Phase transition in the bipartite z-matching

TL;DR

This paper investigates the phase transition in maximum z-matching problems on bipartite random graphs, revealing universal scaling behaviors and phase transitions between saturable and unsaturable states.

Contribution

It introduces a numerical study of z-matching on large bipartite graphs, confirming analytical results and exploring phase transitions and algorithmic complexity.

Findings

Confirmed analytical results for bipartite regular graphs.

Discovered universal finite-size scaling behavior.

Identified phase transition points in Erdős-Rényi graphs.

Abstract

We study numerically the maximum $z$-matching problems on ensembles of bipartite random graphs. The $z$-matching problems describes the matching between two types of nodes, users and servers, where each server may serve up to $z$ users at the same time. By using a mapping to standard maximum-cardinality matching, and because for the latter there exists a polynomial-time exact algorithm, we can study large system sizes of up to $10^6$ nodes. We measure the capacity and the energy of the resulting optimum matchings. First, we confirm previous analytical results for bipartite regular graphs. Next, we study the finite-size behaviour of the matching capacity and find the same scaling behaviour as before for standard matching, which indicates the universality of the problem. Finally, we investigate for bipartite Erd\H{o}s-R\'enyi random graphs the saturability as a function of the average degree, i.e., whether the network allows as many customers as possible to be served, i.e. exploiting the servers in an optimal way. We find phase transitions between unsaturable and saturable phases. These coincide with a strong change of the running time of the exact matching algorithm, as well with the point where a minimum-degree heuristic algorithm starts to fail.