Spectral density of the non-backtracking operator

The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius $\sqrt{\rho}$, where $\rho$ is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero spectral density. That fact that this phase transition is absent in the spectral density of other matrices commonly used for spectral clustering provides a physical justification of the performances of the non-backtracking operator in spectral clustering.