title: Speed of evolution on graphs.
abstract: 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.
authors.affiliation: Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China.
authors.forename: Xiukai
authors.initials: X
authors.lastname: Sui
authors.affiliation: School of Science, Beijing University of Posts and Communications, Beijing 100876, China.|Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann-Strasse 2, 24306 Plön, Germany.
authors.forename: Bin
authors.initials: B
authors.lastname: Wu
authors.affiliation: Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China.
authors.forename: Long
authors.initials: L
authors.lastname: Wang