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

Appears in collections : Spectrum of random graphs / Spectre de graphes aléatoires, Exposés de recherche

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.

Mathematics
and Interactions

Summer School 2026 - ICRA 2026

CEMRACS 2026 : Modeling and AI for Environmental Transition / Centre d'Eté Mathématique de Recherche Avancée en Calcul Scientifique

Coulomb branches and affine quantum groups / Branches de Coulomb et groupes quantiques affines

Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexe

Chapters