Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs

TL;DR

This paper analyzes the spectral properties of the non-backtracking matrix in random graphs, demonstrating its effectiveness for community detection and exploring connections to Ramanujan graphs.

Contribution

It provides new insights into the eigenvalues of the non-backtracking matrix in Erdős-Rényi and stochastic block models, confirming the spectral redemption conjecture.

Findings

Eigenvalues of the non-backtracking matrix are characterized for random graphs.

Community detection is feasible using leading eigenvectors above a certain threshold.

Connections to Ramanujan graphs and Ihara zeta function are explored.

Abstract

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 Erdos-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 confirm the "spectral redemption" conjecture that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.