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The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that cooperator clusters in spatial games are effective to protect cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that assures the prevalence of cooperation in a system. Here, we study the dynamical organization of cooperator clusters in spatial prisoners dilemma game to offer the condition for the dominance of cooperation, finding that a phase transition characterized by the emergence of a large spanning cooperator cluster occurs when the initial fraction of cooperators exceeds a certain threshold. Interestingly, the phase transition belongs to different universality classes of percolation determined by the temptation to defect $b$. Specifically, on square lattices, $1<b<4/3$ leads to a phase transition pertaining to the class of regular site percolation, whereas $3/2<b<2$ gives rise to a phase transition subject to invasion percolation with trapping. Our findings offer deeper understanding of the cooperative behaviors in nature and society.
In real world, individual rationality varies for the sake of the diversity of peoples individuality. In order to investigate how diversity of agents rationality affects the evolution of cooperation, we introduce the individual rationality proportiona
We study the evolution of cooperation in spatial Prisoners dilemma games with and without extortion by adopting aspiration-driven strategy updating rule. We focus explicitly on how the strategy updating manner (whether synchronous or asynchronous) an
We study an evolutionary spatial prisoners dilemma game where the fitness of the players is determined by both the payoffs from the current interaction and their history. We consider the situation where the selection timescale is slower than the inte
We study an evolutionary prisoners dilemma game with two layered graphs, where the lower layer is the physical infrastructure on which the interactions are taking place and the upper layer represents the connections for the strategy adoption (learnin
In the evolutionary Prisoners Dilemma (PD) game, agents play with each other and update their strategies in every generation according to some microscopic dynamical rule. In its spatial version, agents do not play with every other but, instead, inter