Evolutionary games on networks: Phase transition, quasi-equilibrium, and mathematical principles

The stable cooperation ratio of spatial evolutionary games has been widely studied using simulations or approximate analysis methods. However, sometimes such "stable"cooperation ratios obtained via approximate methods might not be actually stable, but correspond to quasi-equilibriums instead. We find that various classic game models, like the evolutionary snowdrift game, evolutionary prisoner's dilemma, and spatial public goods game on square lattices and scale-free networks, exhibit the phase transition in convergence time to the equilibrium state. Moreover, mathematical principles are provided to explain the phase transition of convergence time and quasi-equilibrium of cooperation ratio. The findings explain why and when cooperation and defection have a long-term coexistence. (c) 2023 Elsevier B.V. All rights reserved.