Estimation of the posterior distribution of a W-graph graphon function
Networks have been widely used in many scientific fiels, and in particular in social sciences, in order to represent interactions between objects of interest. Since the earlier work of Moreno in 1934, many random graph models have been proposed to extract knowledge from these structured data sets. For instance, the stochastic block model (SBM) allows the search of groups of vertices sharing homogeneous connection profiles. In this work, we consider the W -graph model which is known to generalize many random graph models but for which very few methods exist to perform inference on real data. First, we recall that the SBM model can be represented as a W-graph with a block-constant graphon function. Using a variational Bayes expectation maximization algorithm, we then approximate the posterior distribution over the model parameters of a SBM model and we show how this variational approximation can be integrated in order to estimate the posterior distribution of W-graph graphon function. We also derive the variational posterior frequency of any motif. The results presented here are tested on simulated data and on the French political blogosphere network.