On the Tree Conjecture for the Network Creation Game

TL;DR

This paper advances understanding of the network creation game by proving that for sufficiently high edge costs, all stable networks are trees, leading to a constant price of anarchy and improved bounds.

Contribution

Introduces a new technique to analyze stable networks at high edge prices, proving all equilibria are trees for n-13, and improves bounds on the price of anarchy.

Findings

All equilibrium networks are trees for n-13

Constant price of anarchy for high edge prices

Improved bounds on the price of anarchy

Abstract

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al. [PODC'03] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of $\alpha$ per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all $\alpha$ and that for $\alpha \geq n$ all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price $\alpha$ and employ it to improve on the best known bounds for both conjectures. In particular we show that for $\alpha > 4n-13$ all equilibrium networks must be trees, which implies a constant price of anarchy for this range of $\alpha$. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.