Spectral Detection on Sparse Hypergraphs

TL;DR

This paper introduces a spectral method for community detection in sparse hypergraphs, demonstrating its effectiveness and advantages over existing methods like belief propagation.

Contribution

A novel spectral approach based on a hypergraph generalization of the non-backtracking matrix, capable of detecting communities in sparse hypergraphs without prior knowledge.

Findings

Spectral method matches belief propagation in detection capability.

Method is simpler, nonparametric, and learns hyperedge generation rules.

Effective in sparse hypergraph regimes.

Abstract

We consider the problem of the assignment of nodes into communities from a set of hyperedges, where every hyperedge is a noisy observation of the community assignment of the adjacent nodes. We focus in particular on the sparse regime where the number of edges is of the same order as the number of vertices. We propose a spectral method based on a generalization of the non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance on a planted generative model and compare it with other spectral methods and with Bayesian belief propagation (which was conjectured to be asymptotically optimal for this model). We conclude that the proposed spectral method detects communities whenever belief propagation does, while having the important advantages to be simpler, entirely nonparametric, and to be able to learn the rule according to which the hyperedges were generated without prior information.