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The stochastic block model (SBM) is a mixture model used for the clusteringof nodes in networks. It has now been employed for more than a decade toanalyze very different types of networks in many scientific fields such asBiology and social sciences. Because of conditional dependency, there is noanalytical expression for the posterior distribution over the latent variables,given the data and model parameters. Therefore, approximation strategies, basedon variational techniques or sampling, have been proposed for clustering.Moreover, two SBM model selection criteria exist for the estimation of thenumber K of clusters in networks but, again, both of them rely on someapproximations. In this paper, we show how an analytical expression can bederived for the integrated complete data log likelihood. We then propose aninference algorithm to maximize this exact quantity. This strategy enables theclustering of nodes as well as the estimation of the number clusters to beperformed at the same time and no model selection criterion has to be computedfor various values of K. The algorithm we propose has a better computationalcost than existing inference techniques for SBM and can be employed to analyzelarge networks with ten thousand nodes. Using toy and true data sets, wecompare our work with other approaches.

Model selection and clustering in stochastic block models with the exact integrated complete data likelihood