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

Charles Bordenave 1 Marc Lelarge 2, 3, 4
3 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
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 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 con rm the "spectral redemption conjecture" that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.
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Submitted on : Tuesday, March 31, 2015 - 5:32:29 PM
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Charles Bordenave, Marc Lelarge. Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs. CoRR abs, 2014, pp.abs/1501.06087. ⟨hal-01137952⟩

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