On Bipartite Sum Basic Equilibria

Abstract

A connected and undirected graph G of size n ≥ 1 is said to be a sum basic equilibrium iff for every edge uv from G and any node v from G, when performing the swap of the edge uv for the edge uv the sum of the distances from u to all the other nodes is not strictly reduced. This concept comes from the so called Network Creation Games, a wide subject inside Algorithmic Game Theory that tries to better understand how Internet-like networks behave. It has been shown that the diameter of sum basic equilibria is 2 O( √ log n) in general and at most 2 for trees. In this paper we see that the upper bound of 2 not only holds for trees but for bipartite graphs, too. Specifically, we show that the only bipartite sum basic equilibrium networks are the complete bipartite graphs K r,s with r, s ≥ 1 .