Community detection with the Bethe-Hessian

TL;DR

This paper provides a rigorous analysis of the Bethe-Hessian spectral method for community detection in sparse stochastic block models, confirming its effectiveness near the Kesten-Stigum threshold and establishing conditions for weak recovery.

Contribution

It offers the first rigorous proof of the Bethe-Hessian spectral method's ability to estimate community number and achieve weak recovery in sparse networks.

Findings

Negative outliers estimate the number of communities above the threshold

Eigenvectors enable weak community recovery for large degrees

Eigenvalue locations concentrate as degree grows large

Abstract

The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.