Pink-noise dynamics in an evolutionary game on a regular graph

We consider an iterated multiplayer prisoner's dilemma game on a square lattice and regular graphs based on the pairwise-Fermi update rule, and obtain heat-maps of the fraction of cooperators and the correlation of neighboring pairs. In the heat-map, there is a mixed region where cooperators and defectors coexist, and in the mixed region the correlation between neighbors is enhanced. Moreover, we observe pink-noise behavior in the mixed region, where the power spectrum can be fitted by a power-law function of frequency. We also find that the pink-noise behavior can be reproduced in a simple random-walk model. In particular, we propose a modified random-walk model which can reproduce not only the pink-noise behavior but also the deviation from it observed in a low-frequency region.