The Information Dynamics Toolkit xl (IDTxl) is a comprehensive software package for efficient inference of networks and their node dynamics from multivariate time series data using information theory. IDTxl provides functionality to estimate the following measures:

1. For network inference:
   • multivariate transfer entropy (TE)/Granger causality (GC)
   • multivariate mutual information (MI)
   • bivariate TE/GC
   • bivariate MI
2. For analysis of node dynamics:
   • active information storage (AIS)
   • partial information decomposition (PID)

IDTxl implements estimators for discrete and continuous data with parallel computing engines for both GPU and CPU platforms. Written for Python3.4.3 .

To get started have a look at the wiki and the documentation. For further discussions, join IDTxl's google group.

P. Wollstadt, J. T. Lizier, R. Vicente, C. Finn, M. Martinez-Zarzuela, P. Mediano, L. Novelli, M. Wibral (2018). IDTxl: The Information Dynamics Toolkit xl: a Python package for the efficient analysis of multivariate information dynamics in networks. Journal of Open Source Software, 4(34), 1081. https://doi.org/10.21105/joss.01081.

Wollstadt, Lizier, Vicente, Finn, Martinez Zarzeula, Lindner, Martinez Mediano, Novelli, Wibral, 2017. "IDTxl - The Information Dynamics Toolkit xl: a Python package for the efficient analysis of multivariate information dynamics in networks", GitHub Repository: https://github.com/pwollstadt/IDTxl.

• Patricia Wollstadt, Brain Imaging Center, MEG Unit, Goethe-University, Frankfurt, Germany; Honda Research Institute Europe GmbH, Offenbach am Main, Germany
• Michael Wibral, Campus Institute for Dynamics of Biological Networks, Georg August University, Göttingen, Germany
• David Alexander Ehrlich, Campus Institute for Dynamics of Biological Networks, Georg August University, Göttingen, Germany; Max Planck Institute for Dynamics and Self-Organization, Goettingen, Germany
• Joseph T. Lizier, Centre for Complex Systems, The University of Sydney, Sydney, Australia
• Raul Vicente, Computational Neuroscience Lab, Institute of Computer Science, University of Tartu, Tartu, Estonia
• Abdullah Makkeh, Campus Institute for Dynamics of Biological Networks, Georg August University, Göttingen, Germany
• Conor Finn, Centre for Complex Systems, The University of Sydney, Sydney, Australia
• Mario Martinez-Zarzuela, Department of Signal Theory and Communications and Telematics Engineering, University of Valladolid, Valladolid, Spain
• Leonardo Novelli, Centre for Complex Systems, The University of Sydney, Sydney, Australia
• Pedro Mediano, Computational Neurodynamics Group, Imperial College London, London, United Kingdom
• Michael Lindner, Campus Institute for Dynamics of Biological Networks, Georg August University, Göttingen, Germany
• Aaron J. Gutknecht, Campus Institute for Dynamics of Biological Networks, Georg August University, Göttingen, Germany

How to contribute? We are happy about any feedback on IDTxl. If you would like to contribute, please open an issue or send a pull request with your feature or improvement. Also have a look at the developer's section in the Wiki for details.

This project has been supported by funding through:

• Universities Australia - Deutscher Akademischer Austauschdienst (German Academic Exchange Service) UA-DAAD Australia-Germany Joint Research Co-operation grant "Measuring neural information synthesis and its impairment", Wibral, Lizier, Priesemann, Wollstadt, Finn, 2016-17
• Australian Research Council Discovery Early Career Researcher Award (DECRA) "Relating function of complex networks to structure using information theory", Lizier, 2016-19
• Deutsche Forschungsgemeinschaft (DFG) Grant CRC 1193 C04, Wibral

• Multivariate transfer entropy: Lizier & Rubinov, 2012, Preprint, Technical Report 25/2012, Max Planck Institute for Mathematics in the Sciences. Available from: http://www.mis.mpg.de/preprints/2012/preprint2012_25.pdf
• Hierarchical statistical testing for multivariate transfer entropy estimation: Novelli et al., 2019, Network Neurosci 3(3)
• Kraskov estimator: Kraskov et al., 2004, Phys Rev E 69, 066138
• Nonuniform embedding: Faes et al., 2011, Phys Rev E 83, 051112
• Faes' compensated transfer entropy: Faes et al., 2013, Entropy 15, 198-219
• PID:
  • Williams & Beer, 2010, arXiv:1004.2515 [cs.IT]
  • Makkeh et al., 2021, Phys Rev E 103, 032149
  • Gutknecht et al., 2021, Proc. R. Soc. A: Math. Phys. Eng, 477(2251), 20210110.
• PID estimators:
  • Bertschinger et al., 2014, Entropy, 16(4)
  • Makkeh et al., 2017, Entropy, 19(10)
  • Makkeh et al., 2018, Entropy, 20(271)
  • Makkeh et al., 2018, Phys. Rev. E 103, 032149
• History-dependence estimator for neural spiking data: Rudelt et al., 2021, PLOS Computational Biology, 17(6)
• Significant subgraph mining: Gutknecht et al., 2021, bioRxiv