Model selection in overlapping stochastic block models

Abstract

Networks are a commonly used mathematical model to describe the rich set of interactions between objects of interest. Many clustering methods have been developed in order to partition such structures, among which several rely on underlying probabilistic models, typically mixture models. The relevant hidden structure may however show overlapping groups in several applications. The Overlapping Stochastic Block Model [Latouche, Birmelé and Ambroise (2011)] has been developed to take this phenomenon into account. Nevertheless, the problem of the choice of the number of classes in the inference step is still open. To tackle this issue, we consider the proposed model in a Bayesian framework and develop a new criterion based on a non asymptotic approximation of the marginal log-likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm, and demonstrate its efficiency by running it on both simulated and real data.

Detecting Overlapping Communities in Networks Using Spectral Methods

“…Model-based methods focus on specifying how node community memberships determine edge probabilities. For example, the overlapping stochastic block model (OSBM) (Latouche et al, 2009) extends the SBM by allowing the entries of the membership matrix Z to be independent Bernoulli variables, thus allowing multiple "1"s in one row, or all "0"s. The mixed membership stochastic block model (Airoldi et al, 2008) draws membership vectors Z i· from a Dirichlet prior. The membership vector is drawn again to generate every edge, instead of being fixed for the node, so the community membership for node i varies depending on which node j it is interacting with.…”

Goodness of Fit of Logistic Regression Models for Random Graphs

“…Given a distribution q(·), we propose to maximize L K (q; ξ) with respect to each variable ξ ij in order to obtain the tightest bound L K (q; ξ) of log p(Y |M K ). This follows the work of Bishop and Svensén (2003) on Bayesian hierarchical mixture of experts and Latouche et al (2011Latouche et al ( , 2014 on the overlapping stochastic block model. As shown in the following proposition, this leads to new estimates ξ ij of ξ ij .…”

“…Thus, each node is associated to a political party from the left wing to the right wing and the status of the writer is also given (political analyst or not). This data set has been studied in a series of works (Zanghi et al, 2008;Latouche et al, 2011Latouche et al, , 2014 where all the authors pointed out the crucial role of the labels in the construction of the network. We considered a set of three covariates x ij = (x 1 ij , x 2 ij , x 3 ij ) ∈ R 3 artificially constructed to analyze the influence of both the political parties and the writer status.…”

Overlapping Stochastic Community Finding

“…The assumption in communityfinding, as opposed to other clustering approaches such as block-modelling [4], is that if a pair of nodes are linked to each other in the network then it is more likely they will both be members of the same community than if they are not linked. This paper introduces a new model, which we call Overlapping Stochastic Community Finding (OSCF) and which is related to the Overlapping Stochastic Block Model (OSBM) [5]. Non-statistical methods are usually described explicitly as a function which takes a network as input and computes a community labelling as a function of the network.…”

“…The OSCF model is similar to the models used in MOSES [3] and in the OSBM [5]. In these models, each node may be be in any number of clusters.…”