Overlapping stochastic block models with application to the French political blogosphere

Complex systems in nature and in society are often represented as networks, describing the rich set of interactions between objects of interest. Many deterministic and probabilistic clustering methods have been developed to analyze such structures. Given a network, almost all of them partition the vertices into disjoint clusters, according to their connection profile. However, recent studies have shown that these techniques were too restrictive and that most of the existing networks contained overlapping clusters. To tackle this issue, we present in this paper the Overlapping Stochastic Block Model. Our approach allows the vertices to belong to multiple clusters, and, to some extent, generalizes the well-known Stochastic Block Model [Nowicki and Snijders (2001)]. We show that the model is generically identifiable within classes of equivalence and we propose an approximate inference procedure, based on global and local variational techniques. Using toy data sets as well as the French Political Blogosphere network and the transcriptional network of Saccharomyces cerevisiae, we compare our work with other approaches.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS382 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

On sparsity, power-law and clustering properties of graphex processes

This paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph