Impossibility of Partial Recovery in the Graph Alignment Problem

Luca Ganassali (1) , Marc Lelarge (1) , Laurent Massoulié (1)

1 DYOGENE - Dynamics of Geometric Networks

Random graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For the correlated Erdős-Rényi model, we prove an impossibility result for partial recovery in the sparse regime, with constant average degree and correlation, as well as a general bound on the maximal reachable overlap. Our bound is tight in the noiseless case (the graph isomorphism problem) and we conjecture that it is still tight with noise. Our proof technique relies on a careful application of the probabilistic method to build automorphisms between tree components of a subcritical Erdős-Rényi graph.

Mots clés: Graph alignment; Probabilistic method; Erdős-Rényi random graphs; Partial recovery

Domaines: Machine Learning [stat.ML]; Apprentissage [cs.LG]; Probabilités [math.PR]; Statistiques [math.ST]

Luca Ganassali, Marc Lelarge, Laurent Massoulié. Impossibility of Partial Recovery in the Graph Alignment Problem. COLT 2021 - 34th Annual Conference on Learning Theory, Aug 2021, Boulder / Virtual, United States. ⟨hal-03410038⟩