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

Charles Bordenave (1) , Marc Lelarge (2, 3, 4) , Laurent Massoulié (4, 5, 6)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 DYOGENE - Dynamics of Geometric Networks
3 LIENS - Laboratoire d'informatique de l'école normale supérieure
4 LINCS - Laboratory of Information, Network and Communication Sciences
5 INFINE - INFormation NEtworks
6 MSR - INRIA - Microsoft Research - Inria Joint Centre

A non-backtracking 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 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-R ˝ enyi 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.

Mots clés: Stochastic Block Model; Ramanujan graphs; non-backtracking matrix.

Domaines: Réseaux et télécommunications [cs.NI]

Charles Bordenave, Marc Lelarge, Laurent Massoulié. Non-backtracking spectrum of random graphs: community detection and non-regular Ramanujan graphs. 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, Oct 2015, Berkeley, United States. ⟨10.1109/FOCS.2015.86⟩. ⟨hal-01226796⟩