Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications

Mixture models are probabilistic models aimed at uncovering and representing latent subgroups within a population. In the realm of network data analysis, the latent subgroups of nodes are typically identified by their connectivity behaviour, with nodes behaving similarly belonging to the same community. In this context, mixture modelling is pursued through stochastic blockmodelling. We consider stochastic blockmodels and some of their variants and extensions from a mixture mo...

Community detection is a fundamental problem in network analysis which is made more challenging by overlaps between communities which often occur in practice. Here we propose a general, flexible, and interpretable generative model for overlapping communities, which can be thought of as a generalization of the degree-corrected stochastic block model. We develop an efficient spectral algorithm for estimating the community memberships, which deals with the overlaps by employing ...

We study the problem of identifying macroscopic structures in networks, characterizing the impact of introducing link directions on the detectability phase transition. To this end, building on the stochastic block model, we construct a class of hardly detectable directed networks. We find closed form solutions by using belief propagation method showing how the transition line depends on the assortativity and the asymmetry of the network. Finally, we numerically identify the e...

This paper provides a selective review of the statistical network analysis literature focused on clustering and inference problems for stochastic blockmodels and their variants. We survey asymptotic normality results for stochastic blockmodels as a means of thematically linking classical statistical concepts to contemporary research in network data analysis. Of note, multiple different forms of asymptotically Gaussian behavior arise in stochastic blockmodels and are useful fo...

The framework of statistical inference has been successfully used to detect the meso-scale structures in complex networks, such as community structure, core-periphery (CP) structure. The main principle is that the stochastic block model (SBM) is used to fit the observed network and the learnt parameters indicate the group assignment, in which the parameters of model are often calculated via an expectation-maximization (EM) algorithm and a belief propagation (BP) algorithm is ...

The stochastic block model is able to generate different network partitions, ranging from traditional assortative communities to disassortative structures. Since the degree-corrected stochastic block model does not specify which mixing pattern is desired, the inference algorithms, which discover the most likely partition of the networks nodes, are likely to get trapped in the local optima of the log-likelihood. Here we introduce a new model constraining nodes' internal degree...

Many inference problems, notably the stochastic block model (SBM) that generates a random graph with a hidden community structure, undergo phase transitions as a function of the signal-to-noise ratio, and can exhibit hard phases in which optimal inference is information-theoretically possible but computationally challenging. In this paper we refine this description by emphasizing the existence of more generic phase diagrams with a hybrid-hard phase in which it is computationa...

We survey some of the concepts, methods, and applications of community detection, which has become an increasingly important area of network science. To help ease newcomers into the field, we provide a guide to available methodology and open problems, and discuss why scientists from diverse backgrounds are interested in these problems. As a running theme, we emphasize the connections of community detection to problems in statistical physics and computational optimization.

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdos-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancement in community d...

Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts, existing approaches to bipartite community detection have drawbacks, including implicit parameter choices, loss of information through one-mode projections, and lack of interpretability. Here we solve the community detection problem for bipa...