Analytical description for the critical fixations of evolutionary coordination games on finite complex structured populations

Dynamical analysis of evolutionary public goods game on signed networks

In evolutionary dynamics, the population structure and multiplayer interactions significantly impact the evolution of cooperation levels. Previous works mainly focus on the theoretical analysis of multiplayer games on regular networks or pairwise games on complex networks. Combining these two factors, complex networks and multiplayer games, we obtain the fixation probability and fixation time of the evolutionary public goods game in a structured population represented by a signed network. We devise a stochastic framework for estimating fixation probability with weak mistrust or strong mistrust mechanisms and develop a deterministic replicator equation to predict the expected density of cooperators when the system evolves to the equilibrium on a signed network. Specifically, the most interesting result is that negative edges diversify the cooperation steady state, evolving in three different patterns of fixed probability in Erdös-Rényi signed and Watts-Strogatz signed networks with the new "strong mistrust" mechanism.

Fixation probabilities of evolutionary coordination games on two coupled populations

Evolutionary forces resulted from competitions between different populations are common, which change the evolutionary behavior of a single population. In an isolated population of coordination games of two strategies (e.g., s_{1} and s_{2}), the previous studies focused on determining the fixation probability that the system is occupied by only one strategy (s_{1}) and their expectation times, given an initial mixture of two strategies. In this work, we propose a model of two interdependent populations, disclosing the effects of the interaction strength on fixation probabilities. In the well-mixing limit, a detailed linear stability analysis is performed, which allows us to find and to classify the different equilibria, yielding a clear picture of the bifurcation patterns in phase space. We demonstrate that the interactions between populations crucially alter the dynamic behavior. More specifically, if the coupling strength is larger than some threshold value, the critical initial density of one strategy (s_{1}) that corresponds to fixation is significantly delayed. Instead, the two populations evolve to the opposite state of all (s_{2}) strategy, which are in favor of the red queen hypothesis. We delineate the extinction time of strategy (s_{1}) explicitly, which is an exponential form. These results are validated by systematic numerical simulations.