Community detection in sparse time-evolving graphs with a dynamical Bethe-Hessian

TL;DR

This paper introduces a spectral algorithm based on an extended Bethe-Hessian matrix for detecting communities in sparse, evolving graphs, achieving optimal detection thresholds and outperforming existing methods.

Contribution

It presents a novel spectral algorithm for community detection in dynamic graphs, extending the Bethe-Hessian approach to handle temporal evolution.

Findings

Achieves community detection at the theoretical detectability threshold.

Provably outperforms existing spectral methods in simulations.

Effective for sparse, time-evolving graphs with community structure.

Abstract

This article considers the problem of community detection in sparse dynamical graphs in which the community structure evolves over time. A fast spectral algorithm based on an extension of the Bethe-Hessian matrix is proposed, which benefits from the positive correlation in the class labels and in their temporal evolution and is designed to be applicable to any dynamical graph with a community structure. Under the dynamical degree-corrected stochastic block model, in the case of two classes of equal size, we demonstrate and support with extensive simulations that our proposed algorithm is capable of making non-trivial community reconstruction as soon as theoretically possible, thereby reaching the optimal detectability threshold and provably outperforming competing spectral methods.