Statistical mechanics of bipartite $z$-matchings

TL;DR

This paper applies the cavity method from statistical mechanics to analyze the bipartite z-matching problem, relevant for resource allocation in networks, and explores its phase diagram on network ensembles.

Contribution

It extends the cavity method to bipartite z-matching problems, providing an exact analytical framework for network capacity analysis.

Findings

Derived the phase diagram of the bipartite z-matching model.

Provided an exact solution for the capacity of user-server networks.

Analyzed the model on various network ensembles.

Abstract

The matching problem has a large variety of applications including the allocation of competitive resources and network controllability. The statistical mechanics approach based on the cavity method has shown to be exact in characterizing this combinatorial problem on locally tree-like networks. Here we use the cavity method to solve the many-to-one bipartite $z$-matching problem that can be considered to be a model for the characterization of the capacity of user-server networks such as wireless communication networks. Finally we study the phase diagram of the model defined in network ensembles.