An Improved Bound for the Tree Conjecture in Network Creation Games

TL;DR

This paper proves that in the network creation game, when the edge cost exceeds three times the number of vertices minus three, all Nash equilibria are spanning trees, advancing understanding of network formation stability.

Contribution

It establishes a new bound, showing the conjecture holds for any edge cost greater than 3n-3, improving previous results in network creation game theory.

Findings

Nash equilibria are spanning trees for α > 3n-3.

Supports the conjecture that high edge costs lead to tree structures.

Provides a tighter bound for the tree conjecture in network creation games.

Abstract

We study Nash equilibria in the network creation game of Fabrikant et al.[10]. In this game a vertex can buy an edge to another vertex for a cost of $\alpha$, and the objective of each vertex is to minimize the sum of the costs of the edges it purchases plus the sum of the distances to every other vertex in the resultant network. A long-standing conjecture states that if $\alpha\ge n$ then every Nash equilibrium in the game is a spanning tree. We prove the conjecture holds for any $\alpha>3n-3$.