Abstract: The stochastic block model (SBM) is a probabilistic model de-signed 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 estima-tors are derived. In particular, the consistency of these estimators is settled for the probability of an edge between two vertices (and for the group pro-portions 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

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