Convergence of multivariate belief propagation, with applications to cuckoo hashing and load balancing

This paper is motivated by two applications, namely i) generalizations of cuckoo hashing,a computationally simple approach to assigning keys to objects, and ii) load balancing incontent distribution networks, where one is interested in determining the impact of contentreplication on performance. These two problems admit a common abstraction: in bothscenarios, performance is characterized by the maximum weight of a generalization of amatching in a bipartite graph, featuring node and edge capacities.Our main result is a law of large numbers characterizing the asymptotic maximum weightmatching in the limit of large bipartite random graphs, when the graphs admit a local weaklimit that is a tree. This result specializes to the two application scenarios, yielding newresults in both contexts. In contrast with previous results, the key novelty is the ability tohandle edge capacities with arbitrary integer values.An analysis of belief propagation algorithms (BP) with multivariate belief vectors un-derlies the proof. In particular, we show convergence of the corresponding BP by exploitingmonotonicity of the belief vectors with respect to the so-called upshifted likelihood ratiostochastic order. This auxiliary result can be of independent interest, providing a new setof structural conditions which ensure convergence of BP.