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Topological features determining the error in the inference of networks using transfer entropy

1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
2 Department of Mechanical and Aerospace Engineering & Department of Biomedical Engineering, New York University, Tandon School of Engineering, Brooklyn, New York 11201, USA

The problem of inferring interactions from observations of individual behavior in networked dynamical systems is ubiquitous in science and engineering. From brain circuits to financial networks, there is a dire need for robust methodologies that can unveil network structures from individual time series. Originally formulated to identify asymmetries in pairs of coupled dynamical systems, transfer entropy has been proposed as a model-free, computationally-inexpensive framework for network inference. While previous studies have cataloged a library of pathological instances in which transfer entropy-based network reconstruction can fail, we presently lack analytical results that can help quantify the accuracy of the identification and pinpoint scenarios where false inferences results are more likely to be registered. Here, we present a detailed analytical study of a Boolean network model of policy diffusion. Through perturbation theory, we establish a closed-form expression for the transfer entropy between any pair of nodes in the network up to the third order in an expansion parameter that is associated with the spontaneous activity of the nodes. While for slowly-varying dynamics transfer entropy is successful in capturing the weight of any link, for faster dynamics the error in the inference is controlled by local topological features of the node pair. Specifically, the error in the inference of a weight between two nodes depends on the mismatch between their weighted indegrees that serves as a common uncertainty bath upon which we must tackle the inference problem. Interestingly, an equivalent result is discovered when numerically studying a network of coupled chaotic tent maps, suggesting that heterogeneity in the in-degree is a critical factor that can undermine the success of transfer entropy-based network inference.
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Keywords boolean network; data-driven; discrete systems; information theory; Markov chain; perturbation theory; policy diffusion

Citation: Roy H. Goodman, Maurizio Porfiri. Topological features determining the error in the inference of networks using transfer entropy. Mathematics in Engineering, 2020, 2(1): 34-54. doi: 10.3934/mine.2020003


  • 1. Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74: 47-97.
  • 2. Anderson RP, Jimenez G, Bae JY, et al. (2016) Understanding policy diffusion in the us: An information-theoretical approach to unveil connectivity structures in slowly evolving complex systems. SIAM J Appl Dyn Syst 15: 1384-1409.
  • 3. Bernstein DS (2018) Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas-Revised and Expanded Edition, Princeton University Press.
  • 4. Boccaletti S, Latora V, Moreno Y, et al. (2006) Complex networks: Structure and dynamics. Phys Rep 424: 175-308.
  • 5. Bollt EM, Sun J, Runge J (2018) Introduction to focus issue: Causation inference and information flow in dynamical systems: Theory and applications. Chaos 28: 075201.
  • 6. Bossomaier T, Barnett L, Harré M, et al. (2016) An Introduction to Transfer Entropy: Information Flow in Complex Systems, Springer International Publishing.
  • 7. Brémaud P (2013) Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer Science & Business Media.
  • 8. Bullmore E, Sporns O (2009) Complex brain networks: Graph theoretical analysis of structural and functional systems. Nat Rev Neurosci 10: 186.
  • 9. Cover T, Thomas J (2012) Elements of Information Theory, Wiley.
  • 10. Dunne JA, Williams RJ, Martinez ND (2002) Food-web structure and network theory: The role of connectance and size. Proc Natl Acad Sci USA 99: 12917-12922.
  • 11. Erdös P, Rényi A (1959) On random graphs, I. Publ Math Debrecen 6: 290-297.
  • 12. Godsil C, Royle GF (2013) Algebraic Graph Theory, Springer Science & Business Media.
  • 13. Grabow C, Macinko J, Silver D, et al. (2016) Detecting causality in policy diffusion processes. Chaos 26: 083113.
  • 14. Keeling MJ, Eames KT (2005) Networks and epidemic models. J R Soc Interface 2: 295-307.
  • 15. Khinchin A (1957) Mathematical Foundations of Information Theory, Dover Publications.
  • 16. Nayfeh AH (2011) Introduction to Perturbation Techniques, John Wiley & Sons.
  • 17. Paninski L (2003) Estimation of entropy and mutual information. Neural Comput 15: 1191-1253.
  • 18. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems, Phys Rev Lett 64: 821.
  • 19. Porfiri M, Ruiz Marín M (2018) Inference of time-varying networks through transfer entropy, the case of a Boolean network model. Chaos 28: 103123.
  • 20. Porfiri M, Ruiz Marín M (2018) Information flow in a model of policy diffusion: An analytical study. IEEE Trans Network Sci Eng 5: 42-54.
  • 21. Prettejohn BJ, Berryman MJ, McDonnell MD, et al. (2011) Methods for generating complex networks with selected structural properties for simulations: A review and tutorial for neuroscientists. Front Comput Neurosci 5: 11.
  • 22. Ramos AM, Builes-Jaramillo A, Poveda G, et al. (2017) Recurrence measure of conditional dependence and applications. Phys Rev E 95: 052206.
  • 23. Roxin A, Hakim V, Brunel N (2008) The statistics of repeating patterns of cortical activity can be reproduced by a model network of stochastic binary neurons. J Neurosci 28: 10734-10745.
  • 24. Runge J (2018) Causal network reconstruction from time series: From theoretical assumptions to practical estimation. Chaos 28: 075310.
  • 25. Schreiber T (2000) Measuring information transfer. Phys Rev Lett 85: 461-464.
  • 26. Squartini T, Caldarelli G, Cimini G, et al. (2018) Reconstruction methods for networks: The case of economic and financial systems. Phys Rep 757: 1-47.
  • 27. Staniek M, Lehnertz K (2008) Symbolic transfer entropy. Phys Rev Lett 100: 158101.
  • 28. Sun J, Taylor D, Bollt EM (2015) Causal network inference by optimal causation entropy. SIAM J Appl Dyn Syst 14: 73-106.
  • 29. Wibral M, Pampu N, Priesemann V, et al. (2013) Measuring information-transfer delays. PLoS One 8: e55809.
  • 30. Wibral M, Vicente R, Lizier JT (2014) Directed Information Measures in Neuroscience, Springer.
  • 31. Yule GU (1925) II.-A mathematical theory of evolution, based on the conclusions of Dr. JC Willis, FRS. Philos Trans R Soc London Ser B 213: 21-87.


This article has been cited by

  • 1. Leonardo Novelli, Fatihcan M. Atay, Jürgen Jost, Joseph T. Lizier, Deriving pairwise transfer entropy from network structure and motifs, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020, 476, 2236, 20190779, 10.1098/rspa.2019.0779

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