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The likelihood that a mutant fixates in the wild population, i.e., fixation probability, has been intensively studied in evolutionary game theory, where individuals' fitness is frequency dependent. However, it is of limited interest when it takes long to take over. Thus the speed of evolution becomes an important issue. In general, it is still unclear how fixation times are affected by the population structure, although the fixation times have already been addressed in the well-mixed populations. Here we theoretically address this issue by pair approximation and diffusion approximation on regular graphs. It is shown (i) that under neutral selection, both unconditional and conditional fixation time are shortened by increasing the number of neighbors; (ii) that under weak selection, for the simplified prisoner's dilemma game, if benefit-to-cost ratio exceeds the degree of the graph, then the unconditional fixation time of a single cooperator is slower than that in the neutral case; and (iii) that under weak selection, for the conditional fixation time, limited neighbor size dilutes the counterintuitive stochastic slowdown which was found in well-mixed populations. Interestingly, we find that all of our results can be interpreted as that in the well-mixed population with a transformed payoff matrix. This interpretation is also valid for both death-birth and birth-death processes on graphs. This interpretation bridges the fixation time in the structured population and that in the well-mixed population. Thus it opens the avenue to investigate the challenging fixation time in structured populations by the known results in well-mixed populations. [ABSTRACT FROM AUTHOR]

Evolutionary game theory on dynamical networks has received much attention. Most of the work has been focused on 2 x 2 games such as prisoner's dilemma and snowdrift, with general n x 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 ONE5, e 11187 (2010)] to n x 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. [ABSTRACT FROM AUTHOR]