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January 2018 Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs
Charles Bordenave, Marc Lelarge, Laurent Massoulié
Ann. Probab. 46(1): 1-71 (January 2018). DOI: 10.1214/16-AOP1142


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 Erdős–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.


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Charles Bordenave. Marc Lelarge. Laurent Massoulié. "Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs." Ann. Probab. 46 (1) 1 - 71, January 2018.


Received: 1 April 2015; Revised: 1 August 2016; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865118
MathSciNet: MR3758726
Digital Object Identifier: 10.1214/16-AOP1142

Primary: 05C80 , 60B20
Secondary: 60J85 , 62M15

Keywords: Community detection , nonbacktracking matrix , Random graphs , spectral gap

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 1 • January 2018
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