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December 2015 Community detection in sparse random networks
Nicolas Verzelen, Ery Arias-Castro
Ann. Appl. Probab. 25(6): 3465-3510 (December 2015). DOI: 10.1214/14-AAP1080


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.


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Nicolas Verzelen. Ery Arias-Castro. "Community detection in sparse random networks." Ann. Appl. Probab. 25 (6) 3465 - 3510, December 2015.


Received: 1 August 2013; Revised: 1 September 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1326.05145
MathSciNet: MR3404642
Digital Object Identifier: 10.1214/14-AAP1080

Primary: 05C80, 62C20

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 6 • December 2015
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