Papers › Spectral Clustering of Graphs with the Bethe Hessian

Spectral Clustering of Graphs with the Bethe Hessian

7 Jun 2014NeurIPS 2014 12arXiv:1406.1880archive 2025-07-28

Alaa Saade, Florent Krzakala, Lenka Zdeborová

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.

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