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Modeling relations between individuals is a classical question in socialsciences and clustering individuals according to the observed patterns ofinteractions allows to uncover a latent structure in the data. Stochastic blockmodel (SBM) is a popular approach for grouping the individuals with respect totheir social comportment. When several relationships of various types can occurjointly between the individuals, the data are represented by multiplex networkswhere more than one edge can exist between the nodes. In this paper, we extendthe SBM to multiplex networks in order to obtain a clustering based on morethan one kind of relationship. We propose to estimate the parameters --such asthe marginal probabilities of assignment to groups (blocks) and the matrix ofprobabilities of connections between groups-- through a variationalExpectation-Maximization procedure. Consistency of the estimates as well asstatistical properties of the model are obtained. The number of groups ischosen thanks to the Integrated Completed Likelihood criteria, a penalizedlikelihood criterion. Multiplex Stochastic Block Model arises in manysituations but our applied example is motivated by a network of French cancerresearchers. The two possible links (edges) between researchers are a directconnection or a connection through their labs. Our results show stronginteractions between these two kinds of connections and the groups that areobtained are discussed to emphasize the common features of researchers groupedtogether.

Stochastic Block Models for Multiplex networks: an application to networks of researchers