TESTING COMMUNITY STRUCTURE FOR HYPERGRAPHS

Document Type

Article

Publication Date

2-1-2022

Abstract

Many complex networks in the real world can be formulated as hypergraphs where community detection has been widely used. However, the fundamental question of whether communities exist or not in an observed hypergraph remains unclear. This work aims to tackle this important problem. Specifically, we systematically study when a hypergraph with community structure can be successfully distinguished from its Erdos-Rényi counterpart, and propose concrete test statistics when the models are distinguishable. The main contribution of this paper is threefold. First, we discover a phase transition in the hyperedge probability for distinguishability. Second, in the bounded-degree regime, we derive a sharp signal-to-noise ratio (SNR) threshold for distinguishability in the special two-community 3- uniform hypergraphs, and derive nearly tight SNR thresholds in the general two-community m-uniform hypergraphs. Third, in the dense regime, we propose a computationally feasible test based on sub-hypergraph counts, obtain its asymptotic distribution, and analyze its power. Our results are further extended to nonuniform hypergraphs in which a new test involving both edge and hyperedge information is proposed. The proofs rely on Janson's contiguity theory (Combin. Probab. Comput. 4 (1995) 369-405), a high-moments driven asymptotic normality result by Gao andWormald (Probab. Theory Related Fields 130 (2004) 368-376), and a truncation technique for analyzing the likelihood ratio.

Identifier

85125830975 (Scopus)

Publication Title

Annals of Statistics

External Full Text Location

https://doi.org/10.1214/21-AOS2099

e-ISSN

21688966

ISSN

00905364

First Page

147

Last Page

169

Issue

1

Volume

50

Grant

DMS-2013789

Fund Ref

National Science Foundation

This document is currently not available here.

Share

COinS