Analysis of ground state in random bipartite matching

TL;DR

This paper investigates the intrinsic properties of the ground state in bipartite matching models, using algorithms and simulations to analyze stability and the effects of connectivity on system behavior.

Contribution

It provides numerical analysis and simulation results on the ground state properties of bipartite matching, highlighting the impact of connectivity on stability.

Findings

The Kuhn-Munkres Algorithm effectively finds the ground state.

Simulation confirms theoretical predictions from the replica method.

Higher connectivity leads to increased system instability.

Abstract

In human society, a lot of social phenomena can be concluded into a mathematical problem called the bipartite matching, one of the most well known model is the marriage problem proposed by Gale and Shapley. In this article, we try to find out some intrinsic properties of the ground state of this model and thus gain more insights and ideas about the matching problem. We apply Kuhn-Munkres Algorithm to find out the numerical ground state solution of the system. The simulation result proves the previous theoretical analysis using replica method. In the result, we also find out the amount of blocking pairs which can be regarded as a representative of the system stability. Furthermore, we discover that the connectivity in the bipartite matching problem has a great impact on the stability of the ground state, and the system will become more unstable if there were more connections between men and women.