Block modelling in dynamic networks with non-homogeneous Poisson processes and exact ICL

Original Article

Abstract

We develop a model in which interactions between nodes of a dynamic network are counted by non-homogeneous Poisson processes. In a block modelling perspective, nodes belong to hidden clusters (whose number is unknown) and the intensity functions of the counting processes only depend on the clusters of nodes. In order to make inference tractable, we move to discrete time by partitioning the entire time horizon in which interactions are observed in fixed-length time sub-intervals. First, we derive an exact integrated classification likelihood criterion and maximize it relying on a greedy search approach. This allows to estimate the memberships to clusters and the number of clusters simultaneously. Then, a maximum likelihood estimator is developed to estimate nonparametrically the integrated intensities. We discuss the over-fitting problems of the model and propose a regularized version solving these issues. Experiments on real and simulated data are carried out in order to assess the proposed methodology.

Keywords

Dynamic network Stochastic block model Exact ICL Non-homogeneous Poisson process 

References

  1. Biernacki C, Celeux G, Govaert G (2000) Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans Pattern Anal Mach Intell 22(7):719–725CrossRefGoogle Scholar
  2. Blondel VD, Guillaume J-L, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech: Theory Experi 2008(10):10008. http://stacks.iop.org/1742-5468/2008/i=10/a=P10008  CrossRefGoogle Scholar
  3. Côme E, Latouche P (2015) Model selection and clustering in stochastic block models based on the exact integrated complete data likelihood. Stat Model 15(6):564–589MathSciNetCrossRefGoogle Scholar
  4. Corneli M, Latouche P, Rossi F (2015) Modelling time evolving interactions in networks through a non stationary extension of stochastic block models. In: Pei J, Silvestri F, Tang J (eds) International conference on advances in social networks analysis and mining ASONAM 2015. IEEE/ACM, ACM, Paris, France, pp 1590–1591. https://hal.archives-ouvertes.fr/hal-01263540
  5. Dubois C, Butts C, Smyth P (2013) Stochastic blockmodelling of relational event dynamics. In: International conference on artificial intelligence and statistics. Volume 31 of the Journal of Machine Learning Research Proceedings, pp 238–246Google Scholar
  6. Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174MathSciNetCrossRefGoogle Scholar
  7. Goldenberg A, Zheng X, Fienberg SE, Airoldi EM (2009) A survey of statistical network models. Mach Learn 2(2):129–133CrossRefMATHGoogle Scholar
  8. Guigourès R, Boullé M, Rossi F (2015) Discovering patterns in time-varying graphs: a triclustering approach. Adv Data Anal Classif. doi:10.1007/s11634-015-0218-6 Google Scholar
  9. Guigourès R, Boullé M, Rossi F (2012) A triclustering approach for time evolving graphs. In: Co-clustering and applications, IEEE 12th international conference on data mining workshops (ICDMW 2012). Brussels, Belgium, pp 115–122Google Scholar
  10. Holland P, Laskey K, Leinhardt S (1983) Stochastic blockmodels: first steps. Soc Netw 5:109–137MathSciNetCrossRefGoogle Scholar
  11. Isella L, Stehl J, Barrat A, Cattuto C, Pinton J, Van den Broeck W (2011) What’s in a crowd? Analysis of face-to-face behavioral networks. J Theor Biol 271(1):166–180CrossRefGoogle Scholar
  12. Leemis LM (1991) Nonparametric estimation of the cumulative intensity function for a nonhomogeneous poisson process. Manag Sci 37(7):886–900. http://www.jstor.org/stable/2632541
  13. Lorrain F, White H (1971) Structural equivalence of individuals in social networks. J Math Sociol 1:49–80CrossRefGoogle Scholar
  14. Matias C, Rebafka T, Villers F (2015) Estimation and clustering in a semiparametric Poisson process stochastic block model for longitudinal networks, HAL (preprint)Google Scholar
  15. Noack A, Rotta R (2008) Multi-level algorithms for modularity clustering. CoRR arXiv:0812.4073
  16. Nouedoui L, Latouche P (2013) Bayesian non parametric inference of discrete valued networks. In: 21th European symposium on artificial neural networks, computational intelligence and machine learning (ESANN 2013). Bruges, Belgium, pp 291–296Google Scholar
  17. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes 3rd edition: the art of scientific computing, 3rd edn. Cambridge University Press, CambridgeMATHGoogle Scholar
  18. Rand WM (1971) Objective criteria for the evaluation of clustering methods. J Am Stat Assoc 66(336):846–850CrossRefGoogle Scholar
  19. Schaeffer SE (2007) Graph clustering. Comput Sci Rev 1(1):27–64MathSciNetCrossRefMATHGoogle Scholar
  20. Wang Y, Wong G (1987) Stochastic blockmodels for directed graphs. J Am Stat Assoc 82:8–19MathSciNetCrossRefMATHGoogle Scholar
  21. Wasserman S, Faust K (1994) Social network analysis: methods and applications, vol 506. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  22. White HC, Boorman S, Breiger R (1976) Social structure from multiple networks: I. Blockmodels of roles and positions. Am J Sociol 81(4):730–780CrossRefGoogle Scholar
  23. Wyse J, Friel N, Latouche P (2014) Inferring structure in bipartite networks using the latent block model and exact icl. arXiv preprint arXiv:1404.2911
  24. Xing EP, Fu W, Song L (2010) A state-space mixed membership blockmodel for dynamic network tomography. Ann Appl Stat 4(2):535–566MathSciNetCrossRefMATHGoogle Scholar
  25. Xu KS, Hero III AO (2013) Dynamic stochastic blockmodels: statistical models for time-evolving networks. In: Proceedings of the 6th International Conference on Social Computing, Behavioral-Cultural Modeling, and Prediction, pp 201–210Google Scholar
  26. Yang T, Chi Y, Zhu S, Gong Y, Jin R (2011) Detecting communities and their evolutions in dynamic social networks—a Bayesian approach. Mach Learn 82(2):157–189MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.Laboratoire SAMMUniversité Paris 1 Panthéon-SorbonneParis Cedex 13France

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