Electronic Journal of Statistics

Consistency of maximum-likelihood and variational estimators in the stochastic block model

Alain Celisse, Jean-Jacques Daudin, and Laurent Pierre

Full-text: Open access

Abstract

The stochastic block model (SBM) is a probabilistic model designed to describe heterogeneous directed and undirected graphs. In this paper, we address the asymptotic inference in SBM by use of maximum-likelihood and variational approaches. The identifiability of SBM is proved while asymptotic properties of maximum-likelihood and variational estimators are derived. In particular, the consistency of these estimators is settled for the probability of an edge between two vertices (and for the group proportions at the price of an additional assumption), which is to the best of our knowledge the first result of this type for variational estimators in random graphs.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 1847-1899.

Dates
First available in Project Euclid: 4 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1349355605

Digital Object Identifier
doi:10.1214/12-EJS729

Mathematical Reviews number (MathSciNet)
MR2988467

Zentralblatt MATH identifier
1295.62028

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 62E17: Approximations to distributions (nonasymptotic) 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Keywords
Random graphs stochastic block model maximum likelihood estimators variational estimators consistency concentration inequalities

Citation

Celisse, Alain; Daudin, Jean-Jacques; Pierre, Laurent. Consistency of maximum-likelihood and variational estimators in the stochastic block model. Electron. J. Statist. 6 (2012), 1847--1899. doi:10.1214/12-EJS729. https://projecteuclid.org/euclid.ejs/1349355605


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References

  • Allman, E., Matias, C., and Rhodes, J. (2009). Identifiability of parameters in latent structure models with many observed variables., Annals of Statistics, 37, 3099–3132.
  • Allman, E., Matias, C., and Rhodes, J. A. (2011). Parameter identifiability in a class of random graph mixture models., Journal of Statistical Planning and Inference, 141(5), 1719–1736.
  • Ambroise, C. and Matias, C. (2012). New consistent and asymptotically normal estimators for random graph mixture models., JRSS. B, 74(1), 3–35.
  • Andrieu, C. and Atchadé, Y. F. (2007). On the efficiency of adaptive mcmc algorithms., Elec. Comm. in Probab., 12, 336–349.
  • Bickel, P. and Chen, A. (2009). A nonparametric view of network models and Newman-Girvan and other modularities., PNAS, pages 1–6.
  • Bickel, P. J., Chen, A., and Levina, E. (2011). The method of moments and degree distributions for network models., Ann. Statist., 39(5), 2280–2301.
  • Boer, P., Huisman, M., Snijders, T., Steglich, C., Wichers, L., and Zeggelink, E. (2006). Stocnet: an open software system for the advanced statistical analysis of social networks, version 1.7. groningen:, Ics/scienceplus.
  • Choi, D. S., Wolfe, P. J., and Airoldi, E. M. (2012). Stochastic blockmodels with growing number of classes., Biometrika, In press (arXiv:1011.4644v2).
  • Daudin, J.-J., Picard, F., and Robin, S. (2008). A mixture model for random graphs., Stat Comput, 18, 173–183.
  • Gazal, S., Daudin, J.-J., and Robin, S. (2011). Accuracy of variational estimates for random graph mixture models., J. Comput. Simul.
  • Holland, P., Laskey, K., and Leinhardt, S. (1983). Stochastic blockmodels: Some first steps., Social Networks, 5, 109–137.
  • Jordan, M. I., Ghahramni, Z., Jaakkola, T. S., and Saul, L. K. (1999). An introduction to variational methods for graphical models., Machine Learning, 37, 183–233.
  • Mariadassou, M., Robin, S., and Vacher, C. (2010). Uncovering latent structure in valued graphs: A variational approach., Ann. Appl. Stat., 4, 715–742.
  • Massart, P. (2007)., Concentration Inequalities and Model Selection, volume 1896 of Lecture Notes in Mathematics. Springer, Berlin. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003, With a foreword by Jean Picard.
  • Mixnet (2009)., http://stat.genopole.cnrs.fr/logiciels/mixnet.
  • Nowicki, K. and Snijders, T. (2001). Estimation and prediction for stochastic block-structures., J. Am. Stat. Assoc., 96, 1077–1087.
  • Picard, F., Miele, V., Daudin, J.-J., Cottret, L., and Robin, S. (2009). Deciphering the connectivity structure of biological networks using mixnet., BMC Bioinformatics, 10.
  • Rohe, K., Chatterjee, S., and Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel., Ann. Statist., 39(4), 1878–1915.
  • Snijders, T. A. B. and Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure., Journal of Classification, 14, 75–100.
  • van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes With Applications to Statistics. Springer Series in Statistics.