Spectral Clustering of Graphs with the Bethe Hessian

TL;DR

This paper introduces the Bethe Hessian matrix for spectral clustering, which achieves optimal cluster detection in stochastic block models while maintaining computational efficiency and simplicity.

Contribution

It proposes using the symmetric Bethe Hessian matrix for spectral clustering, combining the advantages of non-backtracking operators with computational simplicity.

Findings

Bethe Hessian detects clusters down to the theoretical limit in stochastic block models.

It offers computational, theoretical, and memory advantages over non-symmetric operators.

The approach matches the performance of more complex methods while being simpler to implement.

Abstract

Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting clusters, and even to optimal performance for the stochastic block model. Here, we propose to use instead a simpler object, a symmetric real matrix known as the Bethe Hessian operator, or deformed Laplacian. We show that this approach combines the performances of the non-backtracking operator, thus detecting clusters all the way down to the theoretical limit in the stochastic block model, with the computational, theoretical and memory advantages of real symmetric matrices.