Evolutionary dynamics on stochastic evolving networks for multiple-strategy games.

Evolutionary game theory on dynamical networks has received much attention. Most of the work has been focused on 2×2 games such as prisoner's dilemma and snowdrift, with general n×n games seldom addressed. In particular, analytical methods are still lacking. Here we generalize the stochastic linking dynamics proposed by Wu, Zhou, Fu, Luo, Wang, and Traulsen [PLoS ONE 5, e11187 (2010)] to n×n games. We analytically obtain that the fast linking dynamics results in the replicator dynamics with a rescaled payoff matrix. In the rescaled matrix, intuitively, each entry is the product of the original entry and the average duration time of the corresponding link. This result is shown to be robust to a wide class of imitation processes. As applications, we show both analytically and numerically that the biodiversity, modeled as the stability of a zero-sum rock-paper-scissors game, cannot be altered by the fast linking dynamics. In addition, we show that the fast linking dynamics can stabilize tit-for-tat as an evolutionary stable strategy in the repeated prisoner's dilemma game provided the interaction between the identical strategies happens sufficiently often. Our method paves the way for an analytical study of the multiple-strategy coevolutionary dynamics.