Detecting a planted community in an inhomogeneous random graph

TL;DR

This paper introduces a scan test method for detecting small planted communities in inhomogeneous random graphs, demonstrating near-optimal detection capabilities through theoretical bounds and numerical experiments.

Contribution

The paper develops a novel scan test for identifying small communities in inhomogeneous graphs and establishes its near-optimality via information theoretic bounds.

Findings

Scan test effectively detects small communities

Theoretical bounds show near-optimal performance

Numerical experiments validate the method

Abstract

We study the problem of detecting whether an inhomogeneous random graph contains a planted community. Specifically, we observe a single realization of a graph. Under the null hypothesis, this graph is a sample from an inhomogeneous random graph, whereas under the alternative, there exists a small subgraph where the edge probabilities are increased by a multiplicative scaling factor. We present a scan test that is able to detect the presence of such a planted community, even when this community is very small and the underlying graph is inhomogeneous. We also derive an information theoretic lower bound for this problem which shows that in some regimes the scan test is almost asymptotically optimal. We illustrate our results through examples and numerical experiments.