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

Alain A. Celisse (1) , Jean-Jacques J.-J. Daudin (2) , Laurent L. Pierre

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 MIA-Paris - Mathématiques et Informatique Appliquées

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.

Mots clés: concentration inequalities; random graphs; stochastic block model; maxi- mum likelihood estimators; variational estimators; consistency

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Alain A. Celisse, Jean-Jacques J.-J. Daudin, Laurent L. Pierre. Consistency of maximum-likelihood and variational estimators in the stochastic block model. Electronic Journal of Statistics , 2012, 6, pp.1847-1899. ⟨10.1214/12-EJS729⟩. ⟨hal-01000059⟩