Non-backtracking Spectrum of Random Graphs: Community Detection and Non-regular Ramanujan Graphs

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 on-backtracking walks of a given length. It has been used recently in the context of communit...

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Bibliographic Details
Published in2015 IEEE 56th Annual Symposium on Foundations of Computer Science pp. 1347 - 1357
Main Authors Bordenave, Charles, Lelarge, Marc, Massoulie, Laurent
Format Conference Proceeding
LanguageEnglish
Published IEEE 01.10.2015
Subjects
Belief propagation
Context
Eigenvalues and eigenfunctions
Electronic mail
Image edge detection
non-backtracking matrix
Ramanujan graphs
Stochastic Block Model
Stochastic processes
Symmetric matrices
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Summary: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 on-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 eigen valus 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.
ISSN:0272-5428
DOI:10.1109/FOCS.2015.86