Estimation in a binomial stochastic blockmodel for a weighted graph by a variational expectation maximization algorithm

Abstract

Stochastic blockmodels have been widely proposed as a probabilistic random graph model for the analysis of networks data as well as for detecting community structure in these networks. In a number of real-world networks, not all ties among nodes have the same weight. Ties among networks nodes are often associated with weights that differentiate them in terms of their strength, intensity, or capacity. In this paper, we are interested in the case of co-citation networks, where the nodes are words and each edge joining a pair of words is weighted by the number of co-citation of these two words together in the same document. In this type of networks, the weight associated to each edge is an integer value bounded by the the whole number of documents in the considered corpus. Hence, we propose an extension of the stochastic blockmodels to deal with the case of a binomial distribution for the edge’s weights. We provide an inference method through a variational expectation maximization algorithm to estimate the parameters in binomial stochastic blockmodels for weighted networks. To prove the validity of the method and to highlight its main features, we set some applications of the proposed approach by using some simulated data and then some real data sets. Stochastic blockmodels belong to latent classes models. Classes defines a node’s clustering. We compare the clustering found through binomial stochastic blockmodels with the ones found fitting a stochastic blockmodel with Poisson distributed edges’ weights. Inferred Poisson and binomial stochastic blockmodels mainly differs. Moreover, in our examples, the statistical error is lower for binomial stochastic blockmodels.

Keywords:

• Binomial stochastic blockmodel
• Clustering
• Poisson stochastic blockmodel
• Text mining
• Variational inference
• Weighted networks

Acknowledgements

The authors thank the Editor and the referees for their very helpful comments, which led to an improved associated version of the paper.