Analysis of ground state in random bipartite matching

Bipartite matching problems emerge in many human social phenomena. In this paper, we study the ground state of the Gale-Shapley model, which is the most popular bipartite matching model. We apply the Kuhn-Munkres algorithm to compute the numerical ground state of the model. For the first time, we obtain the number of blocking pairs which is a measure of the system instability. We also show that the number of blocking pairs formed by each person follows a geometric distribution. Furthermore, we study how the connectivity in the bipartite matching problems influences the instability of the ground state.