Community detection in sparse random networks

Abstract

We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdős–Rényi graph on $N$ vertices and with connection probability $p_{0}$; under the alternative, there is an unknown subgraph on $n$ vertices where the connection probability is $p_{1}>p_{0}$. In Arias-Castro and Verzelen [Ann. Statist. 42 (2014) 940–969], we focused on the asymptotically dense regime where $p_{0}$ is large enough that $np_{0}>(n/N)^{o(1)}$. We consider here the asymptotically sparse regime where $p_{0}$ is small enough that $np_{0}<(n/N)^{c_{0}}$ for some $c_{0}>0$. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work [Ann. Statist. 42 (2014) 940–969], the arguments for the lower bounds are based on the same technology, but are substantially more technical in the details; also, the methods we study are different: besides a variant of the scan statistic, we study other tests statistics such as the size of the largest connected component, the number of triangles, and the number of subtrees of a given size. Our detection bounds are sharp, except in the Poisson regime where we were not able to fully characterize the constant arising in the bound.