Community Detection in Random Networks

Ery Arias-Castro; Nicolas Verzelen

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arXiv · 1302.7099

Community Detection in Random Networks

Ery Arias-Castro·Nicolas Verzelen

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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.

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Ery Arias-Castro, Nicolas Verzelen. 2013-02-28. Community Detection in Random Networks. https://arxiv.org/abs/1302.7099

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