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