Propagation Connectivity of Random Hypergraphs

Robert Berke, Mikael Onsjö

Abstract: We consider the average-case MLS-3LIN problem, the problem of finding a most likely solution for a given system of perturbed 3LIN-equations generated with equation probability p and perturbation probability q. Our purpose is to investigate the situation for certain message passing algorithms to work for this problem whp We consider the case q = 0 (no perturbation occurs) and analyze the execution of (a simple version of) our algorithm. For characterizing problem instances for which the execution succeeds, we consider their natural 3-uniform hypergraph representation and introduce the notion of “propagation connectivity”, a generalized connectivity property on 3-uniform hypergraphs. The execution succeeds on a given system of 3LIN-equations if and only if the representing hypergraph is propagation connected. We show upper and lower bounds for equation probability p such that propagation connectivity holds whp on random hypergraphs representing MLS-3LIN instances.