Tuesday, November 25, 2014

Spectral Clustering of Graphs with the Bethe Hessian - implementation -

To be filed in the Matrix Factorization Jungle page in the clustering section and its attendant phase transitions.

Spectral Clustering of Graphs with the Bethe Hessian by 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.  
An implementation is available here: http://mode_net.krzakala.org/
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