Dynamic fractals in spatial evolutionary games

TL;DR

This paper explores the critical behavior of a spatial Prisoner's Dilemma game, revealing fractal properties of cluster boundaries and phase transition phenomena through simulations.

Contribution

It introduces the concept of fractal cluster boundaries in spatial evolutionary games and analyzes their properties, which was not previously established.

Findings

Cluster densities exhibit a sudden jump at critical points.

Cluster boundaries are fractal with a dimension close to 2.

Boundaries become space-filling as system size increases.

Abstract

We investigate critical properties of a spatial evolutionary game based on the Prisoner's Dilemma. Simulations demonstrate a jump in the component densities accompanied by drastic changes in average sizes of the component clusters. We argue that the cluster boundary is a random fractal. Our simulations are consistent with the fractal dimension of the boundary being equal to 2, and the cluster boundaries are hence asymptotically space filling as the system size increases.