Recovering communities in weighted stochastic block models

Abstract

We derive sharp thresholds for exact recovery of communities in a weighted stochastic block model, where observations are collected in the form of a weighted adjacency matrix, and the weight of each edge is generated independently from a distribution determined by the community membership of its endpoints. Our main result, characterizing the precise boundary between success and failure of maximum likelihood estimation when edge weights are drawn from discrete distributions, involves the Renyi divergence of order 1/2 between the distributions of within-community and between-community edges. When the Renyi divergence is above a certain threshold, meaning the edge distributions are sufficiently separated, maximum likelihood succeeds with probability tending to 1; when the Renyi divergence is below the threshold, maximum likelihood fails with probability bounded away from 0. In the language of graphical channels, the Renyi divergence pinpoints the information-theoretic capacity of discrete graphical channels with binary inputs. Our results generalize previously established thresholds derived specifically for unweighted block models, and support an important natural intuition relating the intrinsic hardness of community estimation to the problem of edge classification. Along the way, we establish a general relationship between the Renyi divergence and the probability of success of the maximum likelihood estimator for arbitrary edge weight distributions. Finally, we discuss consequences of our bounds for the related problems of censored block models and submatrix localization, which may be seen as special cases of the framework developed in our paper.