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

Mathieu Leconte (1, 2, 3) , Marc Lelarge (1, 3, 4) , Laurent Massoulié (5)

1 LINCS - Laboratory of Information, Network and Communication Sciences
2 Technicolor Paris lab
3 DYOGENE - Dynamics of Geometric Networks
4 LIENS - Laboratoire d'informatique de l'école normale supérieure
5 MSR - INRIA - Microsoft Research - Inria Joint Centre

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 in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP.

Mots clés: random graphs; capacitated b-matchings; local weak convergence; load balancing; cuckoo hashing; belief propagation; upshifted likelihood ratio stochastic order

Domaines: Probabilités [math.PR]; Réseaux et télécommunications [cs.NI]; Algorithme et structure de données [cs.DS]

Mathieu Leconte, Marc Lelarge, Laurent Massoulié. Convergence of multivariate belief propagation, with applications to cuckoo hashing and load balancing.. SODA 2013 - ACM-SIAM Symposium on Discrete Algorithms, Jan 2013, United States. p. 35-46. ⟨hal-00827810⟩