  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.
