Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erd.s.Rényi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the .spectral redemption conjecture. in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.