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Modelling Temporal Networks with Markov Chains, Community Structures and Change Points

  • Tiago P. Peixoto
  • Martin RosvallEmail author
Chapter
Part of the Computational Social Sciences book series (CSS)

Abstract

While temporal networks contain crucial information about the evolving systems they represent, only recently have researchers showed how to incorporate higher-order Markov chains, community structure and abrupt transitions to describe them. However, each approach can only capture one aspect of temporal networks, which often are multifaceted with dynamics taking place concurrently at small and large structural scales and also at short and long timescales. Therefore, these approaches must be combined for more realistic descriptions of empirical systems. Here we present two data-driven approaches developed to capture multiple aspects of temporal network dynamics. Both approaches capture short timescales and small structural scales with Markov chains. Whereas one approach also captures large structural scales with communities, the other instead captures long timescales with change points. Using a nonparametric Bayesian inference framework, we illustrate how the multi-aspect approaches better describe evolving systems by combining different scales, because the most plausible models combine short timescales and small structural scales with large-scale structural and dynamical modular patterns or many change points.

Keywords

Temporal networks Higher-order Markov chains Community structure Change points Bayesian inference 

Notes

Acknowledgement

M.R. was supported by the Swedish Research Council grant 2016-00796.

References

  1. 1.
    Ho, Q., Song, L., Xing, E.P.: Evolving cluster mixed-membership blockmodel for time-varying networks. In: Proceedings of the International Conference on Artificial Intelligence and Statistics, vol. 15, pp. 342–350 (2011)Google Scholar
  2. 2.
    Perra, N., Gonçalves, B., Pastor-Satorras, R., Vespignani, A.: Activity driven modeling of time varying networks. Sci. Rep. 2, 469 (2012)ADSCrossRefGoogle Scholar
  3. 3.
    Rocha, L.E. C., Liljeros, F., Holme, P.: Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual Contacts. PLoS Comput. Biol. 7, e1001109 (2011)ADSCrossRefGoogle Scholar
  4. 4.
    Valdano, E., Ferreri, L., Poletto, C., Colizza, V.: Analytical computation of the epidemic threshold on temporal networks. Phys. Rev. X 5, 021005 (2015)Google Scholar
  5. 5.
    Génois, M., Vestergaard, C.L., Cattuto, C., Barrat, A.: Compensating for population sampling in simulations of epidemic spread on temporal contact networks. Nat. Commun. 6, 8860 (2015)ADSCrossRefGoogle Scholar
  6. 6.
    Ren, G., Wang, X.: Epidemic spreading in time-varying community networks. Chaos: Interdiscip. J. Nonlinear Sci. 24, 023116 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Scholtes, I. et al.: Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks. Nat. Commun. 5, 5024 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    Peixoto, T.P., Rosvall, M.: Modelling sequences and temporal networks with dynamic community structures. Nat. Commun. 8, 582 (2017)ADSCrossRefGoogle Scholar
  9. 9.
    Xu, K.S., Iii, A.O.H.: Dynamic stochastic blockmodels: statistical models for time-evolving networks. In: Greenberg, A.M., Kennedy, W.G., Bos, N.D. (eds.) Social Computing, Behavioral-Cultural Modeling and Prediction. Lecture Notes in Computer Science, vol. 7812, pp. 201–210. Springer, Berlin (2013)Google Scholar
  10. 10.
    Gauvin, L., Panisson, A., Cattuto, C.: Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach. PLoS One 9, e86028 (2014)ADSCrossRefGoogle Scholar
  11. 11.
    Peixoto, T.P.: Inferring the mesoscale structure of layered, edge-valued, and time-varying networks. Phys. Rev. E 92, 042807 (2015)ADSCrossRefGoogle Scholar
  12. 12.
    Stanley, N., Shai, S., Taylor, D., Mucha, P.J. Clustering network layers with the strata multilayer stochastic block model. IEEE Trans. Netw. Sci. Eng. 3, 95–105 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghasemian, A., Zhang, P., Clauset, A., Moore, C., Peel, L.: Detectability thresholds and optimal algorithms for community structure in dynamic networks. Phys. Rev. X 6, 031005 (2016)Google Scholar
  14. 14.
    Zhang, X., Moore, C., Newman, M.E.J.: Random graph models for dynamic networks. Eur. Phys. J. B 90, 200 (2017)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Peel, L., Clauset, A.: Detecting change points in the large-scale structure of evolving networks. In: Twenty-Ninth AAAI Conference on Artificial Intelligence (2015)Google Scholar
  16. 16.
    De Ridder, S., Vandermarliere, B., Ryckebusch, J.: Detection and localization of change points in temporal networks with the aid of stochastic block models. J. Stat. Mech: Theory Exp. 2016, 113302 (2016)Google Scholar
  17. 17.
    Corneli, M., Latouche, P., Rossi, F.: Multiple change points detection and clustering in dynamic networks. Stat. Comput. 28, 989 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gauvin, L., Panisson, A., Cattuto, C., Barrat, A.: Activity clocks: spreading dynamics on temporal networks of human contact. Sci. Rep. 3, 3099 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    Vestergaard, C.L., Génois, M., Barrat, A.: How memory generates heterogeneous dynamics in temporal networks. Phys. Rev. E 90, 042805 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Strelioff, C.C., Crutchfield, J.P., Hübler, A.W.: Inferring Markov chains: Bayesian estimation, model comparison, entropy rate, and out-of-class modeling. Phys. Rev. E 76, 011106 (2007)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Fournet, J., Barrat, A.: Contact patterns among high school students. PLoS One 9, e107878 (2014)ADSCrossRefGoogle Scholar
  22. 22.
    Peixoto, T.P., Gauvin, L.: Change points, memory and epidemic spreading in temporal networks. Sci. Rep. 8, 15511 (2018)ADSCrossRefGoogle Scholar
  23. 23.
    Jaynes, E.T.: Probability Theory: The Logic of Science Cambridge University Press, Cambridge (2003)Google Scholar
  24. 24.
    Mastrandrea, R., Fournet, J., Barrat, A.: Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. PLoS One 10, e0136497 (2015)CrossRefGoogle Scholar
  25. 25.
    Polansky, A.M.: Detecting change-points in Markov chains. Comput. Stat. Data Anal. 51, 6013–6026 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Arnesen, P., Holsclaw, T., Smyth, P.: Bayesian detection of changepoints in finite-state Markov chains for multiple sequences. Technometrics 58, 205–213 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087 (1953)ADSCrossRefGoogle Scholar
  28. 28.
    Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Peixoto, T.P.: Nonparametric Bayesian inference of the microcanonical stochastic block model. Phys. Rev. E 95, 012317 (2017)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.ISI FoundationTorinoItaly
  3. 3.Integrated Science Lab, Department of PhysicsUmeå UniversityUmeåSweden

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