[Paper Review] Network Creation Games: Structure vs Anarchy

Bilò, Davide, Lenzner, Pascal|arXiv (Cornell University)|Jun 28, 2017

Game Theory and Applications8 references14 citations

TL;DR

This paper advances the understanding of network creation games by proving that for α > 17n, every Nash equilibrium is a tree, and for α > 9n, the price of anarchy is constant. It introduces novel coordinate-based analysis of 2-edge-connected components to bound diameter and average degree, significantly extending the range of α where the price of anarchy is known to be constant, while also showing that for α < n/C (C > 4), equilibria are 5-distance-almost-uniform, linking structure to performance.

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 >= 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.

Motivation & Objective

• To close the gap in the known range of α for which the price of anarchy is constant in network creation games.
• To strengthen the tree conjecture by proving that all Nash equilibria are trees for α > 17n.
• To analyze the structural properties of equilibria in the low-α regime (α < n/C, C > 4) and link them to distance-almost-uniform graphs.
• To improve upper bounds on the price of anarchy by leveraging local graph structure and diameter constraints.

Proposed method

• Introduces a coordinate system to precisely analyze cost differences when players deviate, particularly in 2-edge-connected components.
• Derives a lower bound on the average degree of 2-edge-connected components in Nash equilibria.
• Uses diameter bounds on 2-edge-connected components to constrain the overall network diameter and thus the price of anarchy.
• Applies the concept of ǫ-distance-almost-uniform graphs to characterize equilibria in the low-α regime.
• Leverages the fourth power of the graph (G⁴) to collapse distance ranges and relate structural properties to equilibrium stability.
• Combines structural lemmas on node degrees, cycle girth, and distance sets to derive contradictions under assumed suboptimal configurations.

Experimental results

Research questions

• RQ1 For which values of α is the price of anarchy constant in network creation games?
• RQ2 Can the tree conjecture be proven for α > 17n, i.e., are all Nash equilibria trees in this range?
• RQ3 What structural properties characterize Nash equilibria when α < n/C for C > 4?
• RQ4 How do the diameter and average degree of 2-edge-connected components constrain the price of anarchy?
• RQ5 To what extent do ǫ-distance-almost-uniform graphs capture the structure of low-α equilibria?

Key findings

• For α > 17n, every Nash equilibrium is a tree, significantly improving the prior bound of α > 65n.
• For α > 9n, the price of anarchy is at most 5, establishing a constant upper bound in a broader range than previously known.
• The diameter of any 2-edge-connected component in a Nash equilibrium is bounded by 125 when α > 9n and girth g(G) ≥ 20.
• For α < n/C with C > 4, every Nash equilibrium graph G is (5, ǫ)-distance-almost-uniform with ǫ = 4/5(1 + 1/C).
• The fourth power of any Nash equilibrium graph G⁴ is (2, ǫ)-distance-almost-uniform, and its diameter is at most ⌈diam(G)/4⌉.
• The paper establishes that the price of anarchy is bounded by 332 for α > 9n, using diameter and degree constraints on 2-edge-connected components.