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June 2014 Community detection in dense random networks
Ery Arias-Castro, Nicolas Verzelen
Ann. Statist. 42(3): 940-969 (June 2014). DOI: 10.1214/14-AOS1208

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. Specifically, we observe an undirected and unweighted graph on $N$ nodes. Under the null hypothesis, the graph is a realization of an Erdős–Rényi graph with probability $p_{0}$. Under the (composite) alternative, there is an unknown subgraph of $n$ nodes where the probability of connection is $p_{1}>p_{0}$. We derive a detection lower bound for detecting such a subgraph in terms of $N$, $n$, $p_{0}$, $p_{1}$ and exhibit a test that achieves that lower bound. We do this both when $p_{0}$ 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 $np_{0}$ is either bounded away from zero, or tends to zero slowly.

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Ery Arias-Castro. Nicolas Verzelen. "Community detection in dense random networks." Ann. Statist. 42 (3) 940 - 969, June 2014. https://doi.org/10.1214/14-AOS1208

Information

Published: June 2014
First available in Project Euclid: 20 May 2014

zbMATH: 1246.62213
MathSciNet: MR3210992
Digital Object Identifier: 10.1214/14-AOS1208

Subjects:
Primary: 62C20 , 62H30 , 94A13

Keywords: Community detection , dense $k$-subgraph problem , detecting a dense subgraph , Erdős–Rényi random graph , minimax hypothesis testing , planted clique problem , scan statistic , sparse eigenvalue problem

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 3 • June 2014
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