Advertisement

Minimum Entropy Stochastic Block Models Neglect Edge Distribution Heterogeneity

  • Louis DuvivierEmail author
  • Céline Robardet
  • Rémy Cazabet
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 881)

Abstract

The statistical inference of stochastic block models as emerged as a mathematicaly principled method for identifying communities inside networks. Its objective is to find the node partition and the block-to-block adjacency matrix of maximum likelihood i.e. the one which has most probably generated the observed network. In practice, in the so-called microcanonical ensemble, it is frequently assumed that when comparing two models which have the same number and sizes of communities, the best one is the one of minimum entropy i.e. the one which can generate the less different networks. In this paper, we show that there are situations in which the minimum entropy model does not identify the most significant communities in terms of edge distribution, even though it generates the observed graph with a higher probability.

Keywords

Network Community detection Stochastic block model Statistical inference Entropy 

Notes

Acknowledgments

This work was supported by the ACADEMICS grant of the IDEXLYON, project of the Université de Lyon, PIA operated by ANR-16-IDEX-0005, and of the project ANR-18-CE23-0004 (BITUNAM) of the French National Research Agency (ANR).

References

  1. 1.
    Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Nat. Acad. Sci. 99(12), 7821–7826 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fortunato, S., Hric, D.: Community detection in networks: a user guide. Phys. Rep. 659, 1–44 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)CrossRefGoogle Scholar
  4. 4.
    Hastings, M.B.: Community detection as an inference problem. Phys. Rev. E 74(3), 035102 (2006)CrossRefGoogle Scholar
  5. 5.
    Guimera, R., Sales-Pardo, M., Amaral, L.A.N.: Modularity from fluctuations in random graphs and complex networks. Phys. Rev. E 70(2), 025101 (2004)CrossRefGoogle Scholar
  6. 6.
    Peixoto, T.P.: Nonparametric bayesian inference of the microcanonical stochastic block model. Phys. Rev. E 95(1), 012317 (2017)CrossRefGoogle Scholar
  7. 7.
    Cimini, G., Squartini, T., Saracco, F., Garlaschelli, D., Gabrielli, A., Caldarelli, G.: The statistical physics of real-world networks. Nat. Rev. Phys. 1(1), 58 (2019)CrossRefGoogle Scholar
  8. 8.
    Peixoto, T.P.: Entropy of stochastic blockmodel ensembles. Phys. Rev. E 85(5), 056122 (2012)CrossRefGoogle Scholar
  9. 9.
    Peixoto, T.P.: Bayesian stochastic blockmodeling. arXiv preprint. http://arxiv.org/abs/1705.10225 (2017)
  10. 10.
    Fortunato, S., Barthelemy, M.: Resolution limit in community detection. Proc. Nat. Acad. Sci. 104(1), 36–41 (2007)CrossRefGoogle Scholar
  11. 11.
    Peixoto, T.P.: Parsimonious module inference in large networks. Phys. Rev. Lett. 110(14), 148701 (2013)CrossRefGoogle Scholar
  12. 12.
    Decelle, A., Krzakala, F., Moore, C., Zdeborová, L.: Inference and phase transitions in the detection of modules in sparse networks. Phys. Rev. Lett. 107(6), 065701 (2011)CrossRefGoogle Scholar
  13. 13.
    Decelle, A., Krzakala, F., Moore, C., Zdeborová, L.: Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84(6), 066106 (2011)CrossRefGoogle Scholar
  14. 14.
    Dandan, H., Ronhovde, P., Nussinov, Z.: Phase transitions in random Potts systems and the community detection problem: spin-glass type and dynamic perspectives. Philos. Mag. 92(4), 406–445 (2012)CrossRefGoogle Scholar
  15. 15.
    Abbe, E., Sandon, C.: Community detection in general stochastic block models: fundamental limits and efficient algorithms for recovery. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pp. 670–688. IEEE (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Louis Duvivier
    • 1
    Email author
  • Céline Robardet
    • 1
  • Rémy Cazabet
    • 2
  1. 1.Univ Lyon, INSA Lyon, CNRS, LIRIS UMR5205LyonFrance
  2. 2.Univ Lyon, Université Lyon 1, CNRS, LIRIS UMR5205LyonFrance

Personalised recommendations