  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.
