Community Detection in Random Networks

TL;DR

This paper addresses the problem of detecting dense communities within random networks by formulating it as a hypothesis testing problem, deriving detection bounds, and proposing optimal tests for various scenarios.

Contribution

It introduces a formal framework for community detection in random graphs, deriving detection bounds and presenting tests that achieve these bounds, including cases with unknown parameters.

Findings

Derived detection lower bounds for community detection.

Proposed tests that achieve the detection bounds.

Analyzed detection in polynomial-time and related clique detection.

Abstract

We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an Erd\"os-R\'enyi graph with probability p0. Under the (composite) alternative, there is a subgraph of n nodes where the probability of connection is p1 > p0. We derive a detection lower bound for detecting such a subgraph in terms of N, n, p0, p1 and exhibit a test that achieves that lower bound. We do this both when p0 is known and unknown. We also consider the problem of testing in polynomial-time. As an aside, we consider the problem of detecting a clique, which is intimately related to the planted clique problem. Our focus in this paper is in the quasi-normal regime where n p0 is either bounded away from zero, or tends to zero slowly.