Path to fixation of evolutionary processes in graph-structured populations

Abstract

We study the spreading of a single mutant in graph-structured populations with a birth-death update rule. We use a mean-field approach and a Markov chain dynamics to investigate the effect of network topology on the path to fixation. We obtain approximate analytical formulas for average time versus the number of mutants in the fixation process starting with a single mutant for several network structures, namely, cycle, complete graph, two- and three-dimensional lattices, random graph, regular graph, Watts–Strogatz network, and Barabasi–Albert network. In the case of the cycle and complete graph, the results are accurate and in line with the results obtained by other methods. In the case of two- and three-dimensional lattice structures, some efforts are made in other studies to provide an analytical justification for simulation results of the evolutionary process, but they can explain just the onset of the fixation process, not the whole process. The results of the analytical approach of the present paper are well fitted to the simulation results throughout the whole fixation process. Moreover, we analyze the dynamics of evolution for a number of complex structures, and in all cases, we obtain analytical results which are in good agreement with simulations. Our results may shed some light on the process of fixation during the whole path to fixation.