Critical Phenomena and Strategy Ordering with Hub Centrality Approach in the Aspiration-based Coordination Game

TL;DR

This paper analyzes a coordination game on various networks, revealing phase transitions akin to the Ising model, and introduces hub centrality to explain differences between model and real-world networks.

Contribution

It establishes the exact equivalence of the coordination game with the Ising model and introduces the concept of hub centrality to explain network effects.

Findings

Phase transition occurs at critical noise levels, similar to the Ising model.

Critical exponents are independent of clustering and aspiration levels.

Real-world networks exhibit gradual order parameter decrease due to hub centrality effects.

Abstract

We study the coordination game with an aspiration-driven update rule in regular graphs and scale-free networks. We prove that the model coincides exactly with the Ising model and shows a phase transition at the critical selection noise when the aspiration level is zero. It is found that the critical selection noise decreases with clustering in random regular graphs. With a non-zero aspiration level, the model also exhibits a phase transition as long as the aspiration level is smaller than the degree of graphs. We also show that the critical exponents are independent of clustering and aspiration level to confirm that the coordination game belongs to the Ising universality class. As for scale-free networks, the effect of aspiration level on the order parameter at a low selection noise is examined. In model networks (Barab\'{a}si-Albert network and Holme-Kim network), the order parameter abruptly decreases when the aspiration level is the same as the average degree of the network. In real-world networks, in contrast, the order parameter decreases gradually. We explain this difference by proposing the concepts of hub centrality and local hub. The histogram of hub centrality of real-world networks separates into two parts unlike model networks, and local hubs exist only in real-world networks. We conclude that the difference of network structures in model and real-world networks induces qualitatively different behavior in the coordination game.