\"o\">=</span> <span class=\"n\">pp</span><span class=\"o\">.</span><span class=\"n\">MultiOrderModel</span><span class=\"p\">(</span><span class=\"n\">email_paths</span><span class=\"p\">,</span> <span class=\"n\">maxOrder</span><span class=\"o\">=</span><span class=\"mi\">2</span><span class=\"p\">)</span>\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"s1\">&#39;Optimal maximum order = &#39;</span><span class=\"p\">,</span> <span class=\"n\">email_model</span><span class=\"o\">.</span><span class=\"n\">estimateOrder</span><span class=\"p\">(</span><span class=\"n\">email_paths</span><span class=\"p\">))</span>\n</pre></div>\n\n</div>\n</div>\n</div>\n\n<div class=\"output_wrapper\">\n<div class=\"output\">\n\n\n<div class=\"output_area\"><div class=\"prompt\"></div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>2017-03-01 17:24:03 [Severity.INFO]\tReading time-stamped links ...\n2017-03-01 17:24:03 [Severity.INFO]\tBuilding index data structures ...\n2017-03-01 17:24:04 [Severity.INFO]\tSorting time stamps ...\n2017-03-01 17:24:04 [Severity.INFO]\tfinished.\n2017-03-01 17:24:04 [Severity.INFO]\tExtracting time-respecting paths for delta = 30 ...\n2017-03-01 17:24:05 [Severity.INFO]\tCalculating sub path statistics ... \n2017-03-01 17:24:05 [Severity.INFO]\tfinished.\n2017-03-01 17:24:05 [Severity.INFO]\tGenerating 0-th order network layer ...\n2017-03-01 17:24:05 [Severity.INFO]\tGenerating 1-th order network layer ...\n2017-03-01 17:24:06 [Severity.INFO]\tGenerating 2-th order network layer ...\n2017-03-01 17:24:07 [Severity.INFO]\tfinished.\n2017-03-01 17:24:09 [Severity.INFO]\tLikelihood ratio test for K_opt = 2, x = 21871.1297023\n2017-03-01 17:24:09 [Severity.INFO]\tLikelihood ratio test, d_1-d_0 = 329140\n2017-03-01 17:24:09 [Severity.INFO]\tLikelihood ratio test, p = 1.0\nOptimal maximum order =  1\n</pre>\n</div>\n</div>\n\n</div>\n</div>\n\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>For this data set, we find that a <strong>first-order network abstraction is sufficient to explain time-respecting paths</strong>, i.e. here temporal correlations in the data do not justify the added complexity of higher-order graphical models.</p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><a name=\"conclusion\"></a></p>\n<h2 id=\"7.-Conclusion\">7. Conclusion<a class=\"anchor-link\" href=\"#7.-Conclusion\">&#182;</a></h2><p><em><a href=\"#outline\">Back to outline</a></em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\">\n</div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>In conclusion, above I have shown how you can <strong>test whether a network abstraction of sequential data on pathways and temporal networks is justified or not</strong>. With <code>pathpy</code>, answering this crucial question is as simple as it can get. You won't need more than 3-4 lines of simple python code. Hence, a <strong>multi-order analysis with <code>pathpy</code> is an extremely simple but crucial first step that should precede an application of network-based data mining and modeling techniques!</strong>.</p>\n<p>Moreover, <code>pathpy</code> allows to infer multi-order graphical models whose layers can be interpreted as higher-order networks along the lines presented in the <a href=\"#references\">works above</a>. While the argumentation is too long to be included in this tutorial, <a href=\"https://arxiv.org/abs/1702.05499\">this recent work</a> proves that the <strong>inferred maximum order is the optimal order of a graphical abstraction of sequential data</strong>, e.g. when it comes to the calculation of <code>PageRank</code> centralities or prediction tasks. Moreover, <a href=\"http://www.ingoscholtes.net/research/insights/Temporal_Networks.html\">this tutorial</a> illustrates and visualizes that such higher-order network topologies are crucial to accurately model and predict dynamical processes such as diffusion or epidemic spreading.</p>\n<p>I encourage you to get started with <code>pathpy</code> and to apply it to your data! If you have any problems, questions or suggestions, feel free to contact me.</p>\n<p><em>Ingo Scholtes</em></p>\n<p><em>Zurich, February 23 2017</em></p>\n\n</div>\n</div>\n</div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In&nbsp;[&nbsp;]:</div>\n<div class=\"inner_cell\">\n    <div class=\"input_area\">\n<div class=\" highlight hl-ipython3\"><pre><span></span> \n</pre></div>\n\n</div>\n</div>\n</div>\n\n</div>\n    </div>\n  </div>\n</body>\n</html>\n"
  },
  {
    "path": "docs/tutorial.ipynb",
    "content": "{\n \"cells\": [\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"# When is a network a network? Multi-Order Model Selection in Pathways and Temporal Networks\\n\",\n    \"\\n\",\n    \"### An educational tutorial introducing the OpenSource `python` package [`pathpy`](https://github.com/IngoScholtes/pathpy)\\n\",\n    \"\\n\",\n    \"Ingo Scholtes  \\n\",\n    \"[Chair of Systems Design](http://www.sg.ethz.ch)  \\n\",\n    \"ETH Zürich  \\n\",\n    \"\\n\",\n    \"*February 23 2017*\\n\",\n    \"\\n\",\n    \"## Summary\\n\",\n    \"\\n\",\n    \"This educational tutorial introduces the **analysis of sequential data using multi-order graphical models**, based on the python package [`pathpy`](https://github.com/IngoScholtes/pathpy).\\n\",\n    \"\\n\",\n    \"<a name=\\\"references\\\"></a>\\n\",\n    \"\\n\",\n    \"The theoretical foundation of this package has been outlined in the recent paper:  \\n\",\n    \"\\n\",\n    \"- I Scholtes: [When is a network a network? Multi-Order Graphical Model Selection in Pathways and Temporal Networks](https://arxiv.org/abs/1702.05499), arXiv:1702.05499, February 2017\\n\",\n    \"\\n\",\n    \"Moreover, it builds on **higher-order network abstractions** of time-stamped data, as well as **high-order centrality measures** developed in: \\n\",\n    \"\\n\",\n    \"- I Scholtes, N Wider, A Garas: [Higher-Order Aggregate Networks in the Analysis of Temporal Networks: Path structures and centralities](http://link.springer.com/article/10.1140%2Fepjb%2Fe2016-60663-0), The European Physical Journal B, 89:61, March 2016  \\n\",\n    \"- I Scholtes, N Wider, R Pfitzner, A Garas, CJ Tessone, F Schweitzer: [Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks](http://dx.doi.org/10.1140/epjb/e2016-60663-0), Nature Communications, 5, September 2014  \\n\",\n    \"- R Pfitzner, I Scholtes, A Garas, CJ Tessone, F Schweitzer: [Betweenness preference: Quantifying correlations in the topological dynamics of temporal networks](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.198701), Phys Rev Lett, 110(19), 198701, May 2013  \\n\",\n    \"\\n\",\n    \"A key feature of `pathpy` is that it provides a unified approach to the analysis of pathways and temporal networks. Let us first define what we mean by pathway data. We consider sequential data of the form ... \\n\",\n    \"\\n\",\n    \"$(a \\\\rightarrow b)$  \\n\",\n    \"$(b \\\\rightarrow c)$  \\n\",\n    \"$(a \\\\rightarrow b \\\\rightarrow c)$  \\n\",\n    \"$(b \\\\rightarrow c \\\\rightarrow a)$  \\n\",\n    \"$(a \\\\rightarrow b \\\\rightarrow c \\\\rightarrow d)$  \\n\",\n    \"$(c)$  \\n\",\n    \"\\n\",\n    \"... which capture multiple (typically short) paths with varying lengths, observed in a network topology. Such data are relevant in a number of data mining scenarios: Consider, for instance, click streams of multiple users in the Web. Each line above could be the navigation path of a user in a Web graph. Considering biological pathways, each line could be one activation sequence of genes observed in a cell. In social media, paths could be traces of information propagating through a social network. Finally, we will show that pathway data also naturally emerge in time-stamped interaction data, which makes `pathpy` **particularly useful for those studying temporal networks**.\\n\",\n    \"\\n\",\n    \"In this tutorial we show that such sequential data allow us to provide a principled answer to the crucial question: **Is it justified to model a system as a network**, i.e. can we apply graph-theoretic or network-analytic methods to a relational data set?  \\n\",\n    \"\\n\",\n    \"Apart from answering this important question, `pathpy` allows to infer **optimal higher-order graphical models** which generalize the commonly used network abstraction and facilitate the analysis of sequential data. \\n\",\n    \"\\n\",\n    \"The outline of this tutorial is as follows:\\n\",\n    \"\\n\",\n    \"<a name=\\\"outline\\\"></a>\\n\",\n    \"1. [Setting up pathpy](#setup)\\n\",\n    \"2. [Getting started: the Paths object](#paths)\\n\",\n    \"3. [Analyzing temporal networks: the TemporalNetwork class](#temporal)\\n\",\n    \"4. [Importing data on pathways and temporal networks](#data)\\n\",\n    \"5. [Multi-Order graphical models of pathways and temporal networks](#multiorder)\\n\",\n    \"6. [When is a network a network?](#network)\\n\",\n    \"7. [Conclusion](#conclusion)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"setup\\\"></a>\\n\",\n    \"## 1. Setting up `pathpy`\\n\",\n    \"*[Back to outline](#outline)*\\n\",\n    \"\\n\",\n    \"Before diving into the theoretical foundation of our framework, let us first install and setup `pathpy`. The package consist of pure `python` code which means that there are no platform-specific dependencies that complicate the setup. The only requirement is that you have a `python` interpreter (version 3 and above) as well as the packages `numpy` and `scipy`, which are used for mathematical calculations.  \\n\",\n    \"\\n\",\n    \"The code of `pathpy` is available at [gitHub](https://github.com/IngoScholtes/pathpy). Downloading and installing the latest version of `pathpy` is simple. Just fire up a console and type: \\n\",\n    \"\\n\",\n    \"`> pip install git+git://github.com/IngoScholtes/pathpy.git`\\n\",\n    \"\\n\",\n    \"This will download and install the latest development version of `pathpy` and its dependencies. If you want to install a specific [release version](https://github.com/IngoScholtes/pathpy/releases), you can type: \\n\",\n    \"\\n\",\n    \"`> pip install https://github.com/IngoScholtes/pathpy/archive/VERSIONTAG.zip`  \\n\",\n    \"\\n\",\n    \"where `VERSIONTAG` is the tag of the release. In this tutorial, we use the first beta release `v1.0-beta.1`, so we run \\n\",\n    \"\\n\",\n    \"`> pip install https://github.com/IngoScholtes/pathpy/archive/v1.0-beta.1.zip`\\n\",\n    \"\\n\",\n    \"While `pathpy` does not depend on any specific graph library, for illustration purposes this tutorial will use network visualizations generated by `python-igraph`. We can set this up by running:\\n\",\n    \"\\n\",\n    \"`> pip install python-igraph`\\n\",\n    \"\\n\",\n    \"However, `igraph` is not needed to use `pathpy` unless you wan to visualize higher-order graphical models. Now that everything is installed, we can import `pathpy`, `numpy` and `igraph` in our script. We will also use the `IPython.display` function to plot some figures later, so let us import that as well.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 1,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"import igraph\\n\",\n    \"import numpy as np\\n\",\n    \"import pathpy as pp\\n\",\n    \"\\n\",\n    \"from IPython.display import *\\n\",\n    \"from IPython.display import HTML\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"paths\\\"></a>\\n\",\n    \"## 2. Getting started: the `Paths` object\\n\",\n    \"*[Back to outline](#outline)*\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"One of the key objects in `pathpy` is the `Paths` class. It can be used to import, manipulate and analyze pathways like in the example above. As we will see later, we can also use it to generate pathways from temporal networks. For now, let us create an empty `Paths` instance as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 2,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"paths = pp.Paths()\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can manually add paths to this object using the function `addPath`. Since all classes and functions in `pathpy` are documented using so-called `python docstrings` we can use `python`'s interactive help system to print the documentation of this function.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 3,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Help on method addPath in module pathpy.Paths:\\n\",\n      \"\\n\",\n      \"addPath(ngram, separator=',', expandSubPaths=True, pathFrequency=None) method of pathpy.Paths.Paths instance\\n\",\n      \"    Adds the path(s) of a single n-gram to the path statistics object.\\n\",\n      \"    \\n\",\n      \"    @param ngram: An ngram representing a path between nodes, separated by the separator character, e.g. \\n\",\n      \"        the 4-gram a;b;c;d represents a path of length three (with separator ';')\\n\",\n      \"    \\n\",\n      \"    @param separator: The character used as separator for the ngrams (';' by default)\\n\",\n      \"    \\n\",\n      \"    @param expandSubPaths: by default all subpaths of the given ngram are generated, i.e. \\n\",\n      \"        for the trigram a;b;c a path a->b->c of length two will be generated \\n\",\n      \"        as well as two subpaths a->b and b->c of length one\\n\",\n      \"    \\n\",\n      \"    @weight weight: the weight (i.e. frequency) of the ngram\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"help(paths.addPath)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"The function takes a string argument, as well as a separator character. The string is an *n-gram* which consists of $n$ symbols separated by the separator character (default: \\\",\\\"). Each symbol in this n-gram represents a node or vertex traversed by a path of length $n-1$ (we define the length of a path as the number of links it traverses). So the pathways from the example above can be generated as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 4,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"paths.addPath('a,b')\\n\",\n    \"paths.addPath('b,c')\\n\",\n    \"paths.addPath('a,b,c')\\n\",\n    \"paths.addPath('b,c,a')\\n\",\n    \"paths.addPath('a,b,c,d')\\n\",\n    \"paths.addPath('c')\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can print the `Paths` instance to get a human-readable summary:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 5,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Number of paths (unique/sub paths/total):\\t6 (6/23/29)\\n\",\n      \"Nodes:\\t\\t\\t\\t4\\n\",\n      \"Edges:\\t\\t\\t\\t4\\n\",\n      \"Max. path length:\\t\\t3\\n\",\n      \"Avg path length:\\t\\t1.5\\n\",\n      \"Paths of length k = 0\\t\\t1 (1/14/15)\\n\",\n      \"Paths of length k = 1\\t\\t2 (2/7/9)\\n\",\n      \"Paths of length k = 2\\t\\t2 (2/2/4)\\n\",\n      \"Paths of length k = 3\\t\\t1 (1/0/1)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print(paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"This instance contains six paths: One path consists of a single node and has length zero (remember the path length is the number of traverse links). Two paths have length one and two respectively, and a single path has length three. The network in which these paths occur has four nodes $a$, $b$, $c$, and $d$ connected by four directed links/edges $(a,b)$, $(b,c)$, $(c,a)$, and $(c,d)$. \\n\",\n    \"\\n\",\n    \"You will notice a group of three numbers after each path length. The first number counts unique ocurrences of paths. This becomes clear if we add a second occurrence of path $(a \\\\rightarrow b)$:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 6,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Number of paths (unique/sub paths/total):\\t7 (6/25/32)\\n\",\n      \"Nodes:\\t\\t\\t\\t4\\n\",\n      \"Edges:\\t\\t\\t\\t4\\n\",\n      \"Max. path length:\\t\\t3\\n\",\n      \"Avg path length:\\t\\t1.42857142857\\n\",\n      \"Paths of length k = 0\\t\\t1 (1/16/17)\\n\",\n      \"Paths of length k = 1\\t\\t3 (2/7/10)\\n\",\n      \"Paths of length k = 2\\t\\t2 (2/2/4)\\n\",\n      \"Paths of length k = 3\\t\\t1 (1/0/1)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"paths.addPath('a,b')\\n\",\n    \"print(paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We now have a total of seven observations of six unique paths (one path occuring twice). Rather than adding multiple observations by hand, we can actually set frequencies (or weights) of paths. So, rather than adding $(a \\\\rightarrow b)$ twice we could have written the following to get the same result:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 7,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Number of paths (unique/sub paths/total):\\t7 (6/25/32)\\n\",\n      \"Nodes:\\t\\t\\t\\t4\\n\",\n      \"Edges:\\t\\t\\t\\t4\\n\",\n      \"Max. path length:\\t\\t3\\n\",\n      \"Avg path length:\\t\\t1.42857142857\\n\",\n      \"Paths of length k = 0\\t\\t1 (1/16/17)\\n\",\n      \"Paths of length k = 1\\t\\t3 (2/7/10)\\n\",\n      \"Paths of length k = 2\\t\\t2 (2/2/4)\\n\",\n      \"Paths of length k = 3\\t\\t1 (1/0/1)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"paths = pp.Paths()\\n\",\n    \"paths.addPath('a,b', pathFrequency=2)\\n\",\n    \"paths.addPath('b,c')\\n\",\n    \"paths.addPath('a,b,c')\\n\",\n    \"paths.addPath('b,c,a')\\n\",\n    \"paths.addPath('a,b,c,d')\\n\",\n    \"paths.addPath('c')\\n\",\n    \"print(paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"What about the second number in the group of three numbers? It counts the number of so-called sub-paths. For a path $p=(a \\\\rightarrow b \\\\rightarrow c \\\\rightarrow d)$ we call any sequence of nodes $q$ which is contained in $p$ a sub-path of $p$. I.e. $q_1=(a \\\\rightarrow b)$, $q_2=(b \\\\rightarrow c \\\\rightarrow d)$ or the single node $q_3=(b)$ are sub-paths of path $p$.\\n\",\n    \"\\n\",\n    \"Correctly accounting for sub-paths of any length is crucial for our graphical modeling framework. Whenever we add a path (using the default parameter `expandSubPaths=True`) all sub-paths of the added path will be automatically calculated. In the example above, this means that there are a total of $16$ sub-paths of length zero (single nodes). Note that $(a\\\\rightarrow b)$ is occurring twice and $(c)$ is an actual path with length zero for which no additional sub-path is counted. Similarly, there are seven sub-paths of length one, etc.\\n\",\n    \"\\n\",\n    \"We can access paths via a public dictionary `paths`, which contains the list (and frequencies) of paths of any length. For the example above, we can, e.g., access paths of length two as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 8,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"text/plain\": [\n       \"defaultdict(<function pathpy.Paths.Paths.__init__.<locals>.<lambda>.<locals>.<lambda>>,\\n\",\n       \"            {('a', 'b', 'c'): array([1, 1]),\\n\",\n       \"             ('b', 'c', 'a'): array([0, 1]),\\n\",\n       \"             ('b', 'c', 'd'): array([1, 0])})\"\n      ]\n     },\n     \"execution_count\": 8,\n     \"metadata\": {},\n     \"output_type\": \"execute_result\"\n    }\n   ],\n   \"source\": [\n    \"paths.paths[2]\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"This returns a dictionary which contains three different paths of length two, each of which is associated with a tuple consisting of two numbers. The first counts the **actual path observations**, the second number counts the **occurrences of a path as a sub-path** of a longer path observation. In the example above, path $(a \\\\rightarrow b \\\\rightarrow c)$ has been observed one time as an actual path, and one time as a sub-path of the longer path $(a\\\\rightarrow b \\\\rightarrow c \\\\rightarrow d)$. The path $(b \\\\rightarrow c \\\\rightarrow a)$ never occurs as sub-path, while $(b \\\\rightarrow c \\\\rightarrow d)$ only occurs as a sub path of $(a \\\\rightarrow b \\\\rightarrow c \\\\rightarrow d)$.\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"temporal\\\"></a>\\n\",\n    \"## 3. Analyzing temporal networks: the `TemporalNetwork` class\\n\",\n    \"*[Back to outline](#outline)*\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"A key design principle behind `pathpy` is that it unifies the analysis of pathway data (introduced above) and **time-stamped data on temporal networks**. We consider a temporal network as a collection of triplets of the form $(a,b;t)$ which capture that a node $a$ was connected to node $b$ (via a directed link) at a discrete time $t$. Such time-stamped data are of increasing importance, for instance when studying time-stamped interactions in a social network.  \\n\",\n    \"\\n\",\n    \"We can represent such time-stamped network data using `pathpy`'s `TemporalNetwork` class. Let us create an empty instance:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 9,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"t = pp.TemporalNetwork()\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can add time-stamped edges to this temporal network in any order:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 10,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"t.addEdge(source='a', target='b', ts=42)\\n\",\n    \"t.addEdge(source='b', target='c', ts=21)\\n\",\n    \"t.addEdge(source='c', target='d', ts=51)\\n\",\n    \"t.addEdge(source='b', target='c', ts=44)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Again, printing the instance will return a human-readable summary:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 11,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Nodes:\\t\\t\\t4\\n\",\n      \"Time-stamped links:\\t4\\n\",\n      \"Links/Nodes:\\t\\t1.0\\n\",\n      \"Observation period:\\t[21, 51]\\n\",\n      \"Observation length:\\t30\\n\",\n      \"Time stamps:\\t\\t4\\n\",\n      \"Avg. inter-event dt:\\t10.0\\n\",\n      \"Min/Max inter-event dt:\\t2/21\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print(t)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"This temporal network consists of four nodes connected by four time-stamped links. The observation period covers 30 time units and contains four different time stamps with observed edges. The average inter-event time between \\\"events\\\" is 10 time units, the minimum and the maximum difference between any two consecutive events are two and 21 time units respectively.\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now you may ask: How are temporal networks related to the pathways above and how can we study both from the same perspective?\\n\",\n    \"\\n\",\n    \"Well, the point is that temporal networks naturally give rise to **time-respecting paths**, i.e. paths consisting of sequences of time-stamped links which (minimally) satisfy **causality**. Specifically, two time-stamped links $(a,b;t_1)$ and $(b,c;t_2)$ contribute to a time-respecting path $a \\\\rightarrow b \\\\rightarrow c$ if $t_1<t_2$, i.e. if the link $(a,b)$ occurs **before** $(b,c)$. Apart from the condition that links have to occur in the correct order, it is common to impose a **maximum time difference** between consecutive links. I.e. we define a maximum time difference $\\\\delta$ and consider two time-stamped edges $(a,b;t)$ and $(b,c;t')$ to contribute to a time-respecting path if $0 \\\\leq t'-t \\\\leq \\\\delta$. Imposing this additional condition is natural, since we are typically interested in paths which occur at short time scales. Especially, when considering time-stamped data collected over a period of several days, weeks or even years, it is usually not reasonable to consider a path definition where links can be weeks or years apart.\\n\",\n    \"\\n\",\n    \"With this definition of time-respecting paths at hand, we can extract pathways from a sequence of time-stamped edges based on a given value of $\\\\delta$. We can directly do this in `pathpy` using a built-in method to extract time-respecting paths for arbitrary $\\\\delta$. Let us try this for $\\\\delta=\\\\inf$, i.e. we don't impose a constraint for the maximum time difference, but still require that links occur in the right order:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 12,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tExtracting time-respecting paths for delta = inf ...\\n\",\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tCalculating sub path statistics ... \\n\",\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tfinished.\\n\",\n      \"Number of paths (unique/sub paths/total):\\t3 (2/19/22)\\n\",\n      \"Nodes:\\t\\t\\t\\t4\\n\",\n      \"Edges:\\t\\t\\t\\t3\\n\",\n      \"Max. path length:\\t\\t3\\n\",\n      \"Avg path length:\\t\\t2.33333333333\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/10/10)\\n\",\n      \"Paths of length k = 1\\t\\t0 (0/7/7)\\n\",\n      \"Paths of length k = 2\\t\\t2 (1/2/4)\\n\",\n      \"Paths of length k = 3\\t\\t1 (1/0/1)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"trp = pp.Paths.fromTemporalNetwork(t, delta=np.inf)\\n\",\n    \"print(trp)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Here we have two time-respecting paths $(b,c;21) \\\\rightarrow (c,d;51)$ and $(b,c;44) \\\\rightarrow (c,d;51)$ of length two and a path $(a,b;41) \\\\rightarrow (b,c;44) \\\\rightarrow (c,d;51)$ of length three. In addition, all shorter time-respecting paths which are sub paths of the three detected paths are automatically accounted for in the sub-path counts given in the second number in the brackets. If we instead set $\\\\delta$ to a smaller value like $\\\\delta=5$, we get different (time-respecting) paths:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 13,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tExtracting time-respecting paths for delta = 5 ...\\n\",\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tCalculating sub path statistics ... \\n\",\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tfinished.\\n\",\n      \"Number of paths (unique/sub paths/total):\\t3 (3/9/12)\\n\",\n      \"Nodes:\\t\\t\\t\\t4\\n\",\n      \"Edges:\\t\\t\\t\\t3\\n\",\n      \"Max. path length:\\t\\t2\\n\",\n      \"Avg path length:\\t\\t1.33333333333\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/7/7)\\n\",\n      \"Paths of length k = 1\\t\\t2 (2/2/4)\\n\",\n      \"Paths of length k = 2\\t\\t1 (1/0/1)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"trp = pp.Paths.fromTemporalNetwork(t, delta=5)\\n\",\n    \"print(trp)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now we only have two paths of length one (which are just time-stamped edges without a continuation towards a longer time-respecting path) and a single path $(a,b;42) \\\\rightarrow (b,c;44)$ of length two for which the time difference between edges is less than five.\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"In summary, `pathpy` makes it very easy to extract (time-respecting) paths from time-stamped data on temporal networks. Thanks to this, **both pathway and temporal network data can be studied from the same perspective**.\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"data\\\"></a>\\n\",\n    \"## 4. Importing data on pathways and temporal networks\\n\",\n    \"*[Back to outline](#outline)*\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Rather than manually adding paths or time-stamped edges, we can directly read `Paths` and `TemporalNetwork` instances from data files.\\n\",\n    \"\\n\",\n    \"For pathway data, the `Paths` class can be used to read **n-gram files**, i.e. text files where each line contains a path with variable length. Vertices can be arbitrary strings, separated by a special character that can be set when reading the file. `pathpy` supports data where each path is observed multiple times. This either works by reading files where identical paths occur in multiple lines of the file, or by including a special last column in each line which contains the number of observations of a path.\\n\",\n    \"\\n\",\n    \"Such an n-gram file can look like the following excerpt from a file that captures [travel itineraries of passengers of the London Tube](https://tfl.gov.uk/info-for/open-data-users/), recorded via smartcard readers: \\n\",\n    \"\\n\",\n    \"`344,314,445,440,513,346,305,312,289,367,356,299,376,9`  \\n\",\n    \"`339,303,323,376,299,356,367,289,312,305,346,400,465,382,325,7`  \\n\",\n    \"`296,430,474,271,332,331,441,341,280,294,362,528,344,493,29`  \\n\",\n    \"`...`  \\n\",\n    \"\\n\",\n    \"Numbers indicate metro stations passed by on an itinerary, except for the last column which indicates the number of times the given path was observed. If we save such data in a textfile called `tube_paths.ngram`, we can read it as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 14,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:16:53 [Severity.INFO]\\tReading ngram data ... \\n\",\n      \"2017-03-01 17:16:54 [Severity.INFO]\\tfinished. Read 32918 paths with maximum length 35\\n\",\n      \"2017-03-01 17:16:54 [Severity.INFO]\\tCalculating sub path statistics ... \\n\",\n      \"2017-03-01 17:17:15 [Severity.INFO]\\tfinished.\\n\",\n      \"Number of paths (unique/sub paths/total):\\t4295731 (32313/182041358/186337089)\\n\",\n      \"Nodes:\\t\\t\\t\\t276\\n\",\n      \"Edges:\\t\\t\\t\\t663\\n\",\n      \"Max. path length:\\t\\t35\\n\",\n      \"Avg path length:\\t\\t6.86497129359\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/33785801/33785801)\\n\",\n      \"Paths of length k = 1\\t\\t186435 (594/29303635/29490070)\\n\",\n      \"Paths of length k = 2\\t\\t337975 (912/24856364/25194339)\\n\",\n      \"Paths of length k = 3\\t\\t446941 (1288/20638102/21085043)\\n\",\n      \"Paths of length k = 4\\t\\t453255 (1696/16860467/17313722)\\n\",\n      \"Paths of length k = 5\\t\\t467697 (2040/13521645/13989342)\\n\",\n      \"Paths of length k = 6\\t\\t438374 (2316/10679843/11118217)\\n\",\n      \"Paths of length k = 7\\t\\t369090 (2460/8345699/8714789)\\n\",\n      \"Paths of length k = 8\\t\\t317013 (2529/6432722/6749735)\\n\",\n      \"Paths of length k = 9\\t\\t277742 (2572/4876029/5153771)\\n\",\n      \"Paths of length k = 10\\t\\t229761 (2481/3645059/3874820)\\n\",\n      \"Paths of length k = 11\\t\\t181738 (2308/2691873/2873611)\\n\",\n      \"Paths of length k = 12\\t\\t147092 (2080/1955071/2102163)\\n\",\n      \"Paths of length k = 13\\t\\t123089 (1855/1389364/1512453)\\n\",\n      \"Paths of length k = 14\\t\\t85503 (1612/984332/1069835)\\n\",\n      \"Paths of length k = 15\\t\\t70859 (1329/679447/750306)\\n\",\n      \"Paths of length k = 16\\t\\t48461 (1073/467819/516280)\\n\",\n      \"Paths of length k = 17\\t\\t37015 (841/316098/353113)\\n\",\n      \"Paths of length k = 18\\t\\t24104 (655/214303/238407)\\n\",\n      \"Paths of length k = 19\\t\\t16498 (470/144218/160716)\\n\",\n      \"Paths of length k = 20\\t\\t10852 (335/96277/107129)\\n\",\n      \"Paths of length k = 21\\t\\t10416 (294/59624/70040)\\n\",\n      \"Paths of length k = 22\\t\\t5155 (179/38648/43803)\\n\",\n      \"Paths of length k = 23\\t\\t3274 (118/24708/27982)\\n\",\n      \"Paths of length k = 24\\t\\t3136 (98/14180/17316)\\n\",\n      \"Paths of length k = 25\\t\\t2253 (76/7671/9924)\\n\",\n      \"Paths of length k = 26\\t\\t506 (34/5162/5668)\\n\",\n      \"Paths of length k = 27\\t\\t567 (26/3098/3665)\\n\",\n      \"Paths of length k = 28\\t\\t400 (13/1768/2168)\\n\",\n      \"Paths of length k = 29\\t\\t225 (13/1013/1238)\\n\",\n      \"Paths of length k = 30\\t\\t161 (8/547/708)\\n\",\n      \"Paths of length k = 31\\t\\t50 (2/353/403)\\n\",\n      \"Paths of length k = 32\\t\\t12 (1/247/259)\\n\",\n      \"Paths of length k = 33\\t\\t4 (1/161/165)\\n\",\n      \"Paths of length k = 34\\t\\t73 (3/10/83)\\n\",\n      \"Paths of length k = 35\\t\\t5 (1/0/5)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"tube_paths = pp.Paths.readFile('pathpy_tutorial/tube_paths.ngram', separator=',', pathFrequency=True)\\n\",\n    \"print(tube_paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"The large numbers result from the calculation of sub-path statistics outlined above. Here, the fact that this data set contains many long paths results in more than 180 million shorter sub-paths which are contained within longer pathways.\\n\",\n    \"\\n\",\n    \"We finally show how we can read temporal networks from data files. Here we assume that we are given a data file that has the following format \\n\",\n    \"\\n\",\n    \"`time source target`  \\n\",\n    \"`28820 492 938`  \\n\",\n    \"`28860 267 272`  \\n\",\n    \"`29300 181 826`  \\n\",\n    \"`...`  \\n\",\n    \"\\n\",\n    \"Each line captures a directed time-stamped edge from source to target, happening instantaneously at the indicated time stamp. The header column tells which column is which. The ordering of columns can be arbitrary and the character that separates column can be specified. \\n\",\n    \"\\n\",\n    \"The above lines are actually an excerpt from a time-stamped data set released by the [SocioPatterns collaboration](http://www.sociopatterns.org). It captures [face-to-face encounters of workers in a company](http://www.sociopatterns.org/datasets/contacts-in-a-workplace/), recorded via sensor badges. We can read this data file to a `TemporalNetwork` instance as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 15,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:16 [Severity.INFO]\\tReading time-stamped links ...\\n\",\n      \"2017-03-01 17:17:16 [Severity.INFO]\\tBuilding index data structures ...\\n\",\n      \"2017-03-01 17:17:16 [Severity.INFO]\\tSorting time stamps ...\\n\",\n      \"2017-03-01 17:17:16 [Severity.INFO]\\tfinished.\\n\",\n      \"Nodes:\\t\\t\\t92\\n\",\n      \"Time-stamped links:\\t9827\\n\",\n      \"Links/Nodes:\\t\\t106.81521739130434\\n\",\n      \"Observation period:\\t[28820, 1016440]\\n\",\n      \"Observation length:\\t987620\\n\",\n      \"Time stamps:\\t\\t7104\\n\",\n      \"Avg. inter-event dt:\\t139.042658032\\n\",\n      \"Min/Max inter-event dt:\\t20/222680\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"work_t = pp.TemporalNetwork.readFile('pathpy_tutorial/WorkplaceContacts.tedges', sep=' ')\\n\",\n    \"print(work_t)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now we can pick a maximum time difference $\\\\delta$ and extract all time-respecting paths as explained above. Here we choose a maximum time difference of three minutes, which gives the following time-respecting path statistics:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 16,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:16 [Severity.INFO]\\tExtracting time-respecting paths for delta = 180 ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tCalculating sub path statistics ... \\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tfinished.\\n\",\n      \"Number of paths (unique/sub paths/total):\\t10939 (968/31879/42818)\\n\",\n      \"Nodes:\\t\\t\\t\\t92\\n\",\n      \"Edges:\\t\\t\\t\\t755\\n\",\n      \"Max. path length:\\t\\t4\\n\",\n      \"Avg path length:\\t\\t1.28393820276\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/24984/24984)\\n\",\n      \"Paths of length k = 1\\t\\t8467 (669/5578/14045)\\n\",\n      \"Paths of length k = 2\\t\\t1887 (252/1219/3106)\\n\",\n      \"Paths of length k = 3\\t\\t536 (41/98/634)\\n\",\n      \"Paths of length k = 4\\t\\t49 (6/0/49)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"work_paths = pp.Paths.fromTemporalNetwork(work_t, delta=180)\\n\",\n    \"print(work_paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"multiorder\\\"></a>\\n\",\n    \"## 5. Multi-Order Graphical Models of Pathways and Temporal Networks\\n\",\n    \"*[Back to outline](#outline)*\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Now that we can import data on pathways and temporal networks, we turn our attention to the multi-order graphical modeling framework which is the theoretical foundation of our data mining techniques. The mathematical details have been developed in [this recent article](https://arxiv.org/abs/1702.05499). Here we provide a short (and rather high-level) introduction.\\n\",\n    \"\\n\",\n    \"Consider a toy network which consists of five nodes $a,b,c,d,e$ connected by four links $(a,c), (b,c), (c,d), (c,e)$. We further assume that we have a total of 18 observations of the following eight unique paths:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 17,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Number of paths (unique/sub paths/total):\\t19 (6/71/90)\\n\",\n      \"Nodes:\\t\\t\\t\\t5\\n\",\n      \"Edges:\\t\\t\\t\\t4\\n\",\n      \"Max. path length:\\t\\t2\\n\",\n      \"Avg path length:\\t\\t1.57894736842\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/49/49)\\n\",\n      \"Paths of length k = 1\\t\\t8 (4/22/30)\\n\",\n      \"Paths of length k = 2\\t\\t11 (2/0/11)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"paths = pp.Paths()\\n\",\n    \"paths.addPath('a,c', pathFrequency=2)\\n\",\n    \"paths.addPath('b,c', pathFrequency=1)\\n\",\n    \"paths.addPath('c,d', pathFrequency=3)\\n\",\n    \"paths.addPath('c,e', pathFrequency=2)\\n\",\n    \"paths.addPath('a,c,d', pathFrequency=5)\\n\",\n    \"paths.addPath('b,c,e', pathFrequency=6)\\n\",\n    \"print(paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"The question we are going to address is: Considering the statistics of observed paths, is it justified to model the underlying system as a network? And if not, what would be an optimal graphical abstraction for the data set?\\n\",\n    \"\\n\",\n    \"We address these important questions based on [\\\"higher-order\\\" networks](https://www.sg.ethz.ch/team/people/ischoltes/research-insights/temporal-networks-demo/) introduced in [this Nature Communications article](http://www.nature.com/ncomms/2014/140924/ncomms6024/full/ncomms6024.html). Here, we generalize this approach to (i) multi-order models which combine multiple layers of higher-order networks, and (ii) pathway data.\\n\",\n    \"\\n\",\n    \"So what is the problem if we model the system above as a graph or network? The problem is that graph- and network-analytic methods like centrality measures, community detection, etc. are implicitly based on the **assumption that paths in a network are transitive**, i.e. if we observe a path from node $u$ to node $v$ and a path from node $v$ to node $w$, we implicitly assume that there is a transitive path from $u$ via $v$ to $w$. This fundamental assumption is due to the way how graph algorithms (as well as algebraic methods which rely on matrix multiplication or spectral analysis) work. However, what is important to see is that **correlations in the sequence of nodes traversed by paths can invalidate the assumption of transitivity** (see [discussion here](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.198701)).\\n\",\n    \"\\n\",\n    \"Consider the example above: Since the network has two edges $(a,c)$ and $(c,e)$ we would expect that a path of length two from $a$ via $c$ to $e$ exists. But this is actually not the case. Instead, whenever a path traverses from $a$ to $c$, it continues to $d$. Likewise, whenever a path traverses from $b$ to $c$, it continues to $e$. The transitive path $(b \\\\rightarrow c \\\\rightarrow d)$, which we expect based on the link topology of the graph, never occurs. \\n\",\n    \"\\n\",\n    \"So the question really is: Is the topology of the underlying graph enough to explain the statistics of observed paths? Note that the example above is an extreme example, where two transitive paths are completely absent. Rather than being completely absent, we could also have cases where paths are just less (or more) frequent than what we expect. Clearly, such an under- or overrepresentation of paths violates the transitivity assumption of a network abstraction as well (though possibly to a lesser degree).\\n\",\n    \"\\n\",\n    \"While we refer to [the publications above](#references) for a detailed mathematical description of our approach, the key idea is to consider [Markov chain models of different orders](https://en.wikipedia.org/wiki/Markov_chain) which are tailored to reproduce the statistics of paths observed in a given graph topology. We specifically use a graphical construction that resembles [De Bruijn graphs](https://en.wikipedia.org/wiki/De_Bruijn_graph) known from sequence modeling.\\n\",\n    \"\\n\",\n    \"This is how it works: We first consider a \\\"first-order\\\" network abstraction which simply counts the frequencies at which edges are traversed by paths. We can generate such a first-order abstractions of our observed paths using the `HigherOrderNetwork` class provided by `pathpy` as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 18,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"network = pp.HigherOrderNetwork(paths, k=1)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"To visualize this network model, we provide the following helper function, which takes an instance of the `HigherOrderNetwork` class and returns an `igraph` `Graph` object that we can use to plot the network topology. The current version of `pathpy` specifically does not include plotting and visualization tools since (i) we want the package to have minimal dependencies, and (ii) the function below shows that it is very easy to construct `igraph` (and similarly [`graph-tool`](https://graph-tool.skewed.de/)) instances from a `HigherOrderNetwork` object.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 19,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": [\n    \"def Network2igraph(network):\\n\",\n    \"    \\\"\\\"\\\" \\n\",\n    \"    Returns an igraph Graph object which represents \\n\",\n    \"    the k-th layer of a multi-order graphical model.\\n\",\n    \"    \\\"\\\"\\\"\\n\",\n    \"    g = igraph.Graph(directed=True)\\n\",\n    \"\\n\",\n    \"    for e in network.edges:\\n\",\n    \"        if g.vcount()== 0 or e[0] not in g.vs()[\\\"name\\\"]:\\n\",\n    \"            g.add_vertex(e[0])\\n\",\n    \"        if g.vcount()== 0 or e[1] not in g.vs()[\\\"name\\\"]:\\n\",\n    \"            g.add_vertex(e[1])\\n\",\n    \"        g.add_edge(e[0], e[1], weight=network.edges[e].sum())\\n\",\n    \"    return g\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can now use `igraph`'s visual styling and plotting features to plot the network representation.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 20,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"image/png\": 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UmpGxMBAAQGFCHovry8vL9PnLhy5WpFxTtzB6e+DgMMzK3UdfVl5RV+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\\n2SEAAAAASUVORK5CYII=\\n\",\n      \"text/plain\": [\n       \"<IPython.core.display.Image object>\"\n      ]\n     },\n     \"metadata\": {},\n     \"output_type\": \"display_data\"\n    }\n   ],\n   \"source\": [\n    \"g1 = Network2igraph(network)\\n\",\n    \"igraph.plot(g1)\\n\",\n    \"\\n\",\n    \"visual_style = {}\\n\",\n    \"visual_style[\\\"bbox\\\"] = (600, 400)\\n\",\n    \"visual_style[\\\"margin\\\"] = 60\\n\",\n    \"visual_style[\\\"vertex_size\\\"] = 80\\n\",\n    \"visual_style[\\\"vertex_label_size\\\"] = 24\\n\",\n    \"visual_style[\\\"vertex_color\\\"] = \\\"lightblue\\\"\\n\",\n    \"visual_style[\\\"edge_curved\\\"] = 0.2\\n\",\n    \"visual_style[\\\"edge_width\\\"] = 1\\n\",\n    \"visual_style[\\\"edge_arrow_size\\\"] = 2\\n\",\n    \"\\n\",\n    \"visual_style[\\\"layout\\\"] = g1.layout_auto()\\n\",\n    \"visual_style[\\\"vertex_label\\\"] = g1.vs[\\\"name\\\"]\\n\",\n    \"visual_style[\\\"edge_label\\\"] = g1.es[\\\"weight\\\"]\\n\",\n    \"\\n\",\n    \"igraph.plot(g1, 'pathpy_tutorial/g1.png', **visual_style)\\n\",\n    \"display(Image(filename='pathpy_tutorial/g1.png'))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"As you can see, this first-order model corresponds to a weighted network, where link weights  count the number of times an edge has been traversed by paths. Considering that (i) each node in this network is actually a path of length zero, and (ii) each link provides the frequency of paths of length one, we can generalize this to higher-order graphical models. For $k=2$ we get a second-order model, where second-order nodes are paths of length one, and links provide the frequencies of paths of length two.\\n\",\n    \"\\n\",\n    \"We can easily generate this using the `HigherOrderNetwork` class:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 21,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"image/png\": \"iVBORw0KGgoAAAANSUhEUgAAAlgAAAGQCAIAAAD9V4nPAAAABmJLR0QA/wD/AP+gvaeTAAAgAElE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\\nIQgBAIDV/g8E97SdfgQbywAAAABJRU5ErkJggg==\\n\",\n      \"text/plain\": [\n       \"<IPython.core.display.Image object>\"\n      ]\n     },\n     \"metadata\": {},\n     \"output_type\": \"display_data\"\n    }\n   ],\n   \"source\": [\n    \"network2 = pp.HigherOrderNetwork(paths, k=2)\\n\",\n    \"g2 = Network2igraph(network2)\\n\",\n    \"\\n\",\n    \"visual_style[\\\"layout\\\"] = g2.layout_auto()\\n\",\n    \"visual_style[\\\"vertex_label\\\"] = g2.vs[\\\"name\\\"]\\n\",\n    \"visual_style[\\\"edge_label\\\"] = g2.es[\\\"weight\\\"]\\n\",\n    \"\\n\",\n    \"igraph.plot(g2, 'pathpy_tutorial/g2.png', **visual_style)\\n\",\n    \"display(Image(filename='pathpy_tutorial/g2.png'))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"In this representation only two paths of length two actually exist, while - under the assumption that paths in the first-order network are transitive - we would expect four possible paths. If we would have longer paths, we could continue this approach and generate network models with order 3, 4, etc. In a third-order model nodes represent paths of length two, while links capture paths of length three. In general, the link weights of a $k$-th order model capture the statistics of paths of length $k$, thus generalizing the commonly used first-order network view.\\n\",\n    \"\\n\",\n    \"As shown [in our previous work](http://dx.doi.org/10.1140/epjb/e2016-60663-0) and [EPJ B](http://link.springer.com/article/10.1140%2Fepjb%2Fe2016-60663-0) such **higher-order network abstractions** are interesting, since they **capture the temporal-topological topology of sequential data on networks**. Moreover, just like the commonly used first-order abstractions, we can interpret them as network topologies which can be analyzed using network-analytic and algebraic methods. [We have further shown](http://dx.doi.org/10.1140/epjb/e2016-60663-0) that these higher-order graphs can be interpreted as Markov models which capture correlations of a given length $k$ that are hidden in the statistics of pathways.\\n\",\n    \"\\n\",\n    \"Building on this idea, here we go one step further: We combine several layers of higher-order models up to a maximum order of $k$ to a single **multi-order model**. We can fit such a multi-order model to a given set of pathways using the `MultiOrderModel` class of `pathpy`. We can set the maximum order $maxOrder$ up to which higher-order models should be generated. If we don't specify a maximum order, the model will contain all possible higher-order models up to the maximum path length in the data.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 22,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tfinished.\\n\",\n      \"Multi-order model (max. order = 2, DoF (paths/ngrams) = 7/124)\\n\",\n      \"===========================================================================\\n\",\n      \"Layer k = 0\\t6 nodes, 5 links, 49 paths, DoF (paths/ngrams) = 4/4\\n\",\n      \"Layer k = 1\\t5 nodes, 4 links, 30 paths, DoF (paths/ngrams) = 1/20\\n\",\n      \"Layer k = 2\\t4 nodes, 2 links, 11 paths, DoF (paths/ngrams) = 2/100\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m = pp.MultiOrderModel(paths, maxOrder=2)\\n\",\n    \"print(m)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Again, printing the instance prints a human-readable summary of the underlying model. Here it actually combines three layers from order zero (which simply captures \\\"activation frequencies\\\" of nodes) up to the maximum order of two. Each layer $k$ is simply the $k$-th order model introduced above. We can verify this by plotting the corresponding `HigherOrderNetwork` instances which are stored in the dictionary `layers` of the `MultiOrderNetwork` instance.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 23,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"data\": {\n      \"image/png\": 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R0d7e3tFUh9nwmQDjNCgE949+6dprb2lrNRffvpCnos7gqavTdSBb1r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\\n\",\n      \"text/plain\": [\n       \"<IPython.core.display.Image object>\"\n      ]\n     },\n     \"metadata\": {},\n     \"output_type\": \"display_data\"\n    }\n   ],\n   \"source\": [\n    \"g2 = Network2igraph(m.layers[2])\\n\",\n    \"\\n\",\n    \"visual_style[\\\"layout\\\"] = g2.layout_auto()\\n\",\n    \"visual_style[\\\"vertex_label\\\"] = g2.vs[\\\"name\\\"]\\n\",\n    \"visual_style[\\\"edge_label\\\"] = g2.es[\\\"weight\\\"]\\n\",\n    \"\\n\",\n    \"igraph.plot(g2, 'pathpy_tutorial/g2.png', **visual_style)\\n\",\n    \"display(Image(filename='pathpy_tutorial/g2.png'))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"network\\\"></a>\\n\",\n    \"## 6. When is a network a network?\\n\",\n    \"*[Back to outline](#outline)*\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"What can we do with such a multi-order graphical model of a given set of pathways? We can take a **model selection perspective**, and ask how many layers of higher-order models are needed to model a given set of pathways. In other words, we are interested in the optimal maximum order $K_{opt}$ of a multi-order graphical model needed to model a given data set. By \\\"optimal\\\" we refer to the number of layers minimally needed to best explain the observed pathway statistics, however considering the increase in **model complexity** when adding additional layers.\\n\",\n    \"\\n\",\n    \"This optimal maximum order $K_{opt}$ has an interesting interpretation: If for a data set we infer $K_{opt}=1$, this means that there are no significant deviations from the transitivity assumption made by a network representation that would justify the inclusion of higher-order graphical models. In other words: It is - from a model selection perspective - justified to study the underlying system as a network. However, if we find $K_{opt}>1$ this means that the application of a network abstraction (and likewise the use of network-analytic or algebraic methods) is misleading. **Calculating $K_{opt}$ thus allows to answer the crucial question whether a data set should be modeled as a network or not!**\\n\",\n    \"\\n\",\n    \"But how can we calculate what is the \\\"optimal\\\" order? And what do we mean when we say that this optimal order balances \\\"model complexity\\\" and \\\"explanatory power\\\". For this, we take a statistical inference view and calculate the likelihoods of multi-order models with different maximum orders under the observed data. Again, if you you are interested in mathematical details please refer to [this recent research paper](https://arxiv.org/abs/1702.05499). \\n\",\n    \"\\n\",\n    \"Here, it is enough to say that we can conveniently calculate the (log-)likelihood of a multi-order model using the likelihood function of the class `MultiOrderModel`. Let us try this with a multi-order model for our toy example that has a maximum order of one.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 24,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tfinished.\\n\",\n      \"Likelihood =  1.97212806634e-19\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m1 = pp.MultiOrderModel(paths, maxOrder=1)\\n\",\n    \"print('Likelihood = ', m1.getLikelihood(paths, log=False))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We get a likelihood of the model of $\\\\approx 1.97 \\\\cdot 10^{-19}$. Let us compare this to the likelihood of a multi-order model that adds a layer with a second-order model: \"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 25,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tfinished.\\n\",\n      \"Likelihood =  4.03891827985e-16\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m2 = pp.MultiOrderModel(paths, maxOrder=2)\\n\",\n    \"print('Likelihood = ', m2.getLikelihood(paths, log=False))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Quite naturally, adding a second layer increases the likelihood, i.e. we have increased the explanatory power of the model for our data data. This may trick us into thinking that the second model is the better one. However, we should also take into account that, by adding an additional layer, we make the model more complex. Applying [Occam's razor](https://en.wikipedia.org/wiki/Occam's_razor) we should instead search for the **simplest model** which still has reasonable explanatory power, i.e. we should not make the model more complex than neccessary.\\n\",\n    \"\\n\",\n    \"The key to a principled decision about the optimal maximum order is to correctly account for the complexity of the model in terms of its **degrees of freedom**, i.e. the number of free parameters that we have fitted to the data. The correct calculation of this number for any graph topology and any order $k$ is one of the main contributions of [this work](https://arxiv.org/abs/1702.05499). \\n\",\n    \"\\n\",\n    \"Luckily, you won't have to deal with this because `pathpy` automatically takes care of it for you. First of all, we can  check the degrees of freedom of our two candidate models simply by printing the model instances.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 26,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Multi-order model (max. order = 1, DoF (paths/ngrams) = 5/24)\\n\",\n      \"===========================================================================\\n\",\n      \"Layer k = 0\\t6 nodes, 5 links, 49 paths, DoF (paths/ngrams) = 4/4\\n\",\n      \"Layer k = 1\\t5 nodes, 4 links, 30 paths, DoF (paths/ngrams) = 1/20\\n\",\n      \"\\n\",\n      \"Multi-order model (max. order = 2, DoF (paths/ngrams) = 7/124)\\n\",\n      \"===========================================================================\\n\",\n      \"Layer k = 0\\t6 nodes, 5 links, 49 paths, DoF (paths/ngrams) = 4/4\\n\",\n      \"Layer k = 1\\t5 nodes, 4 links, 30 paths, DoF (paths/ngrams) = 1/20\\n\",\n      \"Layer k = 2\\t4 nodes, 2 links, 11 paths, DoF (paths/ngrams) = 2/100\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print(m1)\\n\",\n    \"print(m2)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"The number that matters here is the first number indicated after Dof (corresponding to paths). The model with maximum order one has five degrees of freedom, while the model of maximum order two has seven degrees of freedom. So the model with maximum order two is more complex than the one with maximum order one because we fit two additional parameters (for the layer of order two).\\n\",\n    \"\\n\",\n    \"Again, omitting mathematical details and referring to [this research paper](https://arxiv.org/abs/1702.05499) it turns out that we can apply [Wilk's theorem](https://en.wikipedia.org/wiki/Likelihood-ratio_test#Wilks.27_theorem) to perform a series of [likelihood ratio tests](https://en.wikipedia.org/wiki/Likelihood-ratio_test) in order to determine which maximum order is optimal, while considering the added complexity of higher-order models. And `pathpy` does all of this for you! We can simply call the function `estimateOrder`, which returns the optimal maximum order $K_{opt}$ that we are looking for:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 27,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tLikelihood ratio test for K_opt = 2, x = 22.0\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 2\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tLikelihood ratio test, p = 1.67017007903e-05\\n\",\n      \"Optimal maximum order =  2\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print('Optimal maximum order = ', m2.estimateOrder(paths))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"For our toy example, we see that a (first-order) network abstraction would be misleading. We actually need the second-order graphical model layer, i.e. the added complexity is justified considered the increase in explanatory power for the observed pathways.\\n\",\n    \"\\n\",\n    \"Let us consider another toy example, where pathway statistics are actually *exactly* as we expect it to be under the assumption that paths in the first-order network are transitive:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 28,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Number of paths (unique/sub paths/total):\\t20 (8/76/96)\\n\",\n      \"Nodes:\\t\\t\\t\\t5\\n\",\n      \"Edges:\\t\\t\\t\\t4\\n\",\n      \"Max. path length:\\t\\t2\\n\",\n      \"Avg path length:\\t\\t1.6\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/52/52)\\n\",\n      \"Paths of length k = 1\\t\\t8 (4/24/32)\\n\",\n      \"Paths of length k = 2\\t\\t12 (4/0/12)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"paths = pp.Paths()\\n\",\n    \"paths.addPath('a,c', pathFrequency=2)\\n\",\n    \"paths.addPath('b,c', pathFrequency=2)\\n\",\n    \"paths.addPath('c,d', pathFrequency=2)\\n\",\n    \"paths.addPath('c,e', pathFrequency=2)\\n\",\n    \"paths.addPath('a,c,d', pathFrequency=3)\\n\",\n    \"paths.addPath('b,c,d', pathFrequency=3)\\n\",\n    \"paths.addPath('b,c,e', pathFrequency=3)\\n\",\n    \"paths.addPath('a,c,e', pathFrequency=3)\\n\",\n    \"print(paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"In this special (and quite artificial case) the model with maximum order one and the model with maximum order two have exactly the same likelihoods. I.e., adding the second-order model layer provides no benefit in terms of explanatory power. The reason for this is that there are no correlations in the data that violate the transitivity assumption. We can confirm this as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 29,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tfinished.\\n\",\n      \"Likelihood =  3.28861452936e-20\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m1 = pp.MultiOrderModel(paths, maxOrder=1)\\n\",\n    \"print('Likelihood = ', m1.getLikelihood(paths, log=False))\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 30,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tfinished.\\n\",\n      \"Likelihood =  3.28861452936e-20\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m2 = pp.MultiOrderModel(paths, maxOrder=2)\\n\",\n    \"print('Likelihood = ', m2.getLikelihood(paths, log=False))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Since the model `m2` is more complex, the `estimateOrder` function will actually reject the more complex model, correctly determining that **a network abstraction of this set of pathways is actually justified**.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 31,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tLikelihood ratio test for K_opt = 2, x = -0.0\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 2\\n\",\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tLikelihood ratio test, p = 1.0\\n\",\n      \"Optimal maximum order =  1\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print('Optimal maximum order = ', m2.estimateOrder(paths))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"We can vary this example in a way that we slightly distort the statistics of paths, violating the transitivity assumption, but just by a little bit. Precisely, we overrepresent one of the four paths of length two by one occurrence, compared to what we would expect based on the relative frequencies of links:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 32,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"Number of paths (unique/sub paths/total):\\t21 (8/81/102)\\n\",\n      \"Nodes:\\t\\t\\t\\t5\\n\",\n      \"Edges:\\t\\t\\t\\t4\\n\",\n      \"Max. path length:\\t\\t2\\n\",\n      \"Avg path length:\\t\\t1.61904761905\\n\",\n      \"Paths of length k = 0\\t\\t0 (0/55/55)\\n\",\n      \"Paths of length k = 1\\t\\t8 (4/26/34)\\n\",\n      \"Paths of length k = 2\\t\\t13 (4/0/13)\\n\",\n      \"\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"paths = pp.Paths()\\n\",\n    \"paths.addPath('a,c', pathFrequency=2)\\n\",\n    \"paths.addPath('b,c', pathFrequency=2)\\n\",\n    \"paths.addPath('c,d', pathFrequency=2)\\n\",\n    \"paths.addPath('c,e', pathFrequency=2)\\n\",\n    \"paths.addPath('a,c,d', pathFrequency=3)\\n\",\n    \"paths.addPath('b,c,d', pathFrequency=3)\\n\",\n    \"paths.addPath('b,c,e', pathFrequency=3)\\n\",\n    \"paths.addPath('a,c,e', pathFrequency=4)\\n\",\n    \"print(paths)\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Let us again calculate the likelihoods for the models with maximum order one and two respectively:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 33,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:17 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tfinished.\\n\",\n      \"Likelihood =  2.8147122155e-21\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m1 = pp.MultiOrderModel(paths, maxOrder=1)\\n\",\n    \"print('Likelihood = ', m1.getLikelihood(paths, log=False))\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 34,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tfinished.\\n\",\n      \"Likelihood =  2.91588212235e-21\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"m2 = pp.MultiOrderModel(paths, maxOrder=2)\\n\",\n    \"print('Likelihood = ', m2.getLikelihood(paths, log=False))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Here, the likelihood of the more complex model is slightly larger, because the additional second-order layer captures the overrepresentation of the path $(a\\\\rightarrow c \\\\rightarrow e)$ which is not expected based on the first-order layer. However, since the model is also more complex, our method correctly determines that the small gain in likelihood does not justify the associated increase in model complexity, thus determining that a first-order network abstraction of this data set is optimal.\\n\",\n    \"\\n\",\n    \"In fact, from the output below you can see that we can calculate a $p$-value, which allows us to reject the (alternative) hypothesis that includes a second-order model layer in favor of the (null) hypothesis of the simpler model. We can actually set the significance threshold for the underlying likelihood ratio test as follows:\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 35,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tLikelihood ratio test for K_opt = 2, x = 0.101889947837\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 2\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tLikelihood ratio test, p = 0.950330962083\\n\",\n      \"Optimal maximum order =  1\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"print('Optimal maximum order = ', m2.estimateOrder(paths, significanceThreshold=0.001))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Let us finally go beyond synthetic toy examples and test our method in the data sets which we have imported above. We start with the pathway data capturing travel patterns in the London Tube system. Here our method finds that a **network abstraction of the London Tube is misleading**. It actually finds that we need to consider higher-order model layers up to order six, while the layer of order seven is not significant.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 36,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:17:18 [Severity.INFO]\\tGenerating 3-th order network layer ...\\n\",\n      \"2017-03-01 17:17:19 [Severity.INFO]\\tGenerating 4-th order network layer ...\\n\",\n      \"2017-03-01 17:17:20 [Severity.INFO]\\tGenerating 5-th order network layer ...\\n\",\n      \"2017-03-01 17:17:22 [Severity.INFO]\\tGenerating 6-th order network layer ...\\n\",\n      \"2017-03-01 17:17:25 [Severity.INFO]\\tGenerating 7-th order network layer ...\\n\",\n      \"2017-03-01 17:17:29 [Severity.INFO]\\tfinished.\\n\",\n      \"2017-03-01 17:18:17 [Severity.INFO]\\tLikelihood ratio test for K_opt = 2, x = 46432008.9276\\n\",\n      \"2017-03-01 17:18:17 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 1659\\n\",\n      \"2017-03-01 17:18:17 [Severity.INFO]\\tLikelihood ratio test, p = 0.0\\n\",\n      \"2017-03-01 17:19:19 [Severity.INFO]\\tLikelihood ratio test for K_opt = 3, x = 1484643.01227\\n\",\n      \"2017-03-01 17:19:19 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 5619\\n\",\n      \"2017-03-01 17:19:19 [Severity.INFO]\\tLikelihood ratio test, p = 0.0\\n\",\n      \"2017-03-01 17:20:21 [Severity.INFO]\\tLikelihood ratio test for K_opt = 4, x = 679447.231163\\n\",\n      \"2017-03-01 17:20:21 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 19527\\n\",\n      \"2017-03-01 17:20:21 [Severity.INFO]\\tLikelihood ratio test, p = 0.0\\n\",\n      \"2017-03-01 17:21:30 [Severity.INFO]\\tLikelihood ratio test for K_opt = 5, x = 365515.096382\\n\",\n      \"2017-03-01 17:21:30 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 67834\\n\",\n      \"2017-03-01 17:21:30 [Severity.INFO]\\tLikelihood ratio test, p = 0.0\\n\",\n      \"2017-03-01 17:22:44 [Severity.INFO]\\tLikelihood ratio test for K_opt = 6, x = 406258.608459\\n\",\n      \"2017-03-01 17:22:44 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 246738\\n\",\n      \"2017-03-01 17:22:44 [Severity.INFO]\\tLikelihood ratio test, p = 0.0\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tLikelihood ratio test for K_opt = 7, x = 160119.246725\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 903486\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tLikelihood ratio test, p = 1.0\\n\",\n      \"Optimal maximum order =  6\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"tube_model = pp.MultiOrderModel(tube_paths, maxOrder=7)\\n\",\n    \"print('Optimal maximum order = ', tube_model.estimateOrder(tube_paths))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Despite using a data set with more than 4 million paths and generating higher-order graphical models up to order seven, the whole testing procedure only takes a few minutes (on a six year old laptop computer).\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"What is great about the unified approach to pathway and temporal network analysis, is that we can directly apply this method to temporal network data. Here we can test whether the time-respecting path statistics which we extracted from the workplace data set above justifies a network abstraction. This specifically allows us to **test whether there are temporal correlations in the sequence of time-stamped interactions which invalidate a (static, first-order) network abstraction**.\\n\",\n    \"\\n\",\n    \"We can do this with two (!) lines of python code, testing for the significance of mode layers up to order three.\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 37,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tGenerating 3-th order network layer ...\\n\",\n      \"2017-03-01 17:24:02 [Severity.INFO]\\tfinished.\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tLikelihood ratio test for K_opt = 2, x = 6538.14557914\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 4521\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tLikelihood ratio test, p = 0.0\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tLikelihood ratio test for K_opt = 3, x = 123.324081918\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 20439\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tLikelihood ratio test, p = 1.0\\n\",\n      \"Optimal maximum order =  2\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"work_model = pp.MultiOrderModel(work_paths, maxOrder=3)\\n\",\n    \"print('Optimal maximum order = ', work_model.estimateOrder(work_paths))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Again, we find that a **network abstraction of the time-stamped interactions in this data set is misleading**. Here we actually need to add a second-order model to explain the observed time-respecting paths, while including an additional third-order model layer is not justified!\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"Let us conclude our analysis by studying a third data set, which captures [time-stamped E-Mail exchanges in a Polish manufacturing company](http://link.springer.com/chapter/10.1007/978-3-642-21863-7_17). We again extract time-respecting paths for a given maximum time difference $\\\\delta$, create a multi-order model and detect the optimal maximum order, all in just **four lines of python code which only require six seconds of computation!**\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": 38,\n   \"metadata\": {\n    \"collapsed\": false\n   },\n   \"outputs\": [\n    {\n     \"name\": \"stdout\",\n     \"output_type\": \"stream\",\n     \"text\": [\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tReading time-stamped links ...\\n\",\n      \"2017-03-01 17:24:03 [Severity.INFO]\\tBuilding index data structures ...\\n\",\n      \"2017-03-01 17:24:04 [Severity.INFO]\\tSorting time stamps ...\\n\",\n      \"2017-03-01 17:24:04 [Severity.INFO]\\tfinished.\\n\",\n      \"2017-03-01 17:24:04 [Severity.INFO]\\tExtracting time-respecting paths for delta = 30 ...\\n\",\n      \"2017-03-01 17:24:05 [Severity.INFO]\\tCalculating sub path statistics ... \\n\",\n      \"2017-03-01 17:24:05 [Severity.INFO]\\tfinished.\\n\",\n      \"2017-03-01 17:24:05 [Severity.INFO]\\tGenerating 0-th order network layer ...\\n\",\n      \"2017-03-01 17:24:05 [Severity.INFO]\\tGenerating 1-th order network layer ...\\n\",\n      \"2017-03-01 17:24:06 [Severity.INFO]\\tGenerating 2-th order network layer ...\\n\",\n      \"2017-03-01 17:24:07 [Severity.INFO]\\tfinished.\\n\",\n      \"2017-03-01 17:24:09 [Severity.INFO]\\tLikelihood ratio test for K_opt = 2, x = 21871.1297023\\n\",\n      \"2017-03-01 17:24:09 [Severity.INFO]\\tLikelihood ratio test, d_1-d_0 = 329140\\n\",\n      \"2017-03-01 17:24:09 [Severity.INFO]\\tLikelihood ratio test, p = 1.0\\n\",\n      \"Optimal maximum order =  1\\n\"\n     ]\n    }\n   ],\n   \"source\": [\n    \"email_t = pp.TemporalNetwork.readFile('pathpy_tutorial/CompanyEmails.tedges', sep='\\\\t')\\n\",\n    \"email_paths = pp.Paths.fromTemporalNetwork(email_t, delta=30)\\n\",\n    \"\\n\",\n    \"email_model = pp.MultiOrderModel(email_paths, maxOrder=2)\\n\",\n    \"print('Optimal maximum order = ', email_model.estimateOrder(email_paths))\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"For this data set, we find that a **first-order network abstraction is sufficient to explain time-respecting paths**, i.e. here temporal correlations in the data do not justify the added complexity of higher-order graphical models.\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"<a name=\\\"conclusion\\\"></a>\\n\",\n    \"## 7. Conclusion\\n\",\n    \"*[Back to outline](#outline)*\"\n   ]\n  },\n  {\n   \"cell_type\": \"markdown\",\n   \"metadata\": {},\n   \"source\": [\n    \"In conclusion, above I have shown how you can **test whether a network abstraction of sequential data on pathways and temporal networks is justified or not**. With `pathpy`, answering this crucial question is as simple as it can get. You won't need more than 3-4 lines of simple python code. Hence, a **multi-order analysis with `pathpy` is an extremely simple but crucial first step that should precede an application of network-based data mining and modeling techniques!**.\\n\",\n    \"\\n\",\n    \"Moreover, `pathpy` allows to infer multi-order graphical models whose layers can be interpreted as higher-order networks along the lines presented in the [works above](#references). While the argumentation is too long to be included in this tutorial, [this recent work](https://arxiv.org/abs/1702.05499) proves that the **inferred maximum order is the optimal order of a graphical abstraction of sequential data**, e.g. when it comes to the calculation of `PageRank` centralities or prediction tasks. Moreover, [this tutorial](http://www.ingoscholtes.net/research/insights/Temporal_Networks.html) illustrates and visualizes that such higher-order network topologies are crucial to accurately model and predict dynamical processes such as diffusion or epidemic spreading.\\n\",\n    \"\\n\",\n    \"I encourage you to get started with `pathpy` and to apply it to your data! If you have any problems, questions or suggestions, feel free to contact me.\\n\",\n    \"\\n\",\n    \"*Ingo Scholtes*  \\n\",\n    \"\\n\",\n    \"*Zurich, February 23 2017*\"\n   ]\n  },\n  {\n   \"cell_type\": \"code\",\n   \"execution_count\": null,\n   \"metadata\": {\n    \"collapsed\": true\n   },\n   \"outputs\": [],\n   \"source\": []\n  }\n ],\n \"metadata\": {\n  \"kernelspec\": {\n   \"display_name\": \"Python 3\",\n   \"language\": \"python\",\n   \"name\": \"python3\"\n  },\n  \"language_info\": {\n   \"codemirror_mode\": {\n    \"name\": \"ipython\",\n    \"version\": 3\n   },\n   \"file_extension\": \".py\",\n   \"mimetype\": \"text/x-python\",\n   \"name\": \"python\",\n   \"nbconvert_exporter\": \"python\",\n   \"pygments_lexer\": \"ipython3\",\n   \"version\": \"3.4.5\"\n  }\n },\n \"nbformat\": 4,\n \"nbformat_minor\": 0\n}\n"
  },
  {
    "path": "pathpy/HigherOrderNetwork.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\n    pathpy is an OpenSource python package for the analysis of sequential data on pathways and temporal networks using higher- and multi order graphical models\n\n    Copyright (C) 2016-2017 Ingo Scholtes, ETH Zürich\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU Affero General Public License as published\n    by the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU Affero General Public License for more details.\n\n    You should have received a copy of the GNU Affero General Public License\n    along with this program.  If not, see <http://www.gnu.org/licenses/>.\n\n    Contact the developer:\n    \n    E-mail: ischoltes@ethz.ch\n    Web:    http://www.ingoscholtes.net\n\"\"\"\n\nimport collections as _co\nimport bisect as _bs\nimport itertools as _iter\n\nimport numpy as _np \n\nimport scipy.sparse as _sparse\nimport scipy.sparse.linalg as _sla\nimport scipy.linalg as _la\nimport scipy as _sp\n\nfrom pathpy.Log import Log\nfrom pathpy.Log import Severity\n\n\nclass EmptySCCError(Exception):\n    \"\"\"\n    This exception is thrown whenever a non-empty strongly \n    connected component is needed, but we encounter an empty one\n    \"\"\"\n    pass\n\n\nclass HigherOrderNetwork:\n    \"\"\"\n    Instances of this class capture a k-th-order representation of path statistics. Path statistics \n    can originate from pathway data, temporal networks, or from processes observed on top of a network topology.\n    \"\"\"\n\n\n    def __init__(self, paths, k=1, separator='-', nullModel=False, method='FirstOrderTransitions', lanczosVecs=15, maxiter=1000):\n        \"\"\"\n        Generates a k-th-order representation based on the given path statistics.\n\n        @param paths: An instance of class Paths, which contains the path statistics to be \n            used in the generation of the k-th order representation \n\n        @param k: The order of the network representation to generate. For the default case of \n            k=1, the resulting representation corresponds to the usual (first-order) aggregate network, \n            i.e. links connect nodes and link weights are given by the frequency of each interaction. For \n            k>1, a k-th order node corresponds to a sequence of k nodes. The weight of a k-th order link \n            captures the frequency of a path of length k.\n\n        @param separator: The separator character to be used in higher-order node names.\n\n        @param nullModel: For the default value False, link weights are generated based on the statistics of \n            paths of length k in the underlying path statistics instance. If True, link weights are generated \n            from the first-order model (k=1) based on the assumption of independent links (i.e. corresponding) \n            to a first-order Markov model.\n\n        @param method: specifies how the null model link weights in the k-th order model are calculated. \n            For the default method='FirstOrderTransitions', the weight w('v_1-v_2-...v_k', 'v_2-...-v_k-v_k+1') of \n            a k-order edge is set to the transition probability T['v_k', 'v_k+1'] in the first order network.\n            For method='KOrderPi' the entry pi['v1-...-v_k'] in the stationary distribution of the \n            k-order network is used instead.\n        \"\"\"\n\n        assert nullModel == False or (nullModel and k>1)\n\n        assert method == 'FirstOrderTransitions' or method == 'KOrderPi', 'Error: unknown method to build null model'\n\n        assert len(paths.paths.keys())>0 and max(paths.paths.keys())>=k, 'Error: constructing a model of order k requires paths of at least length k'\n        \n        ## The order of this HigherOrderNetwork\n        self.order = k\n\n        ## The paths object used to generate this instance\n        self.paths = paths\n\n        ## The nodes in this HigherOrderNetwork \n        self.nodes = []\n\n        ## The separator character used to label higher-order nodes. \n        ## For separator '-', a second-order node will be 'a-b'.\n        self.separator = separator\n\n        ## A dictionary containing edges as well as edge weights\n        self.edges = _co.defaultdict( lambda: _np.array([0,0]) )\n\n        ## A dictionary containing the sets of successors of all nodes\n        self.successors = _co.defaultdict( lambda: set() )\n\n        ## A dictionary containing the sets of predecessors of all nodes\n        self.predecessors = _co.defaultdict( lambda: set() )\n\n        if k>1: \n            # For k>1 we need the first-order network to generate the null model\n            # and calculate the degrees of freedom\n\n            # For a multi-order model, the first-order network is generated multiple times!\n            # TODO: Make this more efficient\n            g1 = HigherOrderNetwork(paths, k=1)\n            A = g1.getAdjacencyMatrix(includeSubPaths = True, weighted=False, transposed=True)\n\n        if not nullModel:\n            # Calculate the frequency of all paths of\n            # length k, generate k-order nodes and set\n            # edge weights accordingly\n            for p in paths.paths[k]:\n                # For a 0-order model, we generate a dummy start node\n                if k==0: \n                    v = 'start'\n                    w = p[0]\n                else:\n                    # Generate names of k-order nodes v and w\n                    v = p[0]\n                    w = p[1]\n                    for l in range(1, k):\n                        v = v + separator + p[l]\n                        w = w + separator + p[l+1]\n                self.nodes.append(v)\n                self.nodes.append(w)\n\n                # as edge weights of the k-th order model, we sum the\n                # occurrence of paths of length k as subpath and longest path\n                self.edges[(v,w)] += paths.paths[k][p]\n                self.successors[v].add(w)\n                self.predecessors[w].add(v)\n            self.nodes = list(set(self.nodes)) # removes duplicates, does however randomize the order\n            # Note: For all sequences of length k which (i) have never been observed, but (ii) \n            #       do actually represent paths of length k in the first-order network, we \n            #       may want to include some 'escape' mechanism along the lines of (Cleary and Witten 1994)                        \n\n        else:\n            # generate the *expected* frequencies of all possible \n            # paths based on independently occurring (first-order) links\n            \n            # generate all possible paths of length k \n            # based on edges in the first-order network\n            possiblePaths = list(g1.edges.keys())\n\n            E_new = list()\n            for e1 in possiblePaths:\n                for e2 in g1.edges:\n                    if e1[-1] == e2[0]:\n                        p = e1 + (e2[1],)\n                        E_new.append(p)\n            possiblePaths = E_new\n\n            # validate that the number of unique generated paths corresponds to the sum of entries in A**k            \n            assert (A**k).sum() == len(possiblePaths), 'Expected ' + str((A**k).sum()) + ' paths but got ' + str(len(possiblePaths))\n            \n            if method == 'KOrderPi':\n                # compute stationary distribution of a random walker in the k-th order network\n                g_k = HigherOrderNetwork(paths, k=k, separator = separator, nullModel = False)\n                pi_k = HigherOrderNetwork.getLeadingEigenvector(g_k.getTransitionMatrix(includeSubPaths=True), normalized=True, lanczosVecs=lanczosVecs, maxiter=maxiter)\n            else:\n                # A = g1.getAdjacencyMatrix(includeSubPaths=True, weighted=True, transposed=False)\n                T = g1.getTransitionMatrix(includeSubPaths=True)\n\n            # assign link weights in k-order null model\n            for p in possiblePaths:\n                v = p[0]   \n                w = p[1]\n                # add k-order nodes and edges\n                for l in range(1, k):\n                    v = v + separator + p[l]\n                    w = w + separator + p[l+1]\n                self.nodes.append(v)\n                self.nodes.append(w)\n\n                # NOTE: under the null model's assumption of independent events, we\n                # have P(B|A) = P(A ^ B)/P(A) = P(A)*P(B)/P(A) = P(B)\n                # In other words: we are encoding a k-1-order Markov process in a k-order\n                # Markov model and for the transition probabilities T_AB in the k-order model \n                # we simply have to set the k-1-order probabilities, i.e. T_AB = P(B)\n                \n                # Solution A: Use entries of stationary distribution, \n                # which give stationary visitation frequencies of k-order node w\n                if method == 'KOrderPi':\n                    self.edges[(v,w)] = _np.array( [ 0, pi_k[ g_k.nodes.index(w) ] ] )\n\n                # Solution B: Use relative edge weight in first-order network\n                # Note that A is *not* transposed\n                # self.edges[(v,w)] = A[(g1.nodes.index(p[-2]),g1.nodes.index(p[-1]))] / A.sum()\n\n                # Solution C: Use transition probability in first-order network\n                # Note that T is transposed (!)\n                elif method == 'FirstOrderTransitions':\n                    p_vw = T[ ( g1.nodes.index(p[-1]), g1.nodes.index(p[-2]) ) ]\n                    self.edges[(v,w)] =  _np.array( [0, p_vw] )  \n\n                # Solution D: calculate k-path weights based on entries of squared k-1-order adjacency matrix\n\n                # Note: Solution B and C are equivalent\n                self.successors[v].add(w)\n            self.nodes = list(set(self.nodes))\n\n        # Compute degrees of freedom of models\n        if k==0:\n            # for a zero-order model, we just fit node probabilities (excluding the special 'start' node) \n            # Since probabilities must sum to one, the effective degree of freedom is one less than the number of nodes\n            # This holds for both the paths and the ngrams model\n            self.dof_paths = self.vcount() - 2\n            self.dof_ngrams = self.vcount() - 2\n        else:\n            # for a first-order model, self is the first-order network\n            if k==1:\n                g1 = self\n                A = g1.getAdjacencyMatrix(includeSubPaths = True, weighted=False, transposed=True)\n           \n            # Degrees of freedom in a higher-order ngram model\n            s = g1.vcount()\n\n            ## The degrees of freedom of the higher-order model, under the ngram assumption\n            self.dof_ngrams = (s**k)*(s-1)\n                 \n            # For k>0, the degrees of freedom of a path-based model depend on \n            # the number of possible paths of length k in the first-order network.\n            # Since probabilities in each row must sum to one, the degrees \n            # of freedom must be reduced by one for each k-order node \n            # that has at least one possible transition.\n\n            # (A**k).sum() counts the number of different paths of exactly length k\n            # based on the first-order network, which corresponds to the number of \n            # possible transitions in the transition matrix of a k-th order model. \n            paths_k = (A**k).sum()\n\n            # For the degrees of freedom, we must additionally consider that \n            # rows in the transition matrix must sum to one, i.e. we have to \n            # subtract one degree of freedom for every non-zero row in the (null-model)\n            # transition matrix. In other words, we subtract one for every path of length k-1 \n            # that can possibly be followed by at least one edge to a path of length k\n\n            # This can be calculated by counting the number of non-zero elements in the \n            # vector containing the row sums of A**k\n            non_zero = _np.count_nonzero((A**k).sum(axis=0))\n\n            ## The degrees of freedom of the higher-order model, under the paths assumption\n            self.dof_paths = paths_k - non_zero\n\n\n    def vcount(self):\n        \"\"\" Returns the number of nodes \"\"\"\n        return len(self.nodes)\n\n\n    def ecount(self):\n        \"\"\" Returns the number of links \"\"\"\n        return len(self.edges)\n\n\n    def totalEdgeWeight(self):\n        \"\"\" Returns the sum of all edge weights \"\"\"\n        if len(self.edges)>0:\n            return sum(self.edges.values())\n        else:\n            return _np.array([0,0])\n\n\n    def HigherOrderNodeToPath(self, node):\n        \"\"\"\n        Helper function that transforms a node in a\n        higher-order network of order k into a corresponding \n        path of length k-1. For a higher-order node 'a-b-c-d' \n        this function will return ('a','b','c','d')\n\n        @param node: The higher-order node to be transformed to a path.\n        \"\"\"\n\n        return tuple(node.split(self.separator))\n\n\n    def pathToHigherOrderNodes(self, path, k=None):\n        \"\"\"\n        Helper function that transforms a path into a sequence of k-order nodes \n        using the separator character of the HigherOrderNetwork instance \n\n        Consider an example path (a,b,c,d) with a separator string '-'\n        For k=1, the output will be the list of strings ['a', 'b', 'c', 'd']\n        For k=2, the output will be the list of strings ['a-b', 'b-c', 'c-d']\n        For k=3, the output will be the list of strings ['a-b-c', 'b-c-d']\n        etc. \n\n        @param path: the path tuple to turn into a sequence of higher-order nodes \n\n        @param k: the order of the representation to use (default: order of the HigherOrderNetwork instance)\n        \"\"\"\n        if k == None:\n            k = self.order\n        assert len(path)>k, 'Error: Path must be longer than k'\n\n        if k == 0 and len(path)==1:\n            return ['start', path[0]]\n\n        nodes = []\n    \n        for s in range(0, len(path)-k+1):\n            if s>len(path)-1 or s<0:\n                print(path)\n                print(s)\n                print(k)\n            v = path[s]\n            for l in range(1,k):\n                v = v + self.separator + path[s+l]\n            nodes.append(v)\n\n        return nodes\n\n\n    def getNodeNameMap(self):\n        \"\"\"\n        Returns a dictionary that can be used to map \n        node nodes to matrix/vector indices\n        \"\"\"\n\n        name_map = {}\n        for idx,v in enumerate(self.nodes):\n            name_map[v] = idx\n        return name_map\n\n\n    def getDoF(self, assumption=\"paths\"):\n        \"\"\"\n        Calculates the degrees of freedom (i.e. number of parameters) of \n        this k-order model. Depending on the modeling assumptions, this either\n        corresponds to the number of paths of length k in the first-order network \n        or to the number of all possible k-grams. The degrees of freedom of a model \n        can be used to assess the model complexity when calculating, e.g., the \n        Bayesian Information Criterion (BIC).\n\n        @param assumption: if set to 'paths', for the degree of freedon calculation in the BIC, \n            only paths in the first-order network topology will be considered. This is \n            needed whenever we are interested in a modeling of paths in a given network topology.\n            If set to 'ngrams' all possible n-grams will be considered, independent of whether they \n            are valid paths in the first-order network or not. The 'ngrams' and the 'paths' assumption \n            coincide if the first-order network is fully connected.\n        \"\"\"\n        assert assumption == 'paths' or assumption == 'ngrams', 'Error: Invalid assumption'\n        \n        if assumption == 'paths':            \n            return self.dof_paths\n        else:            \n            return self.dof_ngrams   \n\n\n    def getDistanceMatrix(self):\n        \"\"\"\n        Calculates shortest path distances between all pairs of \n        higher-order nodes using the Floyd-Warshall algorithm.\n        \"\"\"\n\n        Log.add('Calculating distance matrix in higher-order network (k = ' + str(self.order) + ') ...', Severity.INFO)\n\n        dist = _co.defaultdict( lambda: _co.defaultdict( lambda: _np.inf ) )\n\n        for v in self.nodes:\n            dist[v][v] = 0\n\n        for e in self.edges:\n            dist[e[0]][e[1]] = 1\n\n        for v in self.nodes:\n            for w in self.nodes:\n                for k in self.nodes:\n                    if dist[v][w] > dist[v][k] + dist[k][w]:\n                        dist[v][w] = dist[v][k] + dist[k][w]\n\n        Log.add('finished.', Severity.INFO)\n\n        return dist\n\n\n    def getShortestPaths(self):\n        \"\"\"\n        Calculates all shortest paths between all pairs of \n        higher-order nodes using the Floyd-Warshall algorithm.\n        \"\"\"\n        \n        Log.add('Calculating shortest paths in higher-order network (k = ' + str(self.order) + ') ...', Severity.INFO)\n\n        dist = _co.defaultdict( lambda: _co.defaultdict( lambda: _np.inf ) )\n        shortest_paths = _co.defaultdict( lambda: _co.defaultdict( lambda: set() ) )\n\n        for e in self.edges:\n            dist[e[0]][e[1]] = 1\n            shortest_paths[e[0]][e[1]].add(e)\n       \n        for v in self.nodes:\n            for w in self.nodes:\n                if v != w:\n                    for k in self.nodes:\n                        if dist[v][w] > dist[v][k] + dist[k][w]:\n                            dist[v][w] = dist[v][k] + dist[k][w]\n                            shortest_paths[v][w] = set()\n                            for p in list(shortest_paths[v][k]):\n                                for q in list(shortest_paths[k][w]):\n                                    shortest_paths[v][w].add(p+q[1:])\n                        elif dist[v][w] == dist[v][k] + dist[k][w]:\n                            for p in list(shortest_paths[v][k]):\n                                for q in list(shortest_paths[k][w]):\n                                    shortest_paths[v][w].add(p+q[1:])\n        \n        for v in self.nodes:\n            dist[v][v] = 0\n            shortest_paths[v][v].add((v,))\n\n        Log.add('finished.', Severity.INFO)\n\n        return shortest_paths\n\n\n    def getDistanceMatrixFirstOrder(self):\n        \"\"\"\n        Projects a distance matrix from a higher-order to \n        first-order nodes, while path lengths are calculated \n        based on the higher-order topology\n        \"\"\"\n\n        dist = self.getDistanceMatrix()\n        dist_first = _co.defaultdict( lambda: _co.defaultdict( lambda: _np.inf ) )\n\n        # calculate distances between first-order nodes based on distance in higher-order topology \n        for vk in dist:\n            for wk in dist[vk]:\n                v1 = self.HigherOrderNodeToPath(vk)[0]\n                w1 = self.HigherOrderNodeToPath(wk)[-1]\n                if dist[vk][wk] + self.order-1 < dist_first[v1][w1]:\n                    dist_first[v1][w1] = dist[vk][wk] + self.order - 1\n\n        return dist_first\n\n\n    def ClosenessCentrality(self):\n        \"\"\" \n        Calculates the closeness centralities of all nodes.\n        If the order of the higher-order network is larger than one \n        centralities calculated based on the higher-order \n        topology will automatically be projected back to first-order \n        nodes.\n        \"\"\"\n\n        dist_first = self.getDistanceMatrixFirstOrder()\n        node_centralities = _co.defaultdict( lambda: 0 )   \n        \n        Log.add('Calculating closeness centralities (k = ' + str(self.order) + ') ...', Severity.INFO)             \n\n        # calculate closeness values\n        for v1 in dist_first:\n            for w1 in dist_first[v1]:\n                if v1 != w1 and dist_first[v1][w1] < _np.inf:\n                    node_centralities[v1] += 1.0 / dist_first[v1][w1]\n        \n        # assign centrality zero to nodes not occurring on higher-order shortest paths\n        nodes = self.paths.getNodes()\n        for v in nodes:\n            node_centralities[v] += 0\n\n        Log.add('finished.', Severity.INFO)\n\n        return node_centralities    \n\n\n    def EvCent(self, projection='scaled', includeSubPaths=True):\n        \"\"\"\n        Calculates the eigenvector centralities of higher-order nodes. If \n        the order of the HigherOrderNetwork is larger than one, the centralities\n        will be projected to the first-order nodes. \n\n        @param projection: Indicates how the projection from k-th-order nodes (v1, v2, ... , v{k-1})\n            shall be performed. For the method 'all', the eigenvector centrality of the higher-order node \n            will be added to *all* first-order nodes on the path corresponding to the higher-order node. For \n            the method 'last', the centrality of the higher-order node will only be assigned to *last* \n            first-order node v{k-1}. For the method 'scaled' (default), the eigenvector centrality of higher-order \n            nodes will be assigned proportionally to first-order nodes, i.e. each of the three nodes in the \n            third-order node (a,b,c) will receive one third of the eigenvector centrality of (a,b,c).\n        @param includeSubPaths: whether or not to include subpath statistics in the calculation (default True)\n        \"\"\"\n        A = self.getAdjacencyMatrix(includeSubPaths=includeSubPaths, weighted=False, transposed=True)\n\n        # calculate leading eigenvector of A\n        w, v = _sla.eigs(A, k=1, which=\"LM\", ncv=13 )\n        \n        v = v.reshape(v.size,)\n       \n        higher_order_evcent = dict(zip(self.nodes, map(_np.abs, v)))\n\n        # project evcent of higher-order nodes to first-order network\n        first_order_evcent = _co.defaultdict( lambda: 0.0 )\n\n        # sum evcent values based on higher-order nodes \n        # and normalize the result\n        for v in self.nodes:\n            # turns node a-b-c in path tuple (a,b,c)\n            p = self.HigherOrderNodeToPath(v)\n            if projection == 'all':\n                # assign evcent of higher-order node to all first-order nodes\n                for x in p:                    \n                    first_order_evcent[x] += higher_order_evcent[v]\n            elif projection == 'scaled':\n                for x in p:                    \n                    first_order_evcent[x] += higher_order_evcent[v] / float( len(p) )\n            elif projection == 'last':\n                # assign evcent of higher-order node to last first-order node\n                first_order_evcent[p[-1]] += higher_order_evcent[v]\n            elif projection == 'first':\n                # assign evcent of higher-order node to last first-order node\n                first_order_evcent[p[0]] += higher_order_evcent[v]\n\n        # for scaled, values sum to one anyway\n        if projection != 'scaled':\n            for v in first_order_evcent:\n                first_order_evcent[v] /= sum(first_order_evcent.values())\n\n        Log.add('finished.', Severity.INFO)\n\n        return first_order_evcent\n\n        return v\n\n\n\n    def PageRank(self, alpha=0.85, maxIterations=100, convergenceThres=1.0e-6, projection='scaled', includeSubPaths=True):\n        \"\"\"\n        Calculates the PageRank of higher-order nodes based on a \n        power iteration. If the order of the higher-order network is larger than one,\n        the PageRank calculated based on the higher-order\n        topology will automatically be projected back to first-order \n        nodes.\n\n        @param projection: Indicates how the projection from k-th-order nodes (v1, v2, ... , v{k-1})\n            shall be performed. For the method 'all', the pagerank value of the higher-order node \n            will be added to *all* first-order nodes on the path corresponding to the higher-order node. For \n            the method 'last', the PR value of the higher-order node will only be assigned to *last* \n            first-order node v{k-1}. For the method 'scaled' (default), the PageRank of higher-order \n            nodes will be assigned proportionally to first-order nodes, i.e. each of the three nodes in the \n            third-order node (a,b,c) will receive one third of the PageRank of (a,b,c).\n        @param includeSubpaths: whether or not to use subpath statistics in the PageRank calculation\n        \"\"\"\n\n        assert projection == 'all' or projection == 'last' or projection == 'first' or projection == 'scaled', 'Invalid projection method'\n\n        Log.add('Calculating PageRank in ' + str(self.order) + '-th order network...', Severity.INFO)\n\n        higher_order_PR = _co.defaultdict( lambda: 0 )\n\n        n = float(len(self.nodes))\n\n        assert n>0, \"Number of nodes is zero\"\n\n        # entries A[s,t] give directed link s -> t\n        A = self.getAdjacencyMatrix(includeSubPaths=includeSubPaths, weighted=False, transposed=False)\n\n        # sum of outgoing node degrees\n        row_sums = _sp.array(A.sum(axis=1)).flatten()\n        \n        row_sums[row_sums != 0] = 1.0 / row_sums[row_sums != 0]\n        d = _sp.where(row_sums == 0)[0]\n\n        Q = _sparse.spdiags(row_sums.T, 0, *A.shape, format='csr')\n        A = Q * A\n\n        p = _sp.array([1.0 / n] * int(n))\n        pr = p\n       \n        # Power iteration\n        for i in range(maxIterations):\n            last = pr\n            pr = alpha * (pr * A + sum(pr[d]) * p) + (1 - alpha) * p\n\n            if _sp.absolute(pr - last).sum() < n * convergenceThres:\n                higher_order_PR = dict(zip(self.nodes, map(float, pr)))\n                break\n\n        if self.order == 1:\n            return higher_order_PR\n\n        # project PageRank of higher-order nodes to first-order network\n        first_order_PR = _co.defaultdict( lambda: 0.0 )\n\n        # sum PageRank values based on higher-order nodes \n        # and normalize the result\n        for v in self.nodes:\n            # turns node a-b-c in path tuple (a,b,c)\n            p = self.HigherOrderNodeToPath(v)\n            if projection == 'all':\n                # assign PR of higher-order node to all first-order nodes\n                for x in p:                    \n                    first_order_PR[x] += higher_order_PR[v]\n            elif projection == 'scaled':\n                for x in p:\n                    # each node on e.g. a 4-th-order path a-b-c-d receives one fourth of the \n                    # PageRank value, to ensure that the resulting first-order PageRank sums \n                    # to one\n                    first_order_PR[x] += higher_order_PR[v] / float( len(p) )\n            elif projection == 'last':\n                # assign PR of higher-order node to last first-order node\n                first_order_PR[p[-1]] += higher_order_PR[v]\n            elif projection == 'first':\n                # assign PR of higher-order node to last first-order node\n                first_order_PR[p[0]] += higher_order_PR[v]\n\n        # for projection method 'scaled', the values sum to one anyway\n        if projection != 'scaled':\n            for v in first_order_PR:\n                first_order_PR[v] /= sum(first_order_PR.values())\n\n        # assign centrality zero to nodes not occurring in higher-order PR\n        nodes = self.paths.getNodes()\n        for v in nodes:\n            first_order_PR[v] += 0\n\n        Log.add('finished.', Severity.INFO)\n\n        return first_order_PR\n\n\n    def HigherOrderPathToFirstOrder(self, path):\n        \"\"\"\n        Maps a path in the higher-order network \n        to a path in the first-order network. As an \n        example, the second-order path ('a-b', 'b-c', 'c-d')\n        of length two is mapped to the first-order path ('a','b','c','d')\n        of length four. In general, a path of length l in a network of \n        order k is mapped to a path of length l+k-1 in the first-order network. \n\n        @param path: The higher-order path that shall be mapped to the first-order network\n        \"\"\"\n        p1 = self.HigherOrderNodeToPath(path[0])\n        for x in path[1:]:\n            p1 += (self.HigherOrderNodeToPath(x)[-1],)\n        return p1\n\n\n    def BetweennessCentrality(self, normalized=False):\n        \"\"\" \n        Calculates the betweenness centralities of all nodes.\n        If the order of the higher-order network is larger than one \n        centralities calculated based on the higher-order \n        topology will automatically be projected back to first-order \n        nodes.\n\n        @param normalized: If set to True, betweenness centralities of \n            nodes will be scaled by the maximum value (default False)\n        \"\"\"\n\n        shortest_paths = self.getShortestPaths()\n        node_centralities = _co.defaultdict( lambda: 0 )\n\n        shortest_paths_firstorder =  _co.defaultdict( lambda:  _co.defaultdict( lambda: set() ) )\n\n        Log.add('Calculating betweenness centralities (k = ' + str(self.order) + ') ...', Severity.INFO)\n\n        for sk in shortest_paths:\n            for dk in shortest_paths:\n                s1 = self.HigherOrderNodeToPath(sk)[0]\n                d1 = self.HigherOrderNodeToPath(dk)[-1]\n                \n                # we consider a path in a k-th order network \n                # connecting first-order node s1 to d1\n                for pk in shortest_paths[sk][dk]:\n                     # convert k-th order path to first-order path and add\n                    shortest_paths_firstorder[s1][d1].add(self.HigherOrderPathToFirstOrder(pk))\n\n\n        for s1 in shortest_paths_firstorder:\n            for d1 in shortest_paths_firstorder[s1]:                               \n                for p1 in shortest_paths_firstorder[s1][d1]:\n                    # increase betweenness centrality of all intermediary nodes\n                    # on path from s1 to d1\n                    for v in p1[1:-1]:\n                        if s1 != v != d1:\n                            #print('node ' + x + ': ' + str(1.0 / len(shortest_paths[vk][wk])))\n                            node_centralities[v] += 1.0 / (len(shortest_paths_firstorder[s1][d1]) + self.order-1)\n                            #else:\n                            #    node_centralities[v] += 1.0\n        if normalized:\n            m = max(node_centralities.values())\n            for v in node_centralities:\n                node_centralities[v] /= m\n\n        # assign centrality zero to nodes not occurring on higher-order shortest paths\n        nodes = self.paths.getNodes()\n        for v in nodes:\n            node_centralities[v] += 0\n\n        Log.add('finished.', Severity.INFO)\n\n        return node_centralities\n\n\n    def reduceToGCC(self):\n        \"\"\" \n        Reduces the higher-order network to its \n        largest (giant) strongly connected component \n        (using Tarjan's algorithm)\n        \"\"\"\n\n        # nonlocal variables (!)        \n        index = 0\n        S = []\n        indices = _co.defaultdict( lambda: None )\n        lowlink = _co.defaultdict( lambda: None )        \n        onstack = _co.defaultdict( lambda: False )\n\n        # Tarjan's algorithm\n        def strong_connect(v):\n            nonlocal index\n            nonlocal S\n            nonlocal indices\n            nonlocal lowlink            \n            nonlocal onstack\n                        \n            indices[v] = index\n            lowlink[v] = index\n            index += 1\n            S.append(v)\n            onstack[v] = True\n\n            for w in self.successors[v]:\n                if indices[w] == None:\n                    strong_connect(w)\n                    lowlink[v] = min(lowlink[v], lowlink[w])\n                elif onstack[w]:\n                    lowlink[v] = min(lowlink[v], indices[w])\n            \n            # Generate SCC of node v\n            component = set()\n            if lowlink[v] == indices[v]:\n                while True:\n                    w = S.pop()\n                    onstack[w] = False\n                    component.add(w)\n                    if v==w:\n                        break\n            return component\n\n        # Get largest strongly connected component \n        components = _co.defaultdict( lambda: set() )\n        max_size = 0\n        max_head = None\n        for v in self.nodes:\n            if indices[v] == None:\n                components[v] = strong_connect(v)\n                if len(components[v]) > max_size:\n                    max_head = v\n                    max_size = len(components[v])\n        \n        scc = components[max_head]\n\n        # Reduce higher-order network to SCC\n        for v in list(self.nodes):\n            if v not in scc:\n                self.nodes.remove(v)\n                del self.successors[v]\n\n        for (v,w) in list(self.edges):\n            if v not in scc or w not in scc:\n                del self.edges[(v,w)]\n                    \n\n    def summary(self):\n        \"\"\" \n        Returns a string containing basic summary statistics \n        of this higher-order graphical model instance\n        \"\"\"\n\n        summary = 'Graphical model of order k = ' + str(self.order)\n        summary += '\\n'\n        summary += 'Nodes:\\t\\t\\t\\t' +  str(self.vcount()) + '\\n'\n        summary += 'Links:\\t\\t\\t\\t' + str(self.ecount()) + '\\n'\n        summary += 'Total weight (sub/longest):\\t' + str(self.totalEdgeWeight()[0]) + '/' + str(self.totalEdgeWeight()[1]) + '\\n'\n        return summary\n\n\n    def __str__(self):\n        \"\"\"\n        Returns the default string representation of \n        this graphical model instance\n        \"\"\"\n        return self.summary()\n\n\n    def degrees(self, includeSubPaths=True, weighted=True, mode=\"OUT\"):\n        \"\"\"\n        Returns the (weighted) degrees of nodes in the higher-order network\n\n        @param weighted: If true, calculates the sum of weights for each node. If false, the \n            number of links is calculated\n\n        @param mode: either \"IN\", \"OUT\", or \"TOTAL\" \n        \"\"\"\n        degrees = [0]*self.vcount()\n        for v in self.nodes:\n            for u,w in self.edges:\n                if (mode == \"OUT\" and u == v) or (mode == \"IN\" and w == v) or (mode == \"TOTAL\" and (u==v or w==v)):\n                    if weighted:\n                        if includeSubPaths:\n                            degrees[self.nodes.index(v)] += self.edges[(u,w)].sum()\n                        else: \n                            degrees[self.nodes.index(v)] += self.edges[(u,w)][1]\n                    else:\n                        if includeSubPaths:\n                            degrees[self.nodes.index(v)] += 1\n                        else: \n                            if self.edges[(u,w)][1]>0:\n                                degrees[self.nodes.index(v)] += 1\n        return degrees\n\n\n    def getAdjacencyMatrix(self, includeSubPaths = True, weighted = True, transposed=False):\n        \"\"\"\n        Returns a sparse adjacency matrix of the higher-order network. By default, the entry \n            corresponding to a directed link source -> target is stored in row s and column t\n            and can be accessed via A[s,t].\n    \n        @param includeSubPaths: if set to True, the returned adjacency matrix will \n            account for the occurrence of links of order k (i.e. paths of length k-1)\n            as subpaths\n\n        @param weighted: if set to False, the function returns a binary adjacency matrix.\n          If set to True, adjacency matrix entries will contain the weight of an edge.\n      \n        @param transposed: whether to transpose the matrix or not.\n        \"\"\"        \n    \n        row = []\n        col = []\n        data = []\n           \n        if transposed:\n            for s,t in self.edges:\n                row.append(self.nodes.index(t))\n                col.append(self.nodes.index(s))\n        else:\n            for s,t in self.edges:\n                row.append(self.nodes.index(s))\n                col.append(self.nodes.index(t))\n\n        # create array with non-zero entries\n        if not weighted:\n            data = _np.ones(len(self.edges.keys()))\n        else:\n            if includeSubPaths:\n                data = _np.array([float(x.sum()) for x in self.edges.values()])\n            else:\n                data = _np.array([float(x[1]) for x in self.edges.values()])\n\n        return _sparse.coo_matrix( (data, (row, col)), shape=(self.vcount(), self.vcount()) ).tocsr()\n\n\n\n    def getTransitionMatrix(self, includeSubPaths=True):\n        \"\"\"\n        Returns a (transposed) random walk transition matrix \n        corresponding to the higher-order network.\n\n        @param includeSubpaths: whether or not to include subpath statistics in the \n            transition probability calculation (default True)\n        \"\"\"\n        row = []\n        col = []\n        data = []\n        D = self.degrees(includeSubPaths=includeSubPaths, weighted=True, mode='OUT')\n        for (s,t) in self.edges:\n            # either s->t has been observed as a longest path, or we are interested in subpaths as well\n\n            # the following makes sure that we do not accidentially consider zero-weight edges (automatically added by default_dic)\n            if (self.edges[(s,t)][1] > 0) or (includeSubPaths and self.edges[(s,t)][0]>0):\n                row.append(self.nodes.index(t))\n                col.append(self.nodes.index(s))\n                if includeSubPaths:\n                    count = self.edges[(s,t)].sum()\n                else:\n                    count = self.edges[(s,t)][1]\n                #print((s,t))\n                #print(D[self.nodes.index(s)])\n                #print(count)\n                assert D[self.nodes.index(s)]>0, 'Encountered zero out-degree for node ' + str(s) + ' while weight of link (' + str(s) +  ', ' + str(t) + ') is non-zero.'\n                prob = count / D[self.nodes.index(s)]\n                if prob < 0 or prob > 1:\n                    tn.Log.add('Encountered transition probability outside [0,1] range.', Severity.ERROR)\n                    raise ValueError()\n                data.append( prob )\n    \n        data = _np.array(data)\n        data = data.reshape(data.size,)\n\n        return _sparse.coo_matrix( (data, (row, col)), shape=(self.vcount(), self.vcount()) ).tocsr()\n\n\n    @staticmethod\n    def getLeadingEigenvector(A, normalized=True, lanczosVecs = 15, maxiter = 1000):\n        \"\"\"Compute normalized leading eigenvector of a given matrix A.\n\n        @param A: sparse matrix for which leading eigenvector will be computed\n        @param normalized: wheter or not to normalize. Default is C{True}\n        @param lanczosVecs: number of Lanczos vectors to be used in the approximate\n            calculation of eigenvectors and eigenvalues. This maps to the ncv parameter \n            of scipy's underlying function eigs. \n        @param maxiter: scaling factor for the number of iterations to be used in the \n            approximate calculation of eigenvectors and eigenvalues. The number of iterations \n            passed to scipy's underlying eigs function will be n*maxiter where n is the \n            number of rows/columns of the Laplacian matrix.\n        \"\"\"\n\n        if _sparse.issparse(A) == False:\n            raise TypeError(\"A must be a sparse matrix\")\n\n        # NOTE: ncv sets additional auxiliary eigenvectors that are computed\n        # NOTE: in order to be more confident to find the one with the largest\n        # NOTE: magnitude, see https://github.com/scipy/scipy/issues/4987\n        w, pi = _sla.eigs( A, k=1, which=\"LM\", ncv=lanczosVecs, maxiter=maxiter)\n        pi = pi.reshape(pi.size,)\n        if normalized:\n            pi /= sum(pi)\n        return pi\n\n\n    def getLaplacianMatrix(self, includeSubPaths=True):\n        \"\"\"\n        Returns the transposed Laplacian matrix corresponding to the higher-order network.\n\n        @param includeSubpaths: Whether or not subpath statistics shall be included in the \n            calculation of matrix weights\n        \"\"\"   \n    \n        T = self.getTransitionMatrix(includeSubPaths)\n        I  = _sparse.identity( self.vcount() )\n\n        return I-T\n\n\n    def getEigenValueGap(self, includeSubPaths=True, lanczosVecs = 15, maxiter = 20):\n        \"\"\"\n        Returns the eigenvalue gap of the transition matrix.\n\n        @param includeSubPaths: whether or not to include subpath statistics in the \n            calculation of transition probabilities.\n        \"\"\"\n    \n        #NOTE to myself: most of the time goes for construction of the 2nd order\n        #NOTE            null graph, then for the 2nd order null transition matrix   \n    \n        Log.add('Calculating eigenvalue gap ... ', Severity.INFO)\n\n        # Build transition matrices\n        T = self.getTransitionMatrix(includeSubPaths)\n    \n        # Compute the two largest eigenvalues\n        # NOTE: ncv sets additional auxiliary eigenvectors that are computed\n        # NOTE: in order to be more confident to actually find the one with the largest\n        # NOTE: magnitude, see https://github.com/scipy/scipy/issues/4987\n        w2 = _sla.eigs(T, which=\"LM\", k=2, ncv=lanczosVecs, return_eigenvectors=False, maxiter = maxiter)\n        evals2_sorted = _np.sort(-_np.absolute(w2))\n        \n        Log.add('finished.', Severity.INFO)\n    \n        return _np.abs(evals2_sorted[1])    \n\n\n    def getFiedlerVectorSparse(self, normalized = True, lanczosVecs = 15, maxiter = 20):\n        \"\"\"Returns the (sparse) Fiedler vector of the higher-order network. The Fiedler \n        vector can be used for a spectral bisectioning of the network.\n     \n        Note that sparse linear algebra for eigenvalue problems with small eigenvalues \n        is problematic in terms of numerical stability. Consider using the dense version\n        of this method in this case. Note also that the sparse Fiedler vector might be scaled by \n        a factor (-1) compared to the dense version.\n          \n        @param normalized: whether (default) or not to normalize the fiedler vector.\n          Normalization is done such that the sum of squares equals one in order to\n          get reasonable values as entries might be positive and negative.\n        @param lanczosVecs: number of Lanczos vectors to be used in the approximate\n            calculation of eigenvectors and eigenvalues. This maps to the ncv parameter \n            of scipy's underlying function eigs. \n        @param maxiter: scaling factor for the number of iterations to be used in the \n            approximate calculation of eigenvectors and eigenvalues. The number of iterations \n            passed to scipy's underlying eigs function will be n*maxiter where n is the \n            number of rows/columns of the Laplacian matrix.\n        \"\"\"\n    \n        # NOTE: The transposed matrix is needed to get the \"left\" eigenvectors\n        L = self.getLaplacianMatrix()\n\n        # NOTE: ncv sets additional auxiliary eigenvectors that are computed\n        # NOTE: in order to be more confident to find the one with the largest\n        # NOTE: magnitude, see https://github.com/scipy/scipy/issues/4987\n        maxiter = maxiter*L.get_shape()[0]\n        w = _sla.eigs( L, k=2, which=\"SM\", ncv=lanczosVecs, return_eigenvectors=False, maxiter=maxiter )\n    \n        # compute a sparse LU decomposition and solve for the eigenvector \n        # corresponding to the second largest eigenvalue\n        n = L.get_shape()[0]\n        b = _np.ones(n)\n        evalue = _np.sort(_np.abs(w))[1]\n        A = (L[1:n,:].tocsc()[:,1:n] - _sparse.identity(n-1).multiply(evalue))\n        b[1:n] = A[0,:].toarray()\n    \n        lu = _sla.splu(A)\n        b[1:n] = lu.solve(b[1:n])\n\n        if normalized:\n            b /= np.sqrt(np.inner(b, b))\n        return b\n\n\n    def getFiedlerVectorDense(self):\n        \"\"\"\n         Returns the (dense)Fiedler vector of the higher-order network. The Fiedler \n         vector can be used for a spectral bisectioning of the network.             \n        \"\"\"\n    \n        # NOTE: The Laplacian is transposed for the sparse case to get the left\n        # NOTE: eigenvalue.\n        L = self.getLaplacianMatrix()\n        # convert to dense matrix and transpose again to have the untransposed\n        # laplacian again.\n        w, v = _la.eig(L.todense().transpose(), right=False, left=True)\n\n        return v[:,_np.argsort(_np.absolute(w))][:,1]\n\n\n\n    def getAlgebraicConnectivity(self, lanczosVecs = 15, maxiter = 20):\n        \"\"\"\n        Returns the algebraic connectivity of the higher-order network.    \n        \n        @param lanczosVecs: number of Lanczos vectors to be used in the approximate\n            calculation of eigenvectors and eigenvalues. This maps to the ncv parameter \n            of scipy's underlying function eigs. \n        @param maxiter: scaling factor for the number of iterations to be used in the \n            approximate calculation of eigenvectors and eigenvalues. The number of iterations \n            passed to scipy's underlying eigs function will be n*maxiter where n is the\n            number of rows/columns of the Laplacian matrix.         \n        \"\"\"\n    \n        Log.add('Calculating algebraic connectivity ... ', Severity.INFO)\n\n        L = self.getLaplacianMatrix()\n        # NOTE: ncv sets additional auxiliary eigenvectors that are computed\n        # NOTE: in order to be more confident to find the one with the largest\n        # NOTE: magnitude, see https://github.com/scipy/scipy/issues/4987\n        w = _sla.eigs( L, which=\"SM\", k=2, ncv=lanczosVecs, return_eigenvectors=False, maxiter = maxiter )\n        evals_sorted = _np.sort(_np.absolute(w))\n\n        Log.add('finished.', Severity.INFO)\n\n        return _np.abs(evals_sorted[1])\n"
  },
  {
    "path": "pathpy/Log.py",
    "content": "﻿# -*- coding: utf-8 -*-\n\"\"\"\n    pathpy is an OpenSource python package for the analysis of sequential data\n    on pathways and temporal networks using higher- and multi order graphical models\n\n    Copyright (C) 2016-2017 Ingo Scholtes, ETH Zürich\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU Affero General Public License as published\n    by the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the\n    GNU Affero General Public License for more details.\n\n    You should have received a copy of the GNU Affero General Public License\n    along with this program.  If not, see <http://www.gnu.org/licenses/>.\n\n    Contact the developer:\n\n    E-mail: ischoltes@ethz.ch\n    Web:    http://www.ingoscholtes.net\n\"\"\"\n\nimport enum\nimport time\nimport io\nimport sys\n\nclass Severity(enum.IntEnum):\n    \"\"\" An enumeration that can be used to indicate\n        the severity of log messages, and which can be\n        used tpo filter messages based on severities.\n    \"\"\"\n\n    ## Error messages\n    ERROR = 4\n\n    ## Warning messages\n    WARNING = 3\n\n    ## Informational messages (default minimum level)\n    INFO = 2\n\n    ## Messages regarding timing and performance\n    TIMING = 1\n\n    ## Debug messages (really verbose)\n    DEBUG = 0\n\n\nclass Log:\n    \"\"\" A simple logging class, that allows to select what messages should\n        be recorded in the output, and where these message should be directed.\n    \"\"\"\n\n    ## the output stream to which log entries will be written\n    output_stream = sys.stdout\n\n    ## The minimum severity level of messages to be logged\n    min_severity  = Severity.INFO\n\n\n    @staticmethod\n    def setMinSeverity(severity):\n        \"\"\" Sets the minimum sveerity level a message\n        needs to have in order to be recorded in the output stream.\n        By default, any message which has a severity of at least\n        Severity.INFO will be written to the output stream. All messages\n        with lower priority will be surpressed.\n        \"\"\"\n        Log.min_severity = severity\n\n\n    @staticmethod\n    def setOutputStream(stream):\n        \"\"\" Sets the output stream to which all messages will be\n            written. By default, this is sys.stdout, but it can be\n            changed in order to redirect the log to a logfile.\n        \"\"\"\n        output_stream = stream\n\n\n    @staticmethod\n    def add(msg, severity=Severity.INFO):\n        \"\"\" Adds a message with the given severity to the log. This message will be written\n            to the log output stream, which by default is sys.stdout. A newline character\n            will be added to the message by default.\n        \"\"\"\n        if severity >= Log.min_severity:\n            ts = time.strftime(\"%Y-%m-%d %H:%M:%S\", time.gmtime())\n            Log.output_stream.write(ts + ' [' + str(severity) + ']\\t' + msg + '\\n')\n            Log.output_stream.flush()\n"
  },
  {
    "path": "pathpy/MarkovSequence.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\n    pathpy is an OpenSource python package for the analysis of sequential data on pathways and temporal networks using higher- and multi order graphical models\n\n    Copyright (C) 2016-2017 Ingo Scholtes, ETH Zürich\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU Affero General Public License as published\n    by the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU Affero General Public License for more details.\n\n    You should have received a copy of the GNU Affero General Public License\n    along with this program.  If not, see <http://www.gnu.org/licenses/>.\n\n    Contact the developer:\n\n    E-mail: ischoltes@ethz.ch\n    Web:    http://www.ingoscholtes.net\n\"\"\"\n\nimport numpy as _np\nimport collections as _co\nimport bisect as _bs\nimport itertools as _iter\n\nimport scipy.sparse as _sparse\nimport scipy.misc as _misc\nimport scipy.sparse.linalg as _sla\nimport scipy.linalg as _la\n\nfrom scipy.stats import chi2\n\nfrom pathpy.Log import Log\nfrom pathpy.Log import Severity\n\n_np.seterr(all='warn')\n\n\nclass MarkovSequence:\n    \"\"\" Instances of this class can be used to fit\n        standard higher-order Markov models for\n        sequences generated from concatenated paths \"\"\"\n\n    def __init__(self, sequence):\n        \"\"\"\n        Generates a Markov model for a sequence, given\n        as a single list of strings\n        \"\"\"\n\n        ## The sequence to be modeled\n        self.sequence = sequence\n\n        ## The transition probabilities of higher-order Markov chains\n        self.P = {}\n\n        ## the set of states of higher-order Markov chains\n        self.states = {}\n        self.states[1] = set(sequence)\n\n\n    def fitMarkovModel(self, k=1):\n        \"\"\" Generates a k-th order Markov model\n            for the underlying sequence\n        \"\"\"\n\n        # TODO: Add support for k=0\n\n        assert len(self.sequence)>0, \"Error: Empty sequence\"\n\n        # MLE fit of transition probabilities\n        self.P[k] = _co.defaultdict( lambda:  _co.defaultdict( lambda: 0.0 )  )\n\n        Log.add('Fitting Markov model with order k = ' + str(k))\n\n        # Generate initial memory prefix\n        mem = (())\n        for s in self.sequence[:k]:\n            mem += (s,)\n\n        # count state transitions\n        for s in self.sequence[k:]:\n            self.P[k][mem][s] += 1.0\n\n            # shift memory by one element\n            mem = mem[1:] + (s,)\n\n        # normalize transitions\n        for m in self.P[k]:\n            S = float(sum(self.P[k][m].values()))\n            for s in self.P[k][m]:\n                self.P[k][m][s] /= S\n        Log.add('finished.')\n\n\n    def getLikelihood(self, k=1, log=True):\n        \"\"\"\n        Returns the likelihood of the sequence\n        assuming a k-th order Markov model\n        \"\"\"\n\n        if k not in self.P:\n            self.fitMarkovModel(k)\n\n        L = 0\n\n         # Generate initial prefix\n        mem = (())\n        for s in self.sequence[:k]:\n            mem += (s,)\n\n        for s in self.sequence[k:]:\n            L += _np.log(self.P[k][mem][s])\n\n            # shift memory by one element\n            mem = mem[1:] + (s,)\n\n        if log:\n            return L\n        else:\n            return _np.exp(L)\n\n\n    def getBIC(self, k=1, m=1):\n        \"\"\" Returns the Bayesian Information Criterion\n            assuming a k-th order Markov model \"\"\"\n\n        if k not in self.P:\n            self.fitMarkovModel(k)\n\n        if m not in self.P:\n            self.fitMarkovModel(m)\n\n        L_k = self.getLikelihood(k, log=True)\n        L_m = self.getLikelihood(m, log=True)\n\n        s = len(self.states[1])\n        n = len(self.sequence)-k\n\n        # the transition matrix of a first-order model with s states has s**2 entries, subject to the\n        # constraint that entries in each row must sum up to one (thus effectively reducing\n        # the degrees of freedom by a factor of s, i.e. we have s**2-s**1. Generalizing this to order k,\n        # we arrive at s**k * (s-1) = s**(k+1) - s**k derees of freedom\n        bic = _np.log(n) * (s**k - s**m) * (s-1) - 2.0 * (L_k-L_m)\n\n        return bic\n\n\n    def getAIC(self, k=1, m=1):\n        \"\"\" Returns the Aikake Information Criterion\n            assuming a k-th order Markov model \"\"\"\n\n        if k not in self.P:\n            self.fitMarkovModel(k)\n\n        if m not in self.P:\n            self.fitMarkovModel(m)\n\n        L_k = self.getLikelihood(k, log=True)\n        L_m = self.getLikelihood(m, log=True)\n\n        s = len(self.states[1])\n        n = len(self.sequence)\n\n        aic = 2 * (s**k - s**m) * (s-1) - 2.0 * (L_k - L_m)\n\n        return aic\n\n\n    def estimateOrder(self, maxOrder, method='BIC'):\n        \"\"\" Estimates the optimal order of a Markov model\n            based on Likelihood, BIC or AIC \"\"\"\n\n        assert method == 'BIC' or method == 'AIC' or method == 'Likelihood', \"Error: Expecting method 'AIC', 'BIC' or 'Likelihood'\"\n\n        values = []\n        orders = []\n\n        # We need k < m for the BIC and AIC calculation, which\n        # is why we only test up to maxOrder - 1\n        for k in range(1, maxOrder):\n            if k not in self.P:\n                self.fitMarkovModel(k)\n\n            orders.append(k)\n\n            if method == 'AIC':\n                values.append(self.getAIC(k, maxOrder))\n            elif method == 'BIC':\n                values.append(self.getBIC(k, maxOrder))\n            elif method == 'Likelihood':\n                values.append(self.getLikelihood(k, log=True))\n\n        if method == 'Likelihood':\n            values.append(self.getLikelihood(maxOrder, log=True))\n            orders.append(maxOrder)\n\n            # return order at which likelihood is maximized\n            return orders[_np.argmax(values)]\n        else:\n            # return order at which BIC/AIC are minimized\n            return orders[_np.argmin(values)]\n"
  },
  {
    "path": "pathpy/MultiOrderModel.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\n    pathpy is an OpenSource python package for the analysis of sequential data on pathways and temporal networks using higher- and multi order graphical models\n\n    Copyright (C) 2016-2017 Ingo Scholtes, ETH Zürich\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU Affero General Public License as published\n    by the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU Affero General Public License for more details.\n\n    You should have received a copy of the GNU Affero General Public License\n    along with this program.  If not, see <http://www.gnu.org/licenses/>.\n\n    Contact the developer:\n\n    E-mail: ischoltes@ethz.ch\n    Web:    http://www.ingoscholtes.net\n\"\"\"\n\nimport numpy as _np\nimport collections as _co\nimport bisect as _bs\nimport itertools as _iter\n\nimport scipy.sparse as _sparse\nimport scipy.misc as _misc\nimport scipy.sparse.linalg as _sla\nimport scipy.linalg as _la\nfrom scipy.stats import chi2\n\nfrom pathpy.Log import Log\nfrom pathpy.Log import Severity\nfrom pathpy.HigherOrderNetwork import HigherOrderNetwork\n\n_np.seterr(all='warn')\n\n\nclass MultiOrderModel:\n    \"\"\"Instances of this class represent a hierarchy of\n        higher-order networks which collectively represent\n        a multi-order model for path statistics. \"\"\"\n\n\n    def __init__(self, paths, maxOrder=1):\n        \"\"\"\n        Generates a hierarchy of higher-order\n        models for the given path statistics,\n        up to a given maximum order\n\n        @param paths: the paths instance for which the model should be created\n        @param maxOrder: the maximum order of the multi-order model\n        \"\"\"\n\n        ## A dictionary containing the layers of HigherOrderNetworks, where\n        ## layers[k] contains the network of order k\n        self.layers = {}\n\n        ## the maximum order of this multi-order model\n        self.maxOrder = maxOrder\n\n        ## the paths object from which this multi-order model was created\n        self.paths = paths\n\n        ## a dictionary of transition matrices for all layers of the model\n        self.T = {}\n\n        for k in range(maxOrder+1):\n            Log.add('Generating ' + str(k) + '-th order network layer ...')\n            self.layers[k] = HigherOrderNetwork(paths, k, paths.separator, False)\n\n            # compute transition matrices for all layers. In order to use the maximally\n            # available statistics, we always use sub paths in the calculation\n            self.T[k] = self.layers[k].getTransitionMatrix(includeSubPaths=True)\n\n        Log.add('finished.')\n\n\n    def summary(self):\n        \"\"\"\n        Returns a string containing basic summary information\n        of this multi-order model instance.\n        \"\"\"\n        summary = 'Multi-order model (max. order = ' + str(self.maxOrder) + ', DoF (paths/ngrams) = ' + str(self.getDegreesOfFreedom(assumption='paths')) + '/' + str(self.getDegreesOfFreedom(assumption='ngrams')) + ')\\n'\n        summary += '===========================================================================\\n'\n        for k in range(self.maxOrder+1):\n            summary += 'Layer k = ' + str(k) + '\\t' + str(self.layers[k].vcount()) + ' nodes, ' + str(self.layers[k].ecount()) + ' links, ' + str(self.layers[k].totalEdgeWeight().sum()) + ' paths, DoF (paths/ngrams) = ' + str(int(self.layers[k].getDoF('paths'))) + '/' + str(int(self.layers[k].getDoF('ngrams'))) + '\\n'\n        return summary\n\n\n    def __str__(self):\n        \"\"\"\n        Returns the default string representation of\n        this multi-order model instance\n        \"\"\"\n        return self.summary()\n\n\n    def saveStateFile(self, filename, layer=None):\n        \"\"\"\n        Saves the multi-order model in state file format \n        suitable to be used with InfoMap\n\n        @param layer: if none, all layers will be export. If set to k, only \n            the k-th layer of the model will be exported.\n        \"\"\"\n\n        assert layer, 'Error: export of all layers currently not supported'\n\n        name_map = self.layers[layer].getNodeNameMap()\n        first_layer_map = self.layers[1].getNodeNameMap()\n        T = self.layers[layer].getTransitionMatrix()\n\n        file = open(filename,'w')\n\n        file.write('# file generated by pathpy\\n')\n        file.write('*Vertices {0}\\n'.format(self.layers[1].vcount()))\n\n        for i in range(self.layers[1].vcount()):\n            # each line contains uniqueID [name]\n            file.write('{0} \"{1}\"\\n'.format(first_layer_map[self.layers[1].nodes[i]]+1, self.layers[1].nodes[i]))\n\n        # Write higher-order nodes to states section\n        file.write('*States\\n'.format(self.layers[1].vcount()))\n        for i in range(self.layers[layer].vcount()):\n            v = self.layers[layer].nodes[i]\n            v_ix = name_map[v]\n            v_path = self.layers[layer].HigherOrderNodeToPath(v)\n\n            # each line contains uniqueID physicalID [name]\n            file.write('{0} {1} \"{2}\"\\n'.format(v_ix+1, first_layer_map[v_path[-1]]+1, v))\n    \n        file.write('*Links\\n'.format(self.layers[1].vcount()))\n        for e in self.layers[layer].edges:\n            source = e[0]\n            target = e[1]\n\n            # Get source and target paths\n            source_p = self.layers[layer].HigherOrderNodeToPath(source)            \n            source_t = self.layers[layer].HigherOrderNodeToPath(target)\n\n            source_ix = name_map[source]\n            target_ix = name_map[target]\n            \n            # Get edge weight\n            w_st = self.layers[layer].edges[e][1]\n\n            # Get transition probability\n            T_st = T[target_ix, source_ix]\n\n            # Write entry to file\n            # each line contains from to [weight]\n            file.write('{0} {1} {2}\\n'.format(source_ix+1, target_ix+1, T_st))\n\n        file.close()\n\n    def getLikelihood(self, paths, maxOrder=None, log=True):\n        \"\"\"Calculates the likelihood of a multi-order\n        network model up to a maximum order maxOrder based on all\n        path statistics.\n\n        @param paths: the path statistics to be used in the likelihood\n            calculation\n\n        @param maxOrder: the maximum layer order to take into\n            account for the likelihood calculation. For the default\n            value None, all orders will be used for the\n            likelihood calculation.\n\n        @log: Whether or not to return the log likelihood (default: True)\n        \"\"\"\n        if maxOrder == None:\n            maxOrder = self.maxOrder\n        assert maxOrder <= self.maxOrder, 'Error: maxOrder cannot be larger than maximum order of multi-order network'\n\n        # add log-likelihoods of multiple model layers,\n        # assuming that paths are independent\n        L = _np.float64(0)\n\n        for k in range(0, maxOrder+1):\n            if k < maxOrder:\n                p = self.getLayerLikelihood(paths, k, considerLongerPaths=False, log=True)[0]\n            else:\n                p = self.getLayerLikelihood(paths, k, considerLongerPaths=True, log=True)[0]\n            # print('Log L(k=' + str(k) + ') = ' + str(p))\n            assert p <= 0, 'Layer Log-Likelihood out of bounds'\n            L += p\n        assert L <= 0, 'Log-Likelihood out of bounds'\n\n        if log:\n            return L\n        else:\n            return _np.exp(L)\n\n\n\n    def factorial(self, n, log=True):\n        \"\"\"\n        Caclulates (or approximates) the (log of the) factorial n!. The function applies Stirling's approximation if n>20.\n\n        @param n: computes factorial of n\n        @param log: whether or not to return the (natural) logarithm of the factorial\n        \"\"\"\n        f = _np.float64(0)\n        n_ = _np.float64(n)\n        if n > 20: # use Stirling's approximation\n            try:\n                f = n_ * _np.log(n_)-n_ + 0.5 * _np.log(2.0*_np.pi*n_)+1.0/(12.0*n_)-1/(360.0*n_**3.0)\n            except Warning as w:\n                Log.add('Factorial calculation for n = ' + str(n)+ ': ' + str(w), severity=Severity.WARNING)\n\n        else:\n            f = _np.log(_np.math.factorial(n))\n\n        if log:\n            return f\n        else:\n            return _np.exp(f)\n\n\n    def getLayerLikelihood(self, paths, l=1, considerLongerPaths=True, log=True, minL=None):\n        \"\"\"\n        Calculates the (log-)likelihood of the **first** l layers of a multi-order network model\n        using all observed paths of (at least) length l\n\n        @param paths: the path statistics for which to calculate the layer likelihood\n\n        @param l: number of layers for which likelihood shall be calculated\n            Paths of length l (and possibly longer) will be used to calculate the likelihood\n            of model layers for all orders up to l\n\n        @param considerLongerPaths: whether or not to include paths longer than l\n            in the calculation of the likelihood. In general, when calculating the likelihood\n            of a multi-order model which combines orders from 1 to l, this should be set to\n            true only for the value of l that corresponds to the largest order in the model.\n\n        @param log: whether to compute Log-Likelihood (default: True)\n\n        @param minL: minimum length of paths which enter the likelihood calculation. For the\n            default value None, all paths with at least length l will be considered.\n\n        @returns: the (log-)likelihood of the model layer given the path statistics\n        \"\"\"\n\n        # m is the maximum length of any path in the data\n        m = max(paths.paths)\n\n        assert m >= l and len(paths.paths[l])>0, 'Error: there are no paths of length l or longer'\n\n        if minL == None:\n            minL = l\n\n         # Set maximum length of paths to consider in likelihood calculation\n        if considerLongerPaths:\n            maxL = m\n        else:\n            maxL = l\n\n        # For the paths S_k of length k (or longer) that we observe, we need to calculate\n        # the probability of observing all paths in S_k based on the probabilities of\n        # individual paths (which are calculated using the underlying Markov model(s))\n\n        # n is the total number of path observations\n        n = 0\n\n        # Initialize likelihood\n        L = 0\n\n        # compute likelihood for all longest paths\n        # up to the maximum path length maxL\n        for k in range(minL, maxL+1):\n            for p in paths.paths[k]:\n\n                # Only consider observations as *longest* path\n                if paths.paths[k][p][1]>0:\n\n                    # Add m_i observations of path p to total number of observations n\n                    n += paths.paths[k][p][1]\n\n                    # special case: to calculate the likelihood of the path based on a zero-order model we\n                    # use the 'start' -> v transitions in the respective model instance\n                    if l==0:\n                        for s in range(len(p)):\n                            L += _np.log(self.T[0][self.layers[0].nodes.index(p[s]), self.layers[0].nodes.index('start')]) * paths.paths[k][p][1]\n\n                    # general case: compute likelihood of path based on\n                    # hierarchy of higher-order models as follows ...\n                    else:\n\n                        # 1.) transform the path into a sequence of (two or more) l-th-order nodes\n                        nodes = self.layers[l].pathToHigherOrderNodes(p)\n                        # print('l-th order path = ', str(nodes))\n\n                        # 2.) nodes[0] is the prefix of the k-th order transitions, which we\n                        #   can transform into multiple transitions in lower order models.\n                        #   Example: for a path a-b-c-d of length three, the node sequence\n                        #   at order l=3 is ['a-b-c', 'b-c-d'] and thus the prefix is 'a-b-c'.\n                        prefix = nodes[0]\n\n                        # 3.) We extract the transitions for the prefix based on models of\n                        #   orders k_<l. In our example, we have the transitions ...\n                        #   (a-b, b-c) for k_=2\n                        #   (a, b) for k_=1, and\n                        #   (start, a) for k_=0\n                        transitions = {}\n\n                        # for all k_<l in descending order\n                        for k_ in range(l-1, -1, -1):\n                            #print('prefix = ', prefix)\n                            x = prefix.split(self.layers[k_].separator)\n                            transitions[k_] = self.layers[k_].pathToHigherOrderNodes(x)\n                            #print('transition (k_=', k_,') = ', transitions[k_])\n                            prefix = transitions[k_][0]\n\n                        # 4.) Using Bayes theorem, we calculate the likelihood of a path a-b-c-d-e\n                        #   of length four for l=4 as a single transition in a fourth-order model, and\n                        #   four additional transitions in the k_=0, 1, 2 and 3-order models, i.e. we have ...\n                        #   P(a-b-c-d-e) = P(e|a-b-c-d) * [ P(d|a-b-c) * P(c|a-b) * P(b|a) * P(a) ]\n                        #   If we were to model the same path based on model hierarchy with a maximum order of l=2,\n                        #   we instead have three transitions in the second-order model and two additional transitions\n                        #   in the k_=0 and k_=1 order models for the prefix 'a-b' ...\n                        #   P(a-b-c-d-e) = P(e|c-d) * P(d|b-c) * P(c|a-b) * [ P(b|a) * P(a) ]\n\n                        # First multiply the transitions in the l-th order model ...\n                        for s in range(len(nodes)-1):\n                            # print((nodes[s], nodes[s+1]))\n                            # print(T[model.nodes.index(nodes[s+1]), model.nodes.index(nodes[s])])\n                            L += _np.log(self.T[l][self.layers[l].nodes.index(nodes[s+1]), self.layers[l].nodes.index(nodes[s])]) * paths.paths[k][p][1]\n\n                        # ... then multiply additional transition probabilities for the prefix ...\n                        for k_ in range(0, l):\n                            L += _np.log(self.T[k_][self.layers[k_].nodes.index(transitions[k_][1]), self.layers[k_].nodes.index(transitions[k_][0])]) * paths.paths[k][p][1]\n\n            if n == 0:\n                L = 0\n        if log:\n            assert L<=0, 'Log-Likelihood out of bounds'\n            return L, n\n        else:\n            assert L>=0 and L<=1, 'Likelihood out of bounds'\n            return _np.exp(L), n\n\n\n    def getDegreesOfFreedom(self, maxOrder=None, assumption=\"paths\"):\n        \"\"\"\n        Calculates the degrees of freedom of the model based on\n        different assumptions, and taking into account layers up to\n        a maximum order.\n\n        @param: maxOrder: the maximum order up to which model layers shall be\n            taken into account\n\n        @param assumption: if set to 'paths', for the degree of freedom calculation\n            only paths in the first-order network topology will be considered. This is\n            needed whenever we model paths in a *given* network topology.\n            If set to 'ngrams' all possible n-grams will be considered, independent of whether they\n            are valid paths in the first-order network or not. The 'ngrams' and the 'paths' assumption\n            coincide if the first-order network is fully connected, i.e. if all possible paths actually occur.\n        \"\"\"\n        if maxOrder == None:\n            maxOrder = self.maxOrder\n        assert maxOrder <= self.maxOrder, 'Error: maxOrder cannot be larger than maximum order of multi-order network'\n\n        dof = 0\n\n        # Sum degrees of freedom of all model layers up to maxOrder\n        for i in range(0, maxOrder+1):\n           dof += self.layers[i].getDoF(assumption)\n\n        return int(dof)\n\n\n    def likeliHoodRatioTest(self, paths, maxOrderNull=0, maxOrder=1, assumption='paths', significanceThreshold=0.01):\n        \"\"\"\n        Performs a likelihood-ratio test between two multi-order models with given maximum orders, where maxOrderNull serves\n        as null hypothesis and maxOrder serves as alternative hypothesis. The null hypothesis is rejected if the p-value for\n        the observed paths under the null hypothesis is smaller than the given significance threshold.\n\n        Applying this test makes the assumption that we have nested models, i.e. that the null model is contained\n        as a special case in the parameter space of the more complex model. If we assume that the path constraint holds,\n        this is not true for the test of the first- against the zero-order model (since some sequences of the zero order model\n        cannot be generated in the first-order model). However, since the set of possible higher-order transitions is generated\n        based on the first-order model, the nestedness property holds for all higher order models.\n\n        @param paths: the path data to be used in the liklihood ratio test\n        @param maxOrderNull: maximum order of the multi-order model\n                to be used as a null hypothesis\n        @param maxOrder: maximum order of the multi-order model to be used as\n                alternative hypothesis\n        @param assumption: paths or ngrams\n        @param significanceThreshold: the threshold for the p-value\n                below which to accept the alternative hypothesis\n        @returns: a tuple of the format (reject, p) which captures whether or\n                not the null hypothesis is rejected in favor of the alternative\n                hypothesis, as well as the p-value that led to the decision\n        \"\"\"\n\n        assert maxOrderNull < maxOrder, 'Error: order of null hypothesis must be smaller than order of alternative hypothesis'\n        # let L0 be the likelihood for the null model and L1 be the likelihood for the alternative model\n\n        # we first compute a test statistic x = -2 * log (L0/L1) = -2 * (log L0 - log L1)\n        x = -2 * (self.getLikelihood(paths, maxOrder=maxOrderNull, log=True) - self.getLikelihood(paths, maxOrder=maxOrder, log=True))\n\n        # we calculate the additional degrees of freedom in the alternative model\n        dof_diff = self.getDegreesOfFreedom(maxOrder=maxOrder, assumption=assumption) - self.getDegreesOfFreedom(maxOrder=maxOrderNull, assumption=assumption)\n\n        Log.add('Likelihood ratio test for K_opt = ' + str(maxOrder) + ', x = ' + str(x))\n        Log.add('Likelihood ratio test, d_1-d_0 = ' + str(dof_diff))\n\n        # if the p-value is *below* the significance threshold, we reject the null hypothesis\n        p = 1-chi2.cdf(x, dof_diff)\n\n        Log.add('Likelihood ratio test, p = ' + str(p))\n        return (p<significanceThreshold), p\n\n\n    def estimateOrder(self, paths, maxOrder=None, significanceThreshold=0.01):\n        \"\"\"\n        Selects the optimal maximum order of a multi-order network model for the\n        observed paths, based on a likelihood ratio test with p-value threshold of p\n        By default, all orders up to the maximum order of the multi-order model will be tested.\n\n        @param paths: The path statistics for which to perform the order selection\n\n        @param maxOrder: The maximum order up to which the multi-order model shall be tested.\n        \"\"\"\n        if maxOrder == None:\n            maxOrder = self.maxOrder\n        assert maxOrder <= self.maxOrder, 'Error: maxOrder cannot be larger than maximum order of multi-order network'\n        assert maxOrder > 1, 'Error: maxOrder must be larger than one'\n\n        maxAcceptedOrder = 1\n\n        # Test for highest order that passes\n        # likelihood ratio test against null model\n        for k in range(2, maxOrder+1):\n            if self.likeliHoodRatioTest(paths, maxOrderNull=k-1, maxOrder=k, significanceThreshold=significanceThreshold)[0]:\n                maxAcceptedOrder = k\n\n        return maxAcceptedOrder\n"
  },
  {
    "path": "pathpy/Paths.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\n    pathpy is an OpenSource python package for the analysis of\n    sequential data on pathways and temporal networks using higher- and multi order graphical models\n\n    Copyright (C) 2016-2017 Ingo Scholtes, ETH Zürich\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU Affero General Public License as published\n    by the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU Affero General Public License for more details.\n\n    You should have received a copy of the GNU Affero General Public License\n    along with this program.  If not, see <http://www.gnu.org/licenses/>.\n\n    Contact the developer:\n    \n    E-mail: ischoltes@ethz.ch\n    Web:    http://www.ingoscholtes.net\n\"\"\"\n\nimport numpy as _np\nimport collections as _co\nimport bisect as _bs\nimport sys as _sys\nimport operator\n\nimport scipy.sparse as _sparse\nimport scipy.sparse.linalg as _sla\nimport scipy.linalg as _la\n\nfrom pathpy.Log import *\nfrom pathpy.HigherOrderNetwork import HigherOrderNetwork\n\n\nclass Paths:\n    \"\"\"\n    Instances of this class represent path statistics which can be analyzed using higher- and multi-order network\n    models. The origin of the path statistics can be (i) n-gram files which provide us with a list of paths \n    in terms of n-grams of varying lengths, or (ii) a temporal network instance which provides us with a set of\n    time-respecting paths based on a given maximum time difference delta.\n    \"\"\"\n\n    def __init__(self):\n        \"\"\"\n        Creates an empty Paths object\n        \"\"\"\n\n        ## A dictionary of paths that has the following structure:\n        ## - paths[k] is a dictionary containing all paths of length k, indexed by a path tuple p = (u,v,w,...)\n        ## - for each tuple p of length k, paths[k][p] contains a tuple (i,j) where i refers to the number \n        ##       of times p occurs as a subpath of a longer path, and j refers to the number of times p\n        ##       occurs as a *real* or *longest* path (i.e. not being a subpath of a longer path)\n        self.paths = _co.defaultdict( lambda: _co.defaultdict( lambda: _np.array([0,0]) ))\n\n        ## The character used to separate nodes on paths\n        self.separator = ','\n\n\n\n    def summary(self):\n        \"\"\" \n        Returns a string containing basic summary info of this Paths instance\n        \"\"\"\n        sum = 0\n        spsum = 0\n        lpsum = 0\n        maxL = 0\n        avgL = 0\n        nodes = set()\n        for k in self.paths:\n            for p in self.paths[k]:\n                sum += self.paths[k][p].sum()\n                spsum += self.paths[k][p][0]\n                lpsum += self.paths[k][p][1]\n                avgL += self.paths[k][p][1] * k\n                for v in p:\n                    nodes.add(v)\n            if len(self.paths[k])>0:\n                maxL = max(maxL, k)\n        if lpsum>0:\n            avgL = avgL / lpsum\n\n        summary = 'Number of paths (unique/sub paths/total):\\t' +  str(lpsum) + ' (' + str(self.getUniquePaths()) + '/'+ str(spsum) + '/' + str(sum) + ')\\n'\n        summary += 'Nodes:\\t\\t\\t\\t' + str(len(nodes)) + '\\n'\n        summary += 'Edges:\\t\\t\\t\\t' + str(len(self.paths[1])) + '\\n'\n        summary += 'Max. path length:\\t\\t' + str(maxL) + '\\n'\n        summary += 'Avg path length:\\t\\t' + str(avgL) + '\\n'\n        for k in self.paths:\n            sum = 0 \n            spsum = 0\n            lpsum = 0\n            for p in self.paths[k]:\n                sum += self.paths[k][p].sum()          \n                spsum += self.paths[k][p][0]\n                lpsum += self.paths[k][p][1]\n            summary += 'Paths of length k = ' + str(k) + '\\t\\t' + str(lpsum) + ' (' + str(self.getUniquePaths(l=k, considerLongerPaths=False)) + '/' + str(spsum) + '/'+ str(sum) +  ')\\n'\n        return summary\n\n\n    def getSequence(self, stopchar='|'):\n        \"\"\"\n        Returns a single sequence in which all \n        paths have been concatenated. Individual \n        paths are separated by a stop character.\n\n        @stopchar: The character used to separate paths\n        \"\"\"\n\n        Log.add('Concatenating paths to sequence ...')\n\n        sequence = []\n        for l in self.paths:\n            for p in self.paths[l]:\n                segment = []\n                for s in p:\n                    segment.append(s)\n                if not stopchar == '':\n                    segment.append(stopchar)\n                for f in range(self.paths[l][p][1]):\n                    sequence += segment\n\n        Log.add('finished')\n        return sequence\n\n\n    def getUniquePaths(self, l=0, considerLongerPaths=True):\n        \"\"\"\n        Returns the number of unique paths of a given length l (and possibly longer)\n\n        @param l: the (inclusive) maximum length up to which path shall be counted. \n        \"\"\"\n        L = 0\n        lmax = l\n        if considerLongerPaths:\n            if len(self.paths)>0:\n                lmax = max(self.paths)\n            else:\n                lmax = 0\n        for j in range(l, lmax+1):\n            for p in self.paths[j]:\n                if self.paths[j][p][1]>0:\n                    L += 1\n        return L\n\n\n    def __str__(self):\n        \"\"\"\n        Returns the default string representation of \n        this Paths instance\n        \"\"\"\n        return self.summary()\n\n\n    def readEdges(filename=None, separator=',', weight = False, undirected=False, maxlines=_sys.maxsize, expandSubPaths=True):\n        \"\"\"\n        Reads data from a file containing multiple lines of *edges* of the\n        form \"v,w,frequency,X\" (where frequency is optional and X are arbitrary additional columns). The default separating \n        character ',' can be changed. In order to calculate the statistics of paths of any length, \n        by default all subpaths of length 1 (i.e. single nodes) contained in an edge will be considered.\n        \"\"\"\n        p = Paths()\n\n        p.separator = separator\n\n        with open(filename, 'r') as f:\n            Log.add('Reading edge data ... ')\n            line = f.readline()\n            n = 1 \n            while line and n <= maxlines:\n                fields = line.rstrip().split(separator)\n                assert len(fields) >= 2, 'Error: malformed line: ' + line\n\n                path = (fields[0],fields[1])\n\n                if weight:                    \n                    frequency = int(fields[2])\n                else:\n                    frequency = 1\n                p.paths[1][path] += (0,frequency)\n                if undirected:\n                    reverse_path = (fields[1],fields[0])\n                    p.paths[1][reverse_path] += (0,frequency)\n                line = f.readline()\n                n += 1\n        # end of with open()\n        \n        if expandSubPaths:\n            p.expandSubPaths()\n        Log.add('finished.')\n\n        return p\n\n\n    def readFile(filename=None, separator=',', pathFrequency = False, maxlines=_sys.maxsize, maxN=_sys.maxsize, expandSubPaths = True):\n        \"\"\" \n        Reads path data from a file containing multiple lines of n-grams of the \n        form \"a,b,c,d,frequency\" (where frequency is optional). The default separating \n        character ',' can be changed. Each n-gram will be interpreted as a path of length n-1, \n        i.e. bigrams a,b are considered as path of length one, trigrams a,b,c as path of length two, etc.\n        In order to calculate the statistics of paths of any length, by default all subpaths of \n        length k < n-1 contained in an n-gram will be considered. I.e. for n=4 the four-gram a,b,c,d \n        will be considered as a single (longest) path of length n-1 = 3 and three subpaths \n        a->b, b->c, c->d of length k=1 and two subpaths a->b->c amd b->c->d of length k=2 will be \n        additionally counted.\n\n        @param filename: name of the n-gram file to read data from\n\n        @param separator: the character used to separate nodes on the path, i.e. using a \n            separator character of ';' n-grams are represented as a;b;c;...\n\n        @param pathFrequency: if set to true, the last entry in each n-gram will be interpreted as \n            weight (i.e. frequency of the path), e.g. a,b,c,d,4 means that four-gram a,b,c,d has weight four.\n            False by default, which means each path occurrence is assigned a default weight of one (adding weights \n            of multiple occurrences).\n\n        @param maxlines: The maximum number of lines (i.e. ngrams) to read from the input file\n\n        @param maxN: The maximum n for the n-grams to read, i.e. setting maxN to 15 will ignore all n-grams of length \n            16 and longer, which means that only paths up to length n-1 are considered.\n\n        @param expandSubPaths: by default all subpaths of the given ngrams are generated, i.e. \n            for an input file with a single trigram a;b;c a path a->b->c of length two will be generated\n            as well as two subpaths a->b and b->c of length one\n        \"\"\"\n        assert filename is not \"\", 'Empty filename given'\n    \n        # If subpath expansion is applied, we keep the information how many times a path \n        # has been observed as a subpath, and how many times as a \"real\" path        \n\n        p = Paths()\n\n        p.separator = separator\n        maxL = 0 \n\n        with open(filename, 'r') as f:\n            Log.add('Reading ngram data ... ')\n            line = f.readline()\n            n = 1 \n            while line and n <= maxlines:\n                fields = line.rstrip().split(separator)\n                path = ()\n                # Add frequency of \"real\" path to second component of occurrence counter\n                if pathFrequency:\n                    for i in range(0, len(fields)-1):\n                        # Omit empty fields\n                        v = fields[i].strip()\n                        if len(v)>0:\n                            path += (v,)\n                    frequency = int(fields[len(fields)-1])\n                    if len(path) <= maxN:\n                        p.paths[len(path)-1][path] += (0,frequency)\n                        maxL = max(maxL, len(path)-1)\n                    else: # cut path at maxN\n                        p.paths[maxN-1][path[:maxN]] += (0,frequency)\n                        maxL = max(maxL, maxN-1)\n                else:\n                    for i in range(0, len(fields)):\n                        # Omit empty fields\n                        v = fields[i].strip()\n                        if len(v)>0:\n                            path += (v,)\n                    if len(path) <= maxN:\n                        p.paths[len(path)-1][path] += (0,1)\n                        maxL = max(maxL, len(path)-1)\n                    else: # cut path at maxN\n                        p.paths[maxN-1][path[:maxN]] += (0,1)\n                        maxL = max(maxL, maxN-1)\n                line = f.readline()\n                n += 1                \n        # end of with open()\n        Log.add('finished. Read ' + str(n-1) + ' paths with maximum length ' + str(maxL))\n        \n        if expandSubPaths:\n            p.expandSubPaths()\n        Log.add('finished.')\n\n        return p\n\n\n    def writeFile(self, filename, separator=','):\n        \"\"\"\n        Writes path statistics data to a file. \n        Each line in this file captures a longest path \n        (v0,v1,...,vl), as well as its frequency f as follows\n\n        v0,v1,...,vl,f\n\n        @param filename: name of the file to write to\n        @param separator: character that shall be used to \n            separate nodes and frequencies\n        \"\"\"\n        with open(filename, 'w') as f:\n            for l in self.paths:\n                for p in self.paths[l]:\n                    if self.paths[l][p][1]>0:\n                        line = \"\"\n                        for x in p:\n                            line += x\n                            line += separator\n                        line += str(self.paths[l][p][1])\n                        f.write(line+'\\n')\n        f.close()\n\n\n    def ObservationCount(self):\n        \"\"\"\n        Returns the total number of observed pathways of any length \n        (includes multiple observations for paths with a frequency weight)\n        \"\"\"\n\n        sum = 0\n        for k in self.paths:\n            for p in self.paths[k]:\n                sum += self.paths[k][p][1]\n        return sum\n\n\n    def expandSubPaths(self):\n        \"\"\"\n        This function implements the sub path expansion, i.e. \n        for a four-gram a,b,c,d, the paths a->b, b->c, c->d of \n        length one and the paths a->b->c and b->c->d of length \n        two will be counted.\n        \"\"\"             \n\n        # nothing to see here ... \n        if len(self.paths)==0:\n            return\n\n        Log.add('Calculating sub path statistics ... ')\n\n        # the expansion of all subpaths in paths with a maximum path length of maxL\n        # neccessarily generates paths of *any* length up to MaxL.\n        # Forcing the generation of all these indices here, prevents us \n        # from mutating indices during subpath creation. The fact that indices are\n        # immutable allows us to use efficient iterators and prevent unncessary copying\n\n        # Thanks to the use of defaultdict, the following trick will prevent us from \n        # repeatedly testing whether l already exists as a key\n\n        for l in range(max(self.paths)):\n                self.paths[l] = self.paths[l]\n\n        # expand subpaths in all paths of any length ... \n        for pathLength in self.paths:\n            for path in self.paths[pathLength]:\n                                \n                # The frequency is given by the number of occurrences as longest \n                # path, which is stored in the second entry of the numpy array\n                frequency = self.paths[pathLength][path][1]\n\n                # Generate all subpaths of length k for k = 0 to k = pathLength-1 (inclusive)\n                for k in range(0, pathLength):\n                    # Generate subpaths of length k for all start indices s for s = 0 to s = pathLength-k (inclusive)\n                    for s in range(0, pathLength-k+1):\n                        # Add frequency as a subpath to *first* entry of occurrence counter\n                        self.paths[k][path[s:s+k+1]] += (frequency,0)\n\n\n    @staticmethod\n    def fromTemporalNetwork(tempnet, delta=1, maxLength=_sys.maxsize):\n        \"\"\" Calculates the frequency of all time-respecting paths up to maximum length of maxLength, assuming \n        a maximum temporal distance of delta between consecutive time-stamped links on a path. \n        This (static) method returns an instance of the class Paths, which can subsequently be used to \n        generate higher-order network representations based on the path statistics.\n\n        @param delta: Indicates the maximum temporal distance up to which time-stamped links will be \n        considered to contribute to time-respecting paths. For (u,v;3) and (v,w;7) a time-respecting path (u,v)->(v,w) \n        will be inferred for all 0 < delta <= 4, while no time-respecting path will be inferred for all delta > 4. \n        If the max time diff is not set specifically, the default value of delta=1 will be used, meaning that a\n        time-respecting path u -> v -> w will only be inferred if there are *directly consecutive* time-stamped \n        links (u,v;t) (v,w;t+1). Every time-stamped edge is further considered a path of length one, i.e. for maxLength=1 \n        this function will simply return the statistics of time-stamped edges.\n        \n        @param maxLength: Indicates the maximum length up to which time-respecting paths should be calculated, \n             which can be limited due to computational efficiency. A value of k will generate all time-respecting paths \n             consisting of up to k time-stamped links. Note that generating a multi-order model with a maximum order of k \n             requires to extract time-respecting paths with *at least* length k. If a limitation of the maxLength is not \n             required for computational reasons, this parameter should not be set (as it will change the statistics of \n             paths)\n        \"\"\"\n        \n        if maxLength==_sys.maxsize:\n            Log.add('Extracting time-respecting paths for delta = ' + str(delta) + ' ...')\n        else:\n            Log.add('Extracting time-respecting paths up to length ' + str(maxLength) + ' for delta = ' + str(delta) + ' ...')\n\n        # for dictionary p.paths paths[k] contains a list of all \n        # time-respecting paths p with length k and paths[k][p] contains \n        # a two-dimensional counter whose first component counts the number of \n        # occurrences of p as subpath of a longer path and whose second component counts \n        # the number of occurrences of p as \"real\" path\n        p = Paths()\n\n        # a dictionary containing paths that can still be extended \n        # by future time-stamped links\n        # candidates[t][v] is a set of paths which end at time t in node v\n        candidates = _co.defaultdict ( lambda: _co.defaultdict( lambda: set() ) )\n\n        # Note that here we only extract **longest** time-respecting paths, as we will use \n        # the expandSubpaths function later to calculate statistics of shorter paths anyway\n\n        # set of longest time-respecting paths (i.e. those paths which are \n        # NOT sub path of a longer time-respecting path)\n        longest_paths = set()\n\n        # loop over all time stamps t of edges\n        for t in tempnet.ordered_times:\n\n            for e in tempnet.time[t]:\n                # assume that this edge is the root of a longest time-respecting path\n                root = True\n\n                # check whether this edge extends existing candidates\n                for t_prev in list(candidates):\n                    # time stamp of candidate has to be in [t-delta, t) ...\n                    if t_prev >= t-delta and t_prev < t:\n                        # ... and last node has to be e[0] ...\n                        if e[0] in candidates[t_prev]:\n                            for c in list(candidates[t_prev][e[0]]):\n\n                                # c is path (p_0, p_1, ...) which ends in node e[0] at time t_prev\n                                new_path = c + ((e[0], e[1], t),)\n\n                                # we now know that (e[0], e[1]) is not the root of a new longest path\n                                # as it continues a previous path c\n                                root = False\n\n                                # if c has previously been considered a longest path, we discard it \n                                # from the list of longest paths. We also add the extended path as \n                                # a new longest path (possible removing it later if it is further extended)\n                                longest_paths.discard(c)\n                                longest_paths.add(new_path)\n\n                                # we add the newly found path as a candidate for paths\n                                # which can be continued by future edges\n                                if len(new_path)<maxLength:                                    \n                                    candidates[t][e[1]].add(new_path)\n\n                                # delete candidate c, because from now on \n                                # we only extend new_path\n                                candidates[t_prev][e[0]].discard(c)\n                \n                # if edge e does not continue a previous path\n                # we start a new longest path\n                if root:\n                    longest_paths.add( ((e[0], e[1], t),) )\n                    # add edge as candidate path of length one that can be extended by future edges\n                    if maxLength>1:\n                        candidates[t][e[1]].add( ( (e[0], e[1], t), ) )\n            \n            # we finished processing time stamp t, so\n            # we can remove all candidates which finish\n            # at a time smaller than t-delta. Since they cannot \n            # be extended, these are longest paths \n            for t_prev in list(candidates.keys()):\n                if t_prev < t-delta:\n                    del(candidates[t_prev])\n\n        # once we reached the last time stamp, add all candidates                     \n        # as longest paths \n        #for t_prev in candidates:\n        #    for x in candidates[t_prev]:\n        #        for p in candidates[t_prev][x]:\n        #            longest_paths.add(p)\n        \n        # Count occurrences as longest time-respecting path\n        for x in longest_paths:\n            path = (x[0][0],)\n            for edge in x:\n                path += (edge[1],)\n            p.paths[len(x)][path] += _np.array([0,1])\n\n        # expand sub paths of longest paths\n        p.expandSubPaths()\n\n        Log.add('finished.')\n\n        return p\n    \n\n    def addPathTuple(self, path, expandSubPaths=True, frequency=(0,1)):\n        \"\"\"\n        Adds a tuple of elements as a path. If the elements are not strings, \n        a conversion to strings will be made. This function can be used to \n        to set custom subpath statistics, via the frequency tuple (see below).\n\n        @path: The path tuple to be added, e.g. ('a', 'b', 'c')\n        @expandSubPaths: Whether or not to calculate subpath statistics for this path\n        @frequency: A tuple (x,y) indicating the frequency of this path as subpath \n            (first component) and longest path (second component). Default is (0,1).\n        \"\"\"\n        \n        assert len(path)>0, 'Error: paths needs to contain at least one element'        \n\n        if type(path[0]) == str:\n            path_str = path\n        else: \n            path_str = tuple(map(str, path))\n\n        self.paths[len(path)-1][path_str] += frequency\n\n        if expandSubPaths:\n            for k in range(0, len(path_str)-1):\n                for s in range(len(path_str)-k):\n                    # for all start indices from 0 to n-k\n\n                    subpath = ()\n                    # construct subpath\n                    for i in range(s, s+k+1):\n                        subpath += (path_str[i],)\n                    # add subpath weight to first component of occurrences                   \n                    self.paths[k][subpath] += (frequency[1], 0)\n\n\n\n    def getContainedPaths(p, node_filter):\n        \"\"\"\n        Returns the set of maximum-length sub-paths of the path p, which\n        only contain nodes that appear in the node_filter. As an example, \n        for the path (a,b,c,d,e,f,g) and a node_filter [a,b,d,f,g], the method \n        will return [(a,b), (d,), (f,g)].\n\n        @param p: a path tuple to check for contained paths\n        @param node_filter: a set of nodes to which the contained paths should be limited\n        \"\"\"\n        contained_paths = []\n        current_path = ()\n        for k in range(0, len(p)):\n            if p[k] in node_filter:\n                current_path += (p[k],)\n            else:\n                if len(current_path)>0:\n                    contained_paths.append(current_path)\n                    current_path = ()\n        if len(current_path)>0:\n            contained_paths.append(current_path)        \n        \n        return contained_paths\n\n\n    def filterPaths(self, node_filter, minLength=0, maxLength=sys.maxsize):\n        \"\"\"\n        Returns a new paths object which contains only paths between nodes in a given \n        filter set. For each of the paths in the current Paths object, the set of maximally \n        contained subpaths between nodes in node_filter is extracted. This method is useful \n        when studying (sub-)paths passing through a subset of nodes.\n\n        @param node_filter: the nodes for which paths with be extracted from the current\n            set of paths\n        @param minLength: the minimum length of paths to extract (default 0)\n        @param maxLength: the maximum length of paths to extract (default sys.maxsize)\n        \"\"\"                \n    \n        p = Paths()\n        for l in self.paths:\n            for x in self.paths[l]:\n                if self.paths[l][x][1]>0:                    \n                    \n                    # determine all contained subpaths which only pass through nodes in node_filter\n                    contained = Paths.getContainedPaths(x, node_filter)                    \n                    for s in contained:\n                        if len(s)-1>=minLength and len(s)-1<=maxLength:\n                            p.addPathTuple(s, expandSubPaths=True, frequency=(0, self.paths[l][x][1]))\n        return p\n\n\n\n    def projectPaths(self, mapping):\n        \"\"\"\n        Returns a new path object in which nodes have been mapped to different labels\n        given by an arbitrary mapping function. For instance, for the mapping \n        {'a': 'x', 'b': 'x', 'c': 'y', 'd': 'y'} the path (a,b,c,d) is mapped to \n        (x,x,y,y). This is useful, e.g., to map page page click streams to topic \n        click streams, using a mapping from pages to topics.\n        \n        @param mapping: a dictionary that maps nodes to the new labels\n        \"\"\"\n        p = Paths()\n        for l in self.paths:\n            for x in self.paths[l]:\n                if self.paths[l][x][1]>0:\n                    newP = ()\n                    for v in x:\n                        newP += (mapping[v],)\n                    p.addPathTuple(newP, expandSubPaths=True, frequency=(0, self.paths[l][x][1]))\n        return p\n\n\n\n    def addPath(self, ngram, separator=',', expandSubPaths = True, pathFrequency = None):\n        \"\"\"\n        Adds the path(s) of a single n-gram to the path statistics object.\n\n        @param ngram: An ngram representing a path between nodes, separated by the separator character, e.g. \n            the 4-gram a;b;c;d represents a path of length three (with separator ';')\n\n        @param separator: The character used as separator for the ngrams (',' by default)\n\n        @param expandSubPaths: by default all subpaths of the given ngram are generated, i.e. \n            for the trigram a;b;c a path a->b->c of length two will be generated \n            as well as two subpaths a->b and b->c of length one\n\n        @weight pathFrequency: the number of occurrences (i.e. frequency) of the ngram\n        \"\"\"\n\n        fields = ngram.rstrip().split(separator)\n        path = ()        \n        for i in range(0, len(fields)):\n            path += (fields[i],)\n\n        # add the occurrences as *longest* path to the second component of the numpy array\n        if pathFrequency != None:\n            self.paths[len(path)-1][path] += (0, pathFrequency)\n        else:\n            self.paths[len(path)-1][path] += (0, 1)\n\n        if expandSubPaths:\n            for k in range(0, len(path)-1):\n                for s in range(len(path)-k):\n                    # for all start indices from 0 to n-k\n\n                    subpath = ()\n                    # construct subpath\n                    for i in range(s, s+k+1):\n                        subpath += (path[i],)\n                    # add subpath weight to first component of occurrences\n                    if pathFrequency != None:                        \n                        self.paths[k][subpath] += (pathFrequency, 0)\n                    else:\n                        self.paths[k][subpath] += (1, 0)\n\n\n\n    def getSlowDownFactor(self, k=2, lanczosVecs = 15, maxiter = 1000):    \n        \"\"\"\n        Returns a factor S that indicates how much slower (S>1) or faster (S<1)\n        a diffusion process evolves in a k-order model of the path statistics\n        compared to what is expected based on a first-order model. This value captures \n        the effect of order correlations of length k on a diffusion process which evolves \n        based on the observed paths.\n        \"\"\"\n\n        assert k>1, 'Slow-down factor can only be calculated for orders larger than one'\n    \n        #NOTE to myself: most of the time goes for construction of the 2nd order\n        #NOTE            null graph, then for the 2nd order null transition matrix\n    \n        gk = HigherOrderNetwork(self, k=k)\n        gkn = HigherOrderNetwork(self, k=k, nullModel = True)\n    \n        Log.add('Calculating slow down factor ... ', Severity.INFO)\n\n        # Build transition matrices\n        Tk = gk.getTransitionMatrix()\n        Tkn = gkn.getTransitionMatrix()\n    \n        # Compute eigenvector sequences\n        # NOTE: ncv=13 sets additional auxiliary eigenvectors that are computed\n        # NOTE: in order to be more confident to find the one with the largest\n        # NOTE: magnitude, see\n        # NOTE: https://github.com/scipy/scipy/issues/4987\n        w2 = _sla.eigs(Tk, which=\"LM\", k=2, ncv=lanczosVecs, return_eigenvectors=False, maxiter=maxiter)\n        evals2_sorted = _np.sort(-_np.absolute(w2))\n\n        w2n = _sla.eigs(Tkn, which=\"LM\", k=2, ncv=lanczosVecs, return_eigenvectors=False, maxiter=maxiter)\n        evals2n_sorted = _np.sort(-_np.absolute(w2n))\n\n        Log.add('finished.', Severity.INFO)\n    \n        return _np.log(_np.abs(evals2n_sorted[1]))/_np.log(_np.abs(evals2_sorted[1]))\n\n\n\n    def getEntropyGrowthRateRatio(self, method='MLE', k=2, lanczosVecs=15, maxiter=1000):\n        \"\"\"\n        Computes the ratio between the entropy growth rate ratio between\n        the k-order and first-order model of a temporal network t. Ratios smaller\n        than one indicate that the temporal network exhibits non-Markovian characteristics\n        \"\"\"\n    \n        # NOTE to myself: most of the time here goes into computation of the\n        # NOTE            EV of the transition matrix for the bigger of the\n        # NOTE            two graphs below (either 2nd-order or 2nd-order null)\n\n        assert (method == 'MLE' or method=='Miller'), 'Only methods MLE or Miller are supported'\n    \n        # Generate k-order network\n        gk = HigherOrderNetwork(self, k=k)\n        g1 = HigherOrderNetwork(self, k=1)\n    \n        Log.add('Calculating entropy growth rate ratio ... ', Severity.INFO)\n    \n        # Compute entropy growth rate of observed transition matrix\n        A = g1.getAdjacencyMatrix(weighted=False, transposed = True)\n        Tk = gk.getTransitionMatrix()\n        Tk_pi = HigherOrderNetwork.getLeadingEigenvector(Tk, normalized=True, lanczosVecs=lanczosVecs, maxiter=maxiter)\n\n        Tk.data *=  _np.log2(Tk.data)\n\n        # Apply Miller correction to the entropy estimation\n        if method == 'Miller':\n            # Here, K is the number of different k-paths that can exist based on the\n            # observed edges\n            K = (A**k).sum()\n            print('K = ', K)\n\n            # N is the number of observations used to estimate the transition probabilities\n            # in the second-order network. This corresponds to the total edge weight in the \n            # k-order network, or - alternatively - to the number of paths of length k\n            N = 0\n            for p in self.paths[k]:\n                N += self.paths[k][p].sum()\n            print('N = ', N)\n            Hk = _np.sum( Tk * Tk_pi ) + (K-1)/(2*N)\n        else:\n            # simple MLE estimation\n            Hk = -_np.sum( Tk * Tk_pi )\n\n        Hk = _np.absolute(Hk)\n\n        # Compute entropy rate of null model       \n        gk_n = HigherOrderNetwork(self, k=k, nullModel = True)\n\n        # For the entropy rate of the null model, no Miller correction is needed\n        # since we assume that transitions correspond to the true probabilities\n        Tk_n = gk_n.getTransitionMatrix()\n        Tk_n_pi = HigherOrderNetwork.getLeadingEigenvector(Tk_n)\n        Tk_n.data *=  _np.log2(Tk_n.data)\n        Hk_n = -_np.sum( Tk_n * Tk_n_pi )\n        Hk_n = _np.absolute(Hk_n)\n\n        Log.add('finished.', Severity.INFO)\n\n        # Return ratio\n        return Hk/Hk_n   \n\n\n\n    def BWPrefMatrix(self, v):\n        \"\"\"Computes a betweenness preference matrix for a node v\n    \n        @param v: Node for which the betweenness preference matrix shall \n            be calculated\n        \"\"\"\n        # create first-order network \n        g = HigherOrderNetwork(self)\n        \n        indeg = len(g.predecessors[v])\n        outdeg = len(g.successors[v])\n\n        index_succ = {}\n        index_pred = {}\n    \n        B_v = _np.zeros(shape=(indeg, outdeg))\n        \n        # Create an index-to-node mapping for predecessors and successors\n        i = 0\n        for u in g.predecessors[v]:\n            index_pred[u] = i\n            i = i+1\n    \n        i = 0\n        for w in g.successors[v]:\n            index_succ[w] = i\n            i = i+1\n\n        # Calculate entries of betweenness preference matrix\n        for p in self.paths[2]:\n            if p[1] == v:\n                B_v[index_pred[p[0]], index_succ[p[2]]] += self.paths[2][p].sum()\n    \n        return B_v\n\n\n    def __Entropy(prob, K=None, N=None, method='MLE'):\n        \"\"\"\n        Calculates the entropy of an (observed) probability ditribution\n        based on Maximum Likelihood Estimation (MLE) (default) or using \n        a Miller correction. \n\n        @param prob: the observed probabilities\n        @param K: the number of possible outcomes, i.e. outcomes with non-zero probability to be used \n            for the Miller correction (default None)\n        @param N: number of samples based on which observed probabilities where computed. This\n            is needed for the Miller correaction (default None)\n        @param method: The method to be used to calculate entropy. Can be 'MLE' (default) or 'Miller'\n        \"\"\"\n\n        if method == 'MLE':\n            idx = _np.nonzero(prob)\n            return -_np.inner( _np.log2(prob[idx]), prob[idx] )\n        elif method == 'Miller':\n            assert K != None and N != None\n            if N == 0:\n                return 0\n            else:\n                idx = _np.nonzero(prob)\n                return -_np.inner( _np.log2(prob[idx]), prob[idx] ) + (K-1)/(2*N)\n\n\n    def BetweennessPreference(self, v, normalized=False, method = 'MLE'):\n        \"\"\"\n        Calculates the betweenness preferences of a\n        node v based on the mutual information of path \n        statistics of length two.\n\n        @nornalized: whether or not to normalize betweenness preference values\n\n        @method: which method to use for the entropy calculation. The default 'MLE' uses \n            the standard Maximum-Likelihood estimation of entropy. Setting method to \n            'Miller' additionally applies a Miller-correction. see e.g. \n            Liam Paninski: Estimation of Entropy and Mutual Information, Neural Computation 5, 2003 or \n            http://www.nowozin.net/sebastian/blog/estimating-discrete-entropy-part-2.html\n        \"\"\"\n        \n        assert method == 'MLE' or method =='Miller'    \n\n        # If the network is empty, just return zero\n        if len(self.getNodes()) == 0:\n            return 0.0\n\n        # First create the betweenness preference matrix (equation (2) of the paper)\n        B_v = self.BWPrefMatrix(v)\n\n        if B_v.shape[0] == 0 or B_v.shape[1] == 0:\n            return None\n\n        # Normalize matrix (equation (3) of the paper)\n        # NOTE: P_v has the same shape as B_v\n        P_v = _np.zeros(shape=B_v.shape)\n        S = _np.sum(B_v)\n\n        if S>0:\n            P_v = B_v / S\n\n        # Compute marginal probabilities\n        # Marginal probabilities P^v_d = \\sum_s'{P_{s'd}}\n        marginal_d = _np.sum(P_v, axis=0)\n\n        # Marginal probabilities P^v_s = \\sum_d'{P_{sd'}}\n        marginal_s = _np.sum(P_v, axis=1)        \n\n        if method=='Miller':    \n\n            # total number of samples, i.e. observed two-paths\n            N = _np.sum(B_v)\n\n            #print('N = ', N)        \n            #print('B = ', B_v)\n            #print('marginal_s = ', marginal_s)\n            #print('marginal_d = ', marginal_d)\n\n            # marginal entropy H(S)\n            H_s = Paths.__Entropy(marginal_s, len(marginal_s), N, method='Miller')\n\n            # print('H(S) = ', H_s)\n            # marginal entropy H(D)\n\n            H_d = Paths.__Entropy(marginal_d, len(marginal_d), N, method='Miller')\n\n            #print('H(D) = ', H_d)\n            # we need the conditional entropy H(D|S)\n\n            H_ds = 0\n            for s in range(len(marginal_s)):\n\n                # number of two paths s -> v -> * observed in the data\n                N_s = _np.sum(B_v[s,:])\n\n                #print('N(s=' + str(s) + ') = ' +  str(N_s))\n\n                # probabilities of all destinations, given the particular source s\n                p_ds = B_v[s,:]/_np.sum(B_v[s,:])\n\n                #print('P(D|S=' + str(s) + ') = '+ str(p_ds))\n\n                # number of possible destinations d\n                K_s = len(p_ds)\n\n                #print('K(s=' + str(s) + ') = ' +  str(K_s))\n\n                # marginal_s[s] is the overall probability of source s\n                p_s = marginal_s[s]\n\n                # add to conditional entropy\n                H_ds += p_s * Paths.__Entropy(p_ds, K_s, N_s, method='Miller')\n\n            #print('H(D|S) = ', H_ds)\n\n        else: \n            # use MLE estimation\n            H_s = Paths.__Entropy(marginal_s)\n            H_d = Paths.__Entropy(marginal_d)\n            H_ds = 0\n\n            for s in range(len(marginal_s)):\n                p_ds = P_v[s,:]/_np.sum(P_v[s,:])\n                H_ds += marginal_s[s] * Paths.__Entropy(p_ds)\n\n            # Alternative calculation (without explicit entropies)\n            # build mask for non-zero elements\n            # row, col = np.nonzero(P_v)\n            # pv = P_v[(row,col)]\n            # marginal = np.outer(marginal_s, marginal_d)\n            # log_argument = np.divide( pv, marginal[(row,col)] )    \n            # I = np.dot( pv, np.log2(log_argument) )    \n\n        I = H_d - H_ds\n\n        if normalized:\n            I =  I/_np.min([H_s, H_d])\n\n        return I\n\n\n    def getNodes(self):\n        \"\"\"\n        Returns the list of nodes for the underlying \n        set of paths\n        \"\"\"\n        nodes = set()\n        for p in self.paths[0]:\n            nodes.add(p[0])\n        return nodes\n\n\n    def getDistanceMatrix(self):\n        \"\"\"\n        Calculates shortest path distances between all pairs of \n        nodes based on the observed shortest paths (and subpaths)\n        \"\"\"\n        shortest_path_lengths = _co.defaultdict( lambda: _co.defaultdict( lambda: _np.inf ) )\n        \n        Log.add('Calculating distance matrix based on empirical paths ...', Severity.INFO)\n        # Node: no need to initialize shortest_path_lengths[v][v] = 0\n        # since paths of length zero are contained in self.paths\n\n        for l in self.paths:\n            for p in self.paths[l]:\n                start = p[0]\n                end = p[-1]\n                if l < shortest_path_lengths[start][end]:\n                    shortest_path_lengths[start][end] = l\n        \n        Log.add('finished.', Severity.INFO)\n\n        return shortest_path_lengths\n\n\n    def getShortestPaths(self):\n        \"\"\"\n        Calculates all observed shortest paths (and subpaths) between \n        all pairs of nodes\n        \"\"\"\n        shortest_paths = _co.defaultdict( lambda: _co.defaultdict( lambda: set() ) )\n        shortest_path_lengths = _co.defaultdict( lambda: _co.defaultdict( lambda: _np.inf ) )\n        \n        Log.add('Calculating shortest paths based on empirical paths ...', Severity.INFO)\n\n        for l in self.paths:\n            for p in self.paths[l]:\n                s = p[0]\n                d = p[-1]\n                # we found a path of length l from s to d\n                if l < shortest_path_lengths[s][d]:\n                    shortest_path_lengths[s][d] = l\n                    shortest_paths[s][d] = set()\n                    shortest_paths[s][d].add(p)\n                elif l == shortest_path_lengths[s][d]:\n                    shortest_paths[s][d].add(p)\n\n        Log.add('finished.', Severity.INFO)\n        \n        return shortest_paths\n\n\n    def BetweennessCentrality(self, normalized=False):\n        \"\"\"\n        Calculates the betweenness centrality of nodes based on\n        observed shortest paths between all pairs of nodes\n        \"\"\"\n\n        node_centralities = _co.defaultdict( lambda: 0 )\n        shortest_paths = self.getShortestPaths()\n\n        for s in shortest_paths:\n            for d in shortest_paths[s]:\n                for p in shortest_paths[s][d]:\n                    for x in p[1:-1]:\n                        if s != d != x:\n                            # print('node ' + x + ': ' + str(1.0 / len(shortest_paths[start][end])))                            \n                            node_centralities[x] += 1.0 / len(shortest_paths[s][d])\n                            # node_centralities[x] += 1.0\n        if normalized:\n            m = max(node_centralities.values())\n            for v in node_centralities:\n                node_centralities[v] /= m\n\n        # assign zero values to nodes not occurring on shortest paths\n        nodes = self.getNodes()\n        for v in nodes:\n            node_centralities[v] += 0\n\n        return node_centralities\n\n\n    def ClosenessCentrality(self, normalized=False):\n        \"\"\"\n        Calculates the closeness centrality of nodes based on\n        observed shortest paths between all nodes \n        \"\"\"\n\n        node_centralities = _co.defaultdict( lambda: 0 )\n        shortest_path_lengths = self.getDistanceMatrix()                \n\n        for x in shortest_path_lengths:\n            for d in shortest_path_lengths[x]:\n                if x != d and shortest_path_lengths[x][d] < _np.inf:\n                    node_centralities[x] += 1.0 / shortest_path_lengths[x][d]\n\n        # assign zero values to nodes not occurring\n        nodes = self.getNodes()\n        for v in nodes:\n            node_centralities[v] += 0\n       \n        if normalized:\n            m = max(node_centralities.values())\n            for v in nodes:\n                node_centralities[v] /= m\n\n        return node_centralities\n    \n\n    def VisitationProbabilities(self):\n        \"\"\"\n        Calculates the probabilities that randomly chosen paths\n        pass through nodes. If 5 out of 100 paths (of any length) contain \n        node v, it will be assigned a value of 0.05. This measure can be \n        interpreted as path-based ground truth for the notion of importance \n        captured by PageRank applied to a graphical abstraction of the paths.\n        \"\"\"\n\n        Log.add('Calculating path visitation probabilities...', Severity.INFO)\n\n        # entries capture the probability that a given node is visited on an arbitrary path\n        # Note: this is identical to the subpath count of zero-length paths\n        # (i.e. the relative frequencies of nodes across all pathways)\n        visitation_probabilities = _co.defaultdict( lambda: 0 )\n\n        # total number of visits\n        visits = 0.0\n        \n        for l in self.paths:\n            for p in self.paths[l]:\n                for v in p:\n                    # count occurrences in longest paths only!\n                    visitation_probabilities[v] += float(self.paths[l][p][1])\n                    visits += float(self.paths[l][p][1])\n\n        for v in visitation_probabilities:\n            visitation_probabilities[v] /= visits\n                    \n\n        Log.add('finished.', Severity.INFO)\n\n        return visitation_probabilities\n"
  },
  {
    "path": "pathpy/TemporalNetwork.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\n    pathpy is an OpenSource python package for the analysis of sequential data on pathways and temporal networks using higher- and multi order graphical models\n\n    Copyright (C) 2016-2017 Ingo Scholtes, ETH Zürich\n\n    This program is free software: you can redistribute it and/or modify\n    it under the terms of the GNU Affero General Public License as published\n    by the Free Software Foundation, either version 3 of the License, or\n    (at your option) any later version.\n\n    This program is distributed in the hope that it will be useful,\n    but WITHOUT ANY WARRANTY; without even the implied warranty of\n    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n    GNU Affero General Public License for more details.\n\n    You should have received a copy of the GNU Affero General Public License\n    along with this program.  If not, see <http://www.gnu.org/licenses/>.\n\n    Contact the developer:\n    \n    E-mail: ischoltes@ethz.ch\n    Web:    http://www.ingoscholtes.net\n\"\"\"\n\nimport numpy as _np\nimport sys as _sys\nimport collections as _co\nimport bisect as _bs\nimport datetime as _dt\nimport time as _t\n\nfrom pathpy.Log import Log\nfrom pathpy.Log import Severity\nimport pathpy.Paths\n\n\nclass TemporalNetwork:\n    \"\"\" This class represents a sequence of time-stamped edges.\n       Instances of this class can be used to generate path statistics \n       based on the time-respecting paths resulting from a given maximum\n       time difference between consecutive time-stamped edges.\n    \"\"\"\n\n    def __init__(self, tedges = None):\n        \"\"\"\n        Constructor that generates a temporal network instance. \n        \n        @param tedges: an optional list of (possibly unordered time-stamped) links \n            from which to construct a temporal network instance. For the default value None        \n            an empty temporal network will be created.\n        \"\"\"\n\n        ## A list of time-stamped edges of this temporal network\n        self.tedges = []        \n\n        ## A list of nodes of this temporal network\n        self.nodes = []\n\n        ## A dictionary storing all time-stamped links, indexed by time-stamps\n        self.time = _co.defaultdict( lambda: list() )\n\n        ## A dictionary storing all time-stamped links, indexed by time and target node\n        self.targets = _co.defaultdict( lambda: dict() )\n\n        ## A dictionary storing all time-stamped links, indexed by time and source node \n        self.sources = _co.defaultdict( lambda: dict() )\n\n        ## A dictionary storing time stamps at which links (v,*;t) originate from node v\n        self.activities = _co.defaultdict( lambda: list() )\n\n        ## A dictionary storing sets of time stamps at which links (v,*;t) originate from node v\n        ## Note that the insertion into a set is much faster than repeatedly checking whether \n        ## an element already exists in a list!\n        self.activities_sets = _co.defaultdict( lambda: set() )\n\n        ## An ordered list of time-stamps\n        self.ordered_times = []        \n\n        nodes_seen = _co.defaultdict( lambda:False )\n\n        if tedges is not None:\n            Log.add('Building index data structures ...')\n\n            for e in tedges:\n                self.activities_sets[e[0]].add(e[2])\n                self.time[e[2]].append(e)\n                self.targets[e[2]].setdefault(e[1], []).append(e)\n                self.sources[e[2]].setdefault(e[0], []).append(e)\n                if not nodes_seen[e[0]]:\n                    nodes_seen[e[0]] = True\n                if not nodes_seen[e[1]]:\n                    nodes_seen[e[1]] = True\n            self.tedges = tedges\n            self.nodes = list(nodes_seen.keys())\n\n            Log.add('Sorting time stamps ...')\n\n            self.ordered_times = sorted(self.time.keys())\n            for v in self.nodes:\n                self.activities[v] = sorted(self.activities_sets[v])\n            Log.add('finished.')\n\n\n    @staticmethod\n    def readFile(filename, sep=',', timestampformat=\"%Y-%m-%d %H:%M\", maxlines=_sys.maxsize):\n        \"\"\" Reads time-stamped links from a file and returns a new instance \n            of the class TemporalNetwork. The file is assumed to have a header \n\n                source target time \n\n            where columns can be in arbitrary order and separated by arbitrary characters. \n            Each time-stamped link must occur in a separate line and links are assumed to be\n            directed.\n             \n            The time column can be omitted and in this case all links are assumed to occur \n            in consecutive time stamps (that have a distance of one). Time stamps can be simple \n            integers, or strings to be converted to UNIX time stamps via a custom timestamp format. \n            For this, the python function datetime.strptime will be used. \n\n            @param sep: the character that separates columns \n            @param filename: path of the file to read from\n            @param timestampformat: used to convert string timestamps to UNIX timestamps. This parameter is \n                ignored, if the timestamps are digit types (like a simple int).\n            @param maxlines: limit reading of file to certain number of lines, default sys.maxsize\n\n        \"\"\"\n        assert (filename != ''), 'Empty filename given'\n        \n        # Read header\n        with open(filename, 'r') as f:\n            tedges = []\n        \n            header = f.readline()\n            header = header.split(sep)\n\n            # If header columns are included, arbitrary column orders are supported\n            time_ix = -1\n            source_ix = -1\n            mid_ix = -1\n            weight_ix = -1\n            target_ix = -1\n            for i in range(len(header)):\n                header[i] = header[i].strip()\n                if header[i] == 'node1' or header[i] == 'source':\n                    source_ix = i\n                elif header[i] == 'node2' or header[i] == 'target':\n                    target_ix = i\n                elif header[i] == 'time' or header[i] == 'timestamp':\n                    time_ix = i\n\n            assert (source_ix >= 0 and target_ix >= 0), \"Detected invalid header columns: %s\" % header\n\n            if time_ix<0:\n                Log.add('No time stamps found in data, assuming consecutive links', Severity.WARNING)\n        \n            Log.add('Reading time-stamped links ...')\n\n            line = f.readline()\n            n = 1 \n            while line and n <= maxlines:\n                fields = line.rstrip().split(sep)\n                try:\n                    if time_ix >=0:\n                        timestamp = fields[time_ix]\n                        # if the timestamp is a number, we use this \n                        if timestamp.isdigit():\n                            t = int(timestamp)\n                        else:   # if it is a string, we use the timestamp format to convert it to a UNIX timestamp                                \n                            x = _dt.datetime.strptime(timestamp, timestampformat)\n                            t = int(_t.mktime(x.timetuple()))\n                    else:\n                        t = n                \n                    if t>=0:\n                        tedge = (fields[source_ix], fields[target_ix], t)\n                        tedges.append(tedge)\n                    else:\n                        Log.add('Ignoring negative timestamp in line ' + str(n+1) + ': \"' + line.strip() + '\"', Severity.WARNING)\n                except (IndexError, ValueError):\n                    Log.add('Ignoring malformed data in line ' + str(n+1) + ': \"' +  line.strip() + '\"', Severity.WARNING)\n                line = f.readline()\n                n += 1\n        # end of with open()\n\n        return TemporalNetwork(tedges = tedges)\n\n\n\n    def filterEdges(self, edge_filter):\n        \"\"\"Filter time-stamped edges according to a given filter expression. \n\n        @param edge_filter: an arbitrary filter function of the form filter_func(v, w, time) that \n            returns True for time-stamped edges that shall pass the filter, and False for all edges that \n            shall be filtered out.\n        \"\"\"\n\n        Log.add('Starting filtering ...', Severity.INFO)\n        new_t_edges = []\n\n        for (v,w,t) in self.tedges:\n            if edge_filter(v,w,t):\n                new_t_edges.append((v,w,t))\n\n        Log.add('finished. Filtered out ' + str(self.ecount() - len(new_t_edges)) + ' time-stamped edges.', Severity.INFO)\n\n        return TemporalNetwork(tedges=new_t_edges)\n\n\n    def addEdge(self, source, target, ts):\n        \"\"\"Adds a directed time-stamped edge (source,target;time) to the temporal network. To add an undirected \n            time-stamped link (u,v;t) at time t, please call addEdge(u,v;t) and addEdge(v,u;t).\n        \n        @param source: name of the source node of a directed, time-stamped link\n        @param target: name of the target node of a directed, time-stamped link\n        @param ts: (integer) time-stamp of the time-stamped link\n        \"\"\"\n        e = (source, target, ts)\n        self.tedges.append(e)\n        if source not in self.nodes:\n            self.nodes.append(source)\n        if target not in self.nodes:\n            self.nodes.append(target)\n\n        # Add edge to index structures\n        self.time[ts].append(e)\n        self.targets[ts].setdefault(target, []).append(e)\n        self.sources[ts].setdefault(source, []).append(e)\n\n        if ts not in self.activities[source]:\n            self.activities[source].append(ts)\n            self.activities[source].sort()\n\n        # Reorder time stamps\n        self.ordered_times = sorted(self.time.keys())       \n\n\n    def vcount(self):\n        \"\"\"\n        Returns the number of vertices in the temporal network. \n        This number corresponds to the number of nodes in the (first-order) \n        time-aggregated network.\n        \"\"\"\n\n        return len(self.nodes)\n\n        \n    def ecount(self):\n        \"\"\"\n        Returns the number of time-stamped edges (u,v;t) in the temporal network.\n        This number corresponds to the sum of link weights in the (first-order)\n        time-aggregated network.\n        \"\"\"\n\n        return len(self.tedges)\n\n\n    def getObservationLength(self):\n        \"\"\"\n        Returns the length of the observation time in time units.\n        \"\"\"\n\n        return max(self.ordered_times)-min(self.ordered_times)\n    \n\n    def getInterEventTimes(self):\n        \"\"\"\n        Returns an array containing all time differences between any \n        two consecutive time-stamped links (involving any node)\n        \"\"\"\n\n        timediffs = []\n        for i in range(1, len(self.ordered_times)):\n            timediffs += [self.ordered_times[i] - self.ordered_times[i-1]]\n        return _np.array(timediffs)\n\n\n    def getInterPathTimes(self):\n        \"\"\"\n        Returns a dictionary which, for each node v, contains all time differences \n        between any time-stamped link (*,v;t) and the next link (v,*;t') (t'>t)\n        in the temporal network\n        \"\"\"\n\n        interPathTimes = _co.defaultdict( lambda: list() )\n        for e in self.tedges:\n            # Get target v of current edge e=(u,v,t)\n            v = e[1]\n            t = e[2]\n\n            # Get time stamp of link (v,*,t_next) with smallest t_next such that t_next > t\n            i = _bs.bisect_right(self.activities[v], t)\n            if i != len(self.activities[v]):\n                interPathTimes[v].append(self.activities[v][i]-t)\n        return interPathTimes\n\n\n    def summary(self):\n        \"\"\"\n        Returns a string containing basic summary statistics of this temporal network\n        \"\"\"\n\n        summary = ''\n\n        summary += 'Nodes:\\t\\t\\t' +  str(self.vcount()) + '\\n'\n        summary += 'Time-stamped links:\\t' + str(self.ecount()) + '\\n'\n        if self.vcount()>0:\n            summary += 'Links/Nodes:\\t\\t' + str(self.ecount()/self.vcount()) + '\\n'\n        else:\n            summary += 'Links/Nodes:\\t\\tN/A\\n'\n        if len(self.ordered_times)>1:\n            summary += 'Observation period:\\t[' + str(min(self.ordered_times)) + ', ' + str(max(self.ordered_times)) + ']\\n'\n            summary += 'Observation length:\\t' + str(max(self.ordered_times) - min(self.ordered_times)) + '\\n'\n            summary += 'Time stamps:\\t\\t' + str(len(self.ordered_times)) + '\\n'\n\n            d = self.getInterEventTimes()    \n            summary += 'Avg. inter-event dt:\\t' + str(_np.mean(d)) + '\\n'\n            summary += 'Min/Max inter-event dt:\\t' + str(min(d)) + '/' + str(max(d)) + '\\n'\n        \n        return summary       \n\n\n    def __str__(self):\n        \"\"\"\n        Returns the default string representation of \n        this temporal network instance.\n        \"\"\"\n        return self.summary()\n   \n\n    def ShuffleEdges(self, l=0, with_replacement=False):        \n        \"\"\"\n        Generates a shuffled version of the temporal network in which edge statistics (i.e.\n        the frequencies of time-stamped edges) are preserved, while all order correlations are \n        destroyed. The shuffling procedure randomly reshuffles the time-stamps of links.\n        \n        @param l: the length of the sequence to be generated (i.e. the number of time-stamped links.\n            For the default value l=0, the length of the generated shuffled temporal network will be \n            equal to that of the original temporal network. \n        @param with_replacement: Whether or not the sampling should be with replacement (default False)\n        \"\"\"\n\n        tedges = []        \n        \n        timestamps = [e[2] for e in self.tedges]\n        edges = list(self.tedges)\n        \n        if l==0:\n            l = len(self.tedges)\n        for i in range(l):\n            \n            if with_replacement:\n            # Pick random link\n                edge = edges[_np.random.randint(0, len(edges))]\n                # Pick random time stamp\n                time = timestamps[_np.random.randint(0, len(timestamps))]\n            else:\n                # Pick random link\n                edge = edges.pop(_np.random.randint(0, len(edges)))\n            # Pick random time stamp\n                time = timestamps.pop(_np.random.randint(0, len(timestamps)))            \n            \n            # Generate new time-stamped link\n            tedges.append( (edge[0], edge[1], time) )\n\n        # Generate temporal network\n        t = TemporalNetwork(tedges=tedges)\n\n        # Fix node order to correspond to original network\n        t.nodes = self.nodes\n            \n        return t\n\n\n    def exportUnfoldedNetwork(self, filename):\n        \"\"\"\n        Generates a tex file that can be compiled to a time-unfolded \n        representation of the temporal network.\n\n        @param filename: the name of the tex file to be generated.\n        \"\"\"    \n        \n        import os as _os\n\n        output = []\n            \n        output.append('\\\\documentclass{article}\\n')\n        output.append('\\\\usepackage{tikz}\\n')\n        output.append('\\\\usepackage{verbatim}\\n')\n        output.append('\\\\usepackage[active,tightpage]{preview}\\n')\n        output.append('\\\\PreviewEnvironment{tikzpicture}\\n')\n        output.append('\\\\setlength\\PreviewBorder{5pt}%\\n')\n        output.append('\\\\usetikzlibrary{arrows}\\n')\n        output.append('\\\\usetikzlibrary{positioning}\\n')\n        output.append('\\\\begin{document}\\n')\n        output.append('\\\\begin{center}\\n')\n        output.append('\\\\newcounter{a}\\n')\n        output.append(\"\\\\begin{tikzpicture}[->,>=stealth',auto,scale=0.5, every node/.style={scale=0.9}]\\n\")\n        output.append(\"\\\\tikzstyle{node} = [fill=lightgray,text=black,circle]\\n\")\n        output.append(\"\\\\tikzstyle{v} = [fill=black,text=white,circle]\\n\")\n        output.append(\"\\\\tikzstyle{dst} = [fill=lightgray,text=black,circle]\\n\")\n        output.append(\"\\\\tikzstyle{lbl} = [fill=white,text=black,circle]\\n\")\n\n        last = ''\n            \n        for n in _np.sort(self.nodes):\n            if last == '':\n                output.append(\"\\\\node[lbl]                     (\" + n + \"-0)   {$\" + n + \"$};\\n\")\n            else:\n                output.append(\"\\\\node[lbl,right=0.5cm of \"+last+\"-0] (\" + n + \"-0)   {$\" + n + \"$};\\n\")\n            last = n\n            \n        output.append(\"\\\\setcounter{a}{0}\\n\")\n        output.append(\"\\\\foreach \\\\number in {\"+ str(min(self.ordered_times))+ \",...,\" + str(max(self.ordered_times)+1) + \"}{\\n\")\n        output.append(\"\\\\setcounter{a}{\\\\number}\\n\")\n        output.append(\"\\\\addtocounter{a}{-1}\\n\")\n        output.append(\"\\\\pgfmathparse{\\\\thea}\\n\")\n        \n        for n in  _np.sort(self.nodes):\n            output.append(\"\\\\node[v,below=0.3cm of \" + n + \"-\\\\pgfmathresult]     (\" + n + \"-\\\\number) {};\\n\")\n        output.append(\"\\\\node[lbl,left=0.5cm of \" + _np.sort(self.nodes)[0] + \"-\\\\number]    (col-\\\\pgfmathresult) {$t=$\\\\number};\\n\")\n        output.append(\"}\\n\")\n        output.append(\"\\\\path[->,thick]\\n\")\n        i = 1\n        \n        for ts in self.ordered_times:\n            for edge in self.time[ts]:\n                output.append(\"(\" + edge[0] + \"-\" + str(ts) + \") edge (\" + edge[1] + \"-\" + str(ts + 1) + \")\\n\")\n                i += 1                                \n        output.append(\";\\n\")\n        output.append(\n    \"\"\"\\end{tikzpicture}\n    \\end{center}\n    \\end{document}\"\"\")\n        \n        # create directory if necessary to avoid IO errors\n        directory = _os.path.dirname( filename )\n        if directory != '':\n            if not _os.path.exists( directory ):\n                _os.makedirs( directory )\n\n        with open(filename, \"w\") as tex_file:\n            tex_file.write(''.join(output))"
  },
  {
    "path": "pathpy/__init__.py",
    "content": "﻿from .Log import *\nfrom .TemporalNetwork import *\nfrom .Paths import *\nfrom .HigherOrderNetwork import *\nfrom .MultiOrderModel import *\nfrom .MarkovSequence import *\n\nimport pathpy.Log as Log"
  },
  {
    "path": "pathpy.pyproj",
    "content": "﻿<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<Project DefaultTargets=\"Build\" ToolsVersion=\"4.0\" xmlns=\"http://schemas.microsoft.com/developer/msbuild/2003\">\n  <PropertyGroup>\n    <Configuration Condition=\" '$(Configuration)' == '' \">Debug</Configuration>\n    <SchemaVersion>2.0</SchemaVersion>\n    <ProjectGuid>{896991fc-0289-4bae-b680-88e19508c91f}</ProjectGuid>\n    <ProjectHome />\n    <StartupFile>\n    </StartupFile>\n    <SearchPath />\n    <WorkingDirectory>.</WorkingDirectory>\n    <OutputPath>.</OutputPath>\n    <InterpreterId>\n    </InterpreterId>\n    <InterpreterVersion>\n    </InterpreterVersion>\n    <VisualStudioVersion Condition=\"'$(VisualStudioVersion)' == ''\">10.0</VisualStudioVersion>\n    <VSToolsPath Condition=\"'$(VSToolsPath)' == ''\">$(MSBuildExtensionsPath32)\\Microsoft\\VisualStudio\\v$(VisualStudioVersion)</VSToolsPath>\n    <Name>pathpy</Name>\n  </PropertyGroup>\n  <PropertyGroup Condition=\"'$(Configuration)' == 'Debug'\" />\n  <PropertyGroup Condition=\"'$(Configuration)' == 'Release'\" />\n  <PropertyGroup>\n    <PtvsTargetsFile>$(MSBuildExtensionsPath32)\\Microsoft\\VisualStudio\\v$(VisualStudioVersion)\\Python Tools\\Microsoft.PythonTools.targets</PtvsTargetsFile>\n  </PropertyGroup>\n  <ItemGroup>\n    <Compile Include=\"pathpy\\MarkovSequence.py\">\n      <SubType>Code</SubType>\n    </Compile>\n    <Compile Include=\"pathpy\\MultiOrderModel.py\">\n      <SubType>Code</SubType>\n    </Compile>\n    <Compile Include=\"pathpy\\HigherOrderNetwork.py\">\n      <SubType>Code</SubType>\n    </Compile>\n    <Compile Include=\"pathpy\\Log.py\">\n      <SubType>Code</SubType>\n    </Compile>\n    <Compile Include=\"pathpy\\Paths.py\">\n      <SubType>Code</SubType>\n    </Compile>\n    <Compile Include=\"setup.py\" />\n    <Compile Include=\"pathpy\\TemporalNetwork.py\" />\n    <Compile Include=\"pathpy\\__init__.py\" />\n    <Compile Include=\"tests\\conftest.py\" />\n    <Compile Include=\"tests\\test_estimation.py\" />\n    <Compile Include=\"tests\\test_Path.py\" />\n    <Compile Include=\"tests\\test_TemporalNetwork.py\" />\n  </ItemGroup>\n  <ItemGroup>\n    <Folder Include=\"pathpy\\\" />\n    <Folder Include=\"tests\\\" />\n    <Folder Include=\"tests\\test_data\\\" />\n    <Folder Include=\"tests\\__pycache__\\\" />\n  </ItemGroup>\n  <ItemGroup>\n    <Content Include=\".gitignore\" />\n    <Content Include=\"DESCRIPTION.rst\" />\n    <Content Include=\"LICENSE.txt\" />\n    <Content Include=\"README.md\" />\n    <Content Include=\"tests\\README.md\" />\n    <Content Include=\"tests\\test_data\\edge_frequency.edge\" />\n    <Content Include=\"tests\\test_data\\example_int.tedges\" />\n    <Content Include=\"tests\\test_data\\example_timestamp.tedges\" />\n    <Content Include=\"tests\\test_data\\ngram_simple.ngram\" />\n    <Content Include=\"tests\\__pycache__\\conftest.cpython-35-PYTEST.pyc\" />\n    <Content Include=\"tests\\__pycache__\\test_estimation.cpython-35-PYTEST.pyc\" />\n    <Content Include=\"tests\\__pycache__\\test_Path.cpython-35-PYTEST.pyc\" />\n    <Content Include=\"tests\\__pycache__\\test_TemporalNetwork.cpython-35-PYTEST.pyc\" />\n  </ItemGroup>\n  <Import Project=\"$(PtvsTargetsFile)\" Condition=\"Exists($(PtvsTargetsFile))\" />\n  <Import Project=\"$(MSBuildToolsPath)\\Microsoft.Common.targets\" Condition=\"!Exists($(PtvsTargetsFile))\" />\n</Project>"
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  {
    "path": "pathpy.sln",
    "content": "﻿\nMicrosoft Visual Studio Solution File, Format Version 12.00\n# Visual Studio 14\nVisualStudioVersion = 14.0.25420.1\nMinimumVisualStudioVersion = 10.0.40219.1\nProject(\"{888888A0-9F3D-457C-B088-3A5042F75D52}\") = \"pathpy\", \"pathpy.pyproj\", \"{896991FC-0289-4BAE-B680-88E19508C91F}\"\nEndProject\nGlobal\n\tGlobalSection(SolutionConfigurationPlatforms) = preSolution\n\t\tDebug|Any CPU = Debug|Any CPU\n\t\tRelease|Any CPU = Release|Any CPU\n\tEndGlobalSection\n\tGlobalSection(ProjectConfigurationPlatforms) = postSolution\n\t\t{896991FC-0289-4BAE-B680-88E19508C91F}.Debug|Any CPU.ActiveCfg = Debug|Any CPU\n\t\t{896991FC-0289-4BAE-B680-88E19508C91F}.Release|Any CPU.ActiveCfg = Release|Any CPU\n\tEndGlobalSection\n\tGlobalSection(SolutionProperties) = preSolution\n\t\tHideSolutionNode = FALSE\n\tEndGlobalSection\nEndGlobal\n"
  },
  {
    "path": "setup.py",
    "content": "\"\"\"A python package for the analysis of sequential data on pathways and temporal networks from the perspective of higher-order network models.\n\nSee: https://packaging.python.org/en/latest/distributing.html\n\"\"\"\n\n# Always prefer setuptools over distutils\nfrom setuptools import setup, find_packages\n# To use a consistent encoding\nfrom codecs import open\nfrom os import path\nimport sys\n\nhere = path.abspath(path.dirname(__file__))\n\n# Get the long description from the relevant file\nwith open(path.join(here, 'DESCRIPTION.rst'), encoding='utf-8') as f:\n    long_description = f.read()\n\nrequired = ['numpy', 'scipy']\n\nif (sys.version_info.major, sys.version_info.minor) < (3,4):\n        required.append('enum34')\n\nsetup(\n    name='pathpy',\n\n    # Versions should comply with PEP440.  For a discussion on single-sourcing\n    # the version across setup.py and the project code, see\n    # https://packaging.python.org/en/latest/single_source_version.html\n    version='1.1.170316',\n\n    description='A python package for the analysis of sequential data on pathways and temporal networks from the perspective of higher-order network models.',\n    long_description=long_description,\n\n    # The project's main homepage.\n    url='https://github.com/IngoScholtes/pathpy',\n\n    # Author details\n    author='Ingo Scholtes',\n    author_email='ischoltes@ethz.ch',\n\n    # Choose your license\n    license='AGPL-3.0+',\n\n    # See https://pypi.python.org/pypi?%3Aaction=list_classifiers\n    classifiers=[\n        # How mature is this project? Common values are\n        #   3 - Alpha\n        #   4 - Beta\n        #   5 - Production/Stable\n        'Development Status :: 4 - Beta',\n\n        # Indicate who your project is intended for\n        'Intended Audience :: Science/Research',\n        'Topic :: Scientific/Engineering :: Information Analysis',\n\n        # Pick your license as you wish (should match \"license\" above)\n        'License :: OSI Approved :: AGPL-3.0+ License',\n\n        # Specify the Python versions you support here. In particular, ensure\n        # that you indicate whether you support Python 2, Python 3 or both.\n        'Programming Language :: Python :: 3',\n        'Programming Language :: Python :: 3.2',\n        'Programming Language :: Python :: 3.3',\n        'Programming Language :: Python :: 3.4',\n        'Programming Language :: Python :: 3.5',\n    ],\n\n    # What does your project relate to?\n    keywords='network analysis temporal networks pathways sequence modeling graph mining',\n\n    # You can just specify the packages manually here if your project is\n    # simple. Or you can use find_packages().\n    packages=find_packages(exclude=['contrib', 'docs', 'tests*']),\n\n    # List run-time dependencies here.  These will be installed by pip when\n    # your project is installed. For an analysis of \"install_requires\" vs pip's\n    # requirements files see:\n    # https://packaging.python.org/en/latest/requirements.html\n    install_requires=required,    \n\n    # If there are data files included in your packages that need to be\n    # installed, specify them here.  If using Python 2.6 or less, then these\n    # have to be included in MANIFEST.in as well.\n    package_data={\n    }\n)"
  },
  {
    "path": "tests/README.md",
    "content": "# Unit tests for pypath\n\nThis directory contains the unit tests for methods and functions \nin pathpy.\nThe testing framework [pytest](doc.pytest.org/) \nis required to run the tests.\n\nTo run the test suite (without slow tests) run\n```bash\n$ pytest tests\n```\n\n## Slow functions\n\nSlow functions can be decorated with `slow` to mark them \nas skippable if you require only a quick check.\nTo run all tests add the flag `--runslow`:\n```bash\n$ pytest --runslow\n```\n\n## Coverage report\n\nTo compute a coverage report of the tests you need to install \n[coverage.py](https://coverage.readthedocs.io/en/coverage-4.3.4/)\nas well as its `pytest` integration \n[pytest-cov][1]\n```bash\n$ pytest tests/ --runslow --cov=pathpy --cov-report html\n```\nwhich will create an html coverage report in the same directory.\n\n[1]: https://pypi.python.org/pypi/pytest-cov\n"
  },
  {
    "path": "tests/conftest.py",
    "content": "import pathpy as pp\nimport pytest\nimport numpy as np\nimport os\n\ntest_directory = os.path.dirname(os.path.abspath(__file__))\ntest_data_dir = os.path.join(test_directory, 'test_data')\n\n\ndef pytest_addoption(parser):\n    parser.addoption(\"--runslow\", action=\"store_true\",\n        help=\"run slow tests\")\n\n\ndef pytest_runtest_setup(item):\n    if 'slow' in item.keywords and not item.config.getvalue(\"runslow\"):\n        pytest.skip(\"need --runslow option to run\")\n\n\n\n@pytest.fixture()\ndef test_data_directory():\n    return test_data_dir\n\n\n@pytest.fixture()\ndef path_from_ngram_file():\n    \"\"\"load the example file as pypath.Path\"\"\"\n    ngram_file_path = os.path.join(test_data_dir, 'ngram_simple.ngram')\n    path = pp.Paths.readFile(ngram_file_path, pathFrequency=True)\n    return path\n\n\n@pytest.fixture()\ndef path_from_edge_file():\n    file_path = os.path.join(test_data_dir, 'edge_frequency.edge')\n    path = pp.Paths.readEdges(file_path, weight=True)\n\n    return path\n\n\n@pytest.fixture()\ndef path_from_edge_file_undirected():\n    file_path = os.path.join(test_data_dir, 'edge_frequency.edge')\n    path = pp.Paths.readEdges(file_path, weight=True, undirected=True)\n    return path\n\n\ndef generate_random_path(size, rnd_seed):\n    \"\"\"Generate a Path with random path sequences\"\"\"\n    import string\n    node_set = string.ascii_lowercase\n\n    def random_ngram(p_len, nodes):\n        num_elements = len(nodes)\n        sequence = np.random.choice(num_elements, p_len)\n        path = [nodes[i] for i in sequence]\n        return ','.join(path)\n\n    np.random.seed(rnd_seed)\n    paths = pp.Paths()\n    for _ in range(size):\n        frequency = np.random.randint(1, 4)\n        path_length = np.random.randint(1, 10)\n        path_to_add = random_ngram(path_length, node_set)\n        paths.addPath(path_to_add, pathFrequency=frequency)\n\n    return paths\n\n\n@pytest.fixture(scope='function')\ndef random_paths():\n    \"\"\"Generate a Path with random path sequences\"\"\"\n    return generate_random_path\n\n\n@pytest.fixture()\ndef temporal_network_object():\n    t = pp.TemporalNetwork()\n    # Path of length two\n    t.addEdge(\"c\", \"e\", 1)\n    t.addEdge(\"e\", \"f\", 2)\n\n    # Path of length two\n    t.addEdge(\"a\", \"e\", 3)\n    t.addEdge(\"e\", \"g\", 4)\n\n    # Path of length two\n    t.addEdge(\"c\", \"e\", 5)\n    t.addEdge(\"e\", \"f\", 6)\n\n    # Path of length two\n    t.addEdge(\"a\", \"e\", 7)\n    t.addEdge(\"e\", \"g\", 8)\n\n    # Path of length two\n    t.addEdge(\"c\", \"e\", 9)\n    t.addEdge(\"e\", \"f\", 10)\n\n    # The next two edges continue the previous path to ( c-> e-> f-> e -> b )\n    t.addEdge(\"f\", \"e\", 11)\n    t.addEdge(\"e\", \"b\", 12)\n\n    # This is an isolated edge (i.e. path of length one)\n    t.addEdge(\"e\", \"b\", 13)\n\n    # Path of length two\n    t.addEdge(\"c\", \"e\", 14)\n    t.addEdge(\"e\", \"f\", 15)\n\n    # Path of length two\n    t.addEdge(\"b\", \"e\", 16)\n    t.addEdge(\"e\", \"g\", 17)\n\n    # Path of length two\n    t.addEdge(\"c\", \"e\", 18)\n    t.addEdge(\"e\", \"f\", 19)\n\n    # Path of length two\n    t.addEdge(\"c\", \"e\", 20)\n    t.addEdge(\"e\", \"f\", 21)\n\n    return t\n"
  },
  {
    "path": "tests/test_Path.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\nCreated on Fri Feb 20 11:59:22 2015\n@author: Ingo Scholtes\n\n(c) Copyright ETH Zurich, Chair of Systems Design, 2015-2017\n\"\"\"\nimport pathpy as pp\nimport pytest\n\n\nslow = pytest.mark.slow\n\n\ndef test_readfile_import(path_from_ngram_file):\n    levels = list(path_from_ngram_file.paths.keys())\n    max_level = max(levels)\n    expected = 5\n    assert max_level == expected, \\\n        \"The nodes have not been imported correctly\"\n\n    assert path_from_ngram_file.getNodes() == {'a', 'b', 'c', 'd', 'e'}, \\\n        \"Wrong node labels\"\n\n\ndef test_write_file(tmpdir, random_paths):\n    dir_path = tmpdir.mkdir(\"sub\").join(\"test.edges\")\n    p = random_paths(30, 50)\n\n    expected_seq = ''.join(p.getSequence())\n    expected_paths = sorted(expected_seq.split('|'))\n\n    p.writeFile(dir_path.strpath)\n    p2 = pp.Paths.readFile(dir_path.strpath, pathFrequency=True)\n\n    read_back = ''.join(p2.getSequence())\n    read_back_paths = sorted(read_back.split('|'))\n\n    assert expected_paths == read_back_paths\n\n\ndef test_read_edges_import(path_from_edge_file):\n    \"\"\"test if the Paths.readEdges functions works\"\"\"\n    levels = list(path_from_edge_file.paths.keys())\n    max_level = max(levels)\n    expected_max = 1\n    assert expected_max == max_level, \\\n        \"The nodes have not been imported correctly\"\n\n    assert path_from_edge_file.getNodes() == {'1', '2', '3', '5'}, \\\n        \"Nodes not imported correctly\"\n\n\ndef test_read_edges_undirected(path_from_edge_file_undirected):\n    p = path_from_edge_file_undirected\n    layers = list(p.paths.keys())\n    max_layers = max(layers)\n    expected_layers = 1\n    assert max_layers == expected_layers, \\\n        \"The nodes have not been imported correctly\"\n\n    assert p.getNodes() == {'1', '2', '3', '5'}, \\\n        \"Nodes not imported correctly\"\n\n\ndef test_get_sequence(path_from_ngram_file):\n    from collections import Counter\n    \"\"\"Test if the Paths.getSequence function works correctly\"\"\"\n    sequence = path_from_ngram_file.getSequence()\n    sequence = \"\".join(sequence)\n    ct = Counter(sequence.split('|'))\n    assert dict(ct) == {'': 1, 'abcdab': 2, 'dedab': 4}, \\\n        \"Returned the wrong sequence\"\n\n\ndef test_get_unique_paths(random_paths):\n    p = random_paths(90, 90)\n    assert p.getUniquePaths() == 87, \\\n        \"Wrong number of paths detected\"\n\n\ndef test_observation_count_file(path_from_ngram_file):\n    assert path_from_ngram_file.ObservationCount() == 6, \\\n        \"Wrong number of observations detected\"\n\n\ndef test_observation_count_large(random_paths):\n    p = random_paths(90, 90)\n    assert p.ObservationCount() == 193, \\\n        \"Wrong number of observations detected\"\n\n\ndef test_path_summary(random_paths):\n    p = random_paths(90, 90)\n    print(p)\n\n\ndef test_summary_multi_order_model(random_paths):\n    p = random_paths(90, 90)\n    multi = pp.MultiOrderModel(paths=p, maxOrder=3)\n    print(multi)\n\n\ndef test_get_shortest_paths(path_from_ngram_file):\n    path_from_ngram_file.getShortestPaths()\n    paths_dict = path_from_ngram_file.getShortestPaths()\n    expected_paths = {('d', 'a'): {('d', 'a')},\n                      ('b', 'd'): {('b', 'c', 'd')},\n                      ('d', 'e'): {('d', 'e')},\n                      ('a', 'c'): {('a', 'b', 'c')},\n                      ('a', 'a'): {('a',)},\n                      ('e', 'a'): {('e', 'd', 'a')},\n                      ('e', 'b'): {('e', 'd', 'a', 'b')},\n                      ('e', 'e'): {('e',)},\n                      ('a', 'b'): {('a', 'b')},\n                      ('b', 'b'): {('b',)},\n                      ('c', 'd'): {('c', 'd')},\n                      ('d', 'b'): {('d', 'a', 'b')},\n                      ('c', 'a'): {('c', 'd', 'a')},\n                      ('b', 'a'): {('b', 'c', 'd', 'a')},\n                      ('c', 'b'): {('c', 'd', 'a', 'b')},\n                      ('e', 'd'): {('e', 'd')},\n                      ('a', 'd'): {('a', 'b', 'c', 'd')},\n                      ('d', 'd'): {('d',)},\n                      ('c', 'c'): {('c',)},\n                      ('b', 'c'): {('b', 'c')}\n                      }\n    paths_to_check = dict()\n    for k in paths_dict:\n        for p in paths_dict[k]:\n            paths_to_check[(k, p)] = paths_dict[k][p]\n    assert paths_to_check == expected_paths\n\n\ndef test_get_contained_paths():\n    path_to_check = ('a', 'b', 'c', 'd', 'e', 'f', 'g')\n    node_filter = ('a', 'b', 'd', 'f', 'g')\n    cont_paths = pp.Paths.getContainedPaths(path_to_check, node_filter)\n    expected = [('a', 'b'), ('d',), ('f', 'g')]\n    assert cont_paths == expected\n\n\ndef test_filter_paths(path_from_ngram_file):\n    from collections import Counter\n    p = path_from_ngram_file\n    new_paths = p.filterPaths(node_filter=['a', 'b', 'c'])\n    expected_sequence = {'': 1, 'ab': 6, 'abc': 2}\n\n    new_sequence = ''.join(new_paths.getSequence())\n    ct = Counter(new_sequence.split('|'))\n    assert dict(ct) == expected_sequence\n\n\ndef test_project_paths(path_from_ngram_file):\n    from collections import Counter\n    p = path_from_ngram_file\n    mapping = {'a': 'x', 'b': 'x', 'c': 'y', 'd': 'y', 'e': 'y'}\n    new_p = p.projectPaths(mapping=mapping)\n    new_sequence = ''.join(new_p.getSequence())\n    ct = Counter(new_sequence.split('|'))\n    expected_sequence = {'': 1, 'xxyyxx': 2, 'yyyxx': 4}\n    assert dict(ct) == expected_sequence\n\n\ndef test_get_nodes(random_paths):\n    p = random_paths(3, 9)\n    rest = p.getNodes()\n    expected = {'b', 'o', 'u', 'v', 'w', 'y'}\n    assert rest == expected\n"
  },
  {
    "path": "tests/test_TemporalNetwork.py",
    "content": "import pathpy as pp\nimport os\nimport numpy as np\n\n\ndef test_read_temporal_file_int(test_data_directory,):\n    file_path = os.path.join(test_data_directory, 'example_int.tedges')\n    t = pp.TemporalNetwork.readFile(file_path)\n    times = t.ordered_times\n    expected_times = [0, 2, 4, 5, 6, 8]\n    assert times == expected_times\n\n    activities = sorted(list(t.activities.values()))\n    expected_activities = [[], [], [], [], [0, 2, 5], [2], [4], [6], [8]]\n    assert expected_activities == activities\n\n\ndef test_read_temporal_file_time_stamp(test_data_directory,):\n    file_path = os.path.join(test_data_directory, 'example_timestamp.tedges')\n    t = pp.TemporalNetwork.readFile(file_path, timestampformat=\"%Y-%m-%d %H:%M\")\n    times = t.ordered_times\n    time_diffs = [j-i for i, j in zip(times[:-1], times[1:])]\n    expected_diffs = [10800, 15060, 264960]\n    # TODO: The actual time number depends on local set by the user\n    assert time_diffs == expected_diffs\n\n\ndef test_filter_temporal_edges(temporal_network_object):\n    t = temporal_network_object\n\n    def filter_func(v, w, time):\n        return time % 2 == 0\n\n    filtered = t.filterEdges(filter_func)\n    times = filtered.ordered_times\n    expected = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]\n    assert times == expected\n\n\ndef test_get_interpath_times(temporal_network_object):\n    t = temporal_network_object\n    inter_time = dict(t.getInterPathTimes())\n    expected = {'e': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n                'b': [4, 3], 'f': [9, 5, 1]\n                }\n    assert inter_time == expected\n\n\ndef test_shuffle_edges(temporal_network_object):\n    t = temporal_network_object\n    np.random.seed(90)\n    t1 = t.ShuffleEdges(with_replacement=True)\n    times1 = len(t1.tedges)\n    expected1 = len(t.tedges)\n    assert times1 == expected1\n    np.random.seed(90)\n    t2 = t.ShuffleEdges(l=4, with_replacement=False)\n    edges2 = len(t2.tedges)\n    expected2 = 4\n    assert edges2 == expected2\n\n\ndef test_inter_event_times(temporal_network_object):\n    time_diffs = temporal_network_object.getInterEventTimes()\n    # all time differences are 1\n    assert (time_diffs == 1).all()\n\n\ndef test_inter_path_times(temporal_network_object):\n    t = temporal_network_object\n    path_times = dict(t.getInterPathTimes())\n    expected = {'f': [9, 5, 1],\n                'e': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n                'b': [4, 3]}\n    assert path_times == expected\n\n\ndef test_temporal_summary(temporal_network_object):\n    print(temporal_network_object)\n"
  },
  {
    "path": "tests/test_data/edge_frequency.edge",
    "content": "1,2,45,3,3,3\n1,3,2,23,2,2\n1,5,5,12,5,2\n3,5,2,11,45,2\n5,3,4,12,2,2\n5,2,1,12,4,1\n"
  },
  {
    "path": "tests/test_data/example_int.tedges",
    "content": "source,target,time\n 1,2,0\n 1,2,2\n 1,3,5\n 3,2,6\n 2,1,8\n 4,5,2\n 5,3,4"
  },
  {
    "path": "tests/test_data/example_timestamp.tedges",
    "content": "source,target,time\n1,4,2000-03-04 12:45\n2,4,2000-03-04 15:45\n5,2,2000-03-04 19:56\n8,2,2000-03-07 21:32"
  },
  {
    "path": "tests/test_data/ngram_simple.ngram",
    "content": "a,b,c,d,a,b,2\nd,e,d,a,b,4\n"
  },
  {
    "path": "tests/test_estimation.py",
    "content": "# -*- coding: utf-8 -*-\n\"\"\"\nCreated on Fri Feb 20 11:59:22 2015\n@author: Ingo Scholtes\n\n(c) Copyright ETH Zurich, Chair of Systems Design, 2015-2017\n\"\"\"\n\nimport pathpy as pp\nimport numpy as _np\nimport pytest\n\n# mark to be used as decorator on slow functions such that they are only run\n# when explicitly called with `$ pytest --runslow`\nslow = pytest.mark.slow\n\n\ndef test_estimate_order_1():\n    \"\"\"Example without second-order correlations\"\"\"\n    paths = pp.Paths()\n\n    paths.addPath('a,c')\n    paths.addPath('b,c')\n    paths.addPath('c,d')\n    paths.addPath('c,e')\n\n    for k in range(4):\n        paths.addPath('a,c,d')\n        paths.addPath('b,c,e')\n        paths.addPath('b,c,d')\n        paths.addPath('a,c,e')\n\n    m = pp.MultiOrderModel(paths, maxOrder=2)\n    assert m.estimateOrder(\n        paths) == 1, \"Error, wrongly detected higher-order correlations\"\n\n\ndef test_estimate_order_2():\n    # Example with second-order correlations\n    paths = pp.Paths()\n\n    paths.addPath('a,c')\n    paths.addPath('b,c')\n    paths.addPath('c,d')\n    paths.addPath('c,e')\n\n    for k in range(4):\n        paths.addPath('a,c,d')\n        paths.addPath('b,c,e')\n\n    m = pp.MultiOrderModel(paths, maxOrder=2)\n    assert m.estimateOrder(\n        paths) == 2, \"Error, did not detect second-order correlations\"\n\n    x = list(map(str, _np.random.choice(range(10), 100000)))\n    ms = pp.MarkovSequence(x)\n    assert ms.estimateOrder(maxOrder=2, method='BIC') == 1, \\\n        \"Error, wrongly detected higher-order correlations\"\n    assert ms.estimateOrder(maxOrder=2, method='AIC') == 1, \\\n        \"Error, wrongly detected higher-order correlations\"\n\n    g1 = pp.HigherOrderNetwork(paths, k=1)\n    assert g1.vcount() == 5, \\\n        \"Error, wrong number of nodes in first-order network\"\n    assert g1.ecount() == 4, \\\n        \"Error, wrong number of links in first-order network\"\n\n    g2 = pp.HigherOrderNetwork(paths, k=2)\n    assert g2.vcount() == 4, \\\n        \"Error, wrong number of nodes in second-order network\"\n    assert g2.ecount() == 2, \\\n        \"Error, wrong number of links in second-order network\"\n\n    g2.reduceToGCC()\n    assert g2.vcount() == 1, \\\n        \"Error, wrong number of nodes in giant connected component\"\n    assert g2.ecount() == 0, \\\n        \"Error, wrong number of links in giant connected component\"\n\n\ndef test_estimate_order_strongly_connected():\n    \"\"\"\n    Example with single strongly connected component in first- \n    and two connected components in second-order network\n    \"\"\"\n    paths = pp.Paths()\n\n    ngram_list = ['a,b,c', 'b,c,b', 'c,b,a', \n                  'b,a,b', 'e,b,f', 'b,f,b', \n                  'f,b,e', 'b,e,b']\n\n    for ngram in ngram_list:\n        paths.addPath(ngram)\n\n    g1 = pp.HigherOrderNetwork(paths, k=1)\n    g1.reduceToGCC()\n    assert g1.vcount() == 5, \"Error, wrong number of nodes in first-order network\"\n    assert g1.ecount() == 8, \"Error, wrong number of links in first-order network\"\n\n    g2 = pp.HigherOrderNetwork(paths, k=2)\n    g2.reduceToGCC()\n    assert g2.vcount() == 4, \"Error, wrong number of nodes in second-order network\"\n    assert g2.ecount() == 4, \"Error, wrong number of links in second-order network\"\n\n    # test mapping of higher-order nodes and paths\n    assert g2.HigherOrderNodeToPath('a-b') == ('a', 'b'), \\\n        \"Error: mapping from higher-order node to first-order path failed\"\n    assert g2.HigherOrderPathToFirstOrder(('a-b', 'b-c')) == ('a', 'b', 'c'), \\\n        \"Error: mapping from higher-order path to first-order path failed\"\n\n\ndef test_temp_net_extraction(temporal_network_object):\n    t = temporal_network_object\n    paths = pp.Paths.fromTemporalNetwork(t, delta=1)\n\n    assert paths.ObservationCount() == 10, \\\n        \"Extracted wrong number of time-respecting paths\"\n\n\ndef test_betweenness_preference_empty():\n    t = pp.TemporalNetwork()\n    paths = pp.Paths.fromTemporalNetwork(t, delta=3)\n    assert len(paths.getNodes()) == 0\n\n    betweenness_pref = paths.BetweennessPreference('e', method='MLE')\n    expected = 0.0\n    assert betweenness_pref == pytest.approx(expected)\n\n\ndef test_betweenness_preference_mle(temporal_network_object):\n    t = temporal_network_object\n\n    # Extract (time-respecting) paths\n    paths = pp.Paths.fromTemporalNetwork(t, delta=1)\n    betweenness_pref = paths.BetweennessPreference('e', method='MLE')\n    expected = 1.2954618442383219\n    assert betweenness_pref == pytest.approx(expected)\n\n\ndef test_betweenness_preference_miller(temporal_network_object):\n    t = temporal_network_object\n    paths = pp.Paths.fromTemporalNetwork(t, delta=1)\n\n    betweenness_pref = paths.BetweennessPreference('e', method='Miller')\n    expected = 0.99546184423832196\n    assert betweenness_pref == pytest.approx(expected)\n\n\ndef test_betweenness_preference_normalized(temporal_network_object):\n    t = temporal_network_object\n    paths = pp.Paths.fromTemporalNetwork(t, delta=1)\n    # test normalize\n    betweenness_pref_norm = paths.BetweennessPreference('e', normalized=True)\n    expected_norm = 1\n    assert betweenness_pref_norm == pytest.approx(expected_norm)\n\n\ndef test_slow_down_factor_random(random_paths):\n    paths = random_paths(90, 90)\n    slow_down_factor = paths.getSlowDownFactor()\n    expected = 4.05\n    assert slow_down_factor == pytest.approx(expected, rel=1e-2), \\\n        \"Got %f slowdown factor expected %f +- 1e-2\" % (slow_down_factor, expected)\n\n\ndef test_get_distance_matrix_temporal(temporal_network_object):\n    p = pp.Paths.fromTemporalNetwork(temporal_network_object)\n    shortest_paths_dict = p.getDistanceMatrix()\n\n    path_distances = dict()\n    for k in shortest_paths_dict:\n        for p in shortest_paths_dict[k]:\n            path_distances[(k, p)] = shortest_paths_dict[k][p]\n\n    expected_distances = {\n        ('c', 'e'): 1,\n        ('c', 'f'): 2,\n        ('c', 'c'): 0,\n        ('b', 'g'): 2,\n        ('f', 'e'): 1,\n        ('c', 'b'): 4,\n        ('a', 'a'): 0,\n        ('a', 'g'): 2,\n        ('g', 'g'): 0,\n        ('e', 'g'): 1,\n        ('e', 'e'): 0,\n        ('b', 'b'): 0,\n        ('e', 'b'): 1,\n        ('e', 'f'): 1,\n        ('f', 'b'): 2,\n        ('a', 'e'): 1,\n        ('f', 'f'): 0,\n        ('b', 'e'): 1\n    }\n    assert path_distances == expected_distances\n\n\ndef test_get_distance_matrix_empty():\n    p = pp.Paths()\n    shortest_paths_dict = p.getDistanceMatrix()\n    assert len(shortest_paths_dict) == 0\n\n\ndef test_betweenness_centrality(path_from_ngram_file):\n    p = path_from_ngram_file\n    betweenness_centrality = p.BetweennessCentrality(normalized=False)\n    betweenness = {n: c for n, c in betweenness_centrality.items()}\n    expected = {'b': 2.0, 'a': 3.0, 'e': 0, 'c': 3.0, 'd': 5.0}\n    assert betweenness == expected\n\n\ndef test_betweenness_centrality_norm(path_from_ngram_file):\n    p = path_from_ngram_file\n    betweenness_centrality = p.BetweennessCentrality(normalized=True)\n    betweenness = max(c for c in betweenness_centrality.values())\n    expected_norm_max = 1\n    assert pytest.approx(betweenness) == expected_norm_max\n\n\ndef test_closeness_centrality(path_from_ngram_file):\n    p = path_from_ngram_file\n    closeness_centrality = p.ClosenessCentrality(normalized=False)\n    closeness_sum = sum(c for c in closeness_centrality.values())\n    expected_sum = 9.833333333333332\n    assert closeness_sum == pytest.approx(expected_sum)\n\n    nodes = {n for n in closeness_centrality}\n    expected_nodes = {'a', 'b', 'c', 'd', 'e'}\n    assert nodes == expected_nodes\n\n\ndef test_closeness_centrality_norm(path_from_ngram_file):\n    p = path_from_ngram_file\n    closeness_centrality = p.ClosenessCentrality(normalized=True)\n    closeness_max = max(c for c in closeness_centrality.values())\n    expected_max = 1\n    assert closeness_max == pytest.approx(expected_max)\n\n\ndef test_visitation_probabilities(path_from_ngram_file):\n    p = path_from_ngram_file\n    v_prob = p.VisitationProbabilities()\n    prob_sum = sum(p for p in v_prob.values())\n    assert prob_sum == pytest.approx(1)\n\n    max_prob = max(p for p in v_prob.values())\n    expected_max = 0.3125\n    assert max_prob == pytest.approx(expected_max)\n\n\n@slow\ndef test_entropy_growth_rate_ratio_mle(random_paths):\n    p = random_paths(100, 500)\n    mle_ratio = p.getEntropyGrowthRateRatio(method=\"MLE\")\n    mle_expected = 0.10515408343772015\n    assert mle_ratio == pytest.approx(mle_expected)\n\n@slow\ndef test_entropy_growth_rate_ratio_miller(random_paths):\n    p = random_paths(100, 500)\n    miller_ratio = p.getEntropyGrowthRateRatio(method=\"Miller\")\n    miller_expected = 0.6765478705937058\n    assert miller_ratio == pytest.approx(miller_expected)\n"
  }
]