## The Annals of Statistics

### Community detection in dense random networks

#### 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.

#### Article information

Source
Ann. Statist., Volume 42, Number 3 (2014), 940-969.

Dates
First available in Project Euclid: 20 May 2014

https://projecteuclid.org/euclid.aos/1400592648

Digital Object Identifier
doi:10.1214/14-AOS1208

Mathematical Reviews number (MathSciNet)
MR3210992

Zentralblatt MATH identifier
1246.62213

#### Citation

Arias-Castro, Ery; Verzelen, Nicolas. Community detection in dense random networks. Ann. Statist. 42 (2014), no. 3, 940--969. doi:10.1214/14-AOS1208. https://projecteuclid.org/euclid.aos/1400592648

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