Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs
Résumé
A non-backtracking walk on a graph is a directed path such that no edge is the inverse of its
preceding edge. The non-backtracking matrix of a graph is indexed by its directed edges and
can be used to count non-backtracking 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 non-backtracking matrix of the Erd}os-Renyi 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 conrm the "spectral redemption conjecture" that community detection
can be made on the basis of the leading eigenvectors above the feasibility threshold.