Abstract
The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that the cooperator clusters in spatial games are effective in protecting the cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that ensures the prevalence of cooperation in a system. Here, we study the dynamical organization of the cooperator clusters in spatial prisoner's 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 the 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 a deeper understanding of cooperative behavior in nature and society.
Highlights
The paradox of cooperation among selfish individuals still puzzles scientific communities
And (b), we obtain the critical exponents β/ν 0.051 and γ /ν 0.908, which are very close to the critical exponents of the regular site percolation (β/ν 0.052 and γ /ν 0.896) [67]. These results indicate that the cooperation percolation belongs to the same universality class as the regular site percolation when 1 < b < 4/3
And (b), we obtain the critical exponents β/ν 0.082 and γ /ν 0.856, which are close to the critical exponents of the invasion percolation with trapping (β/ν 0.084 and γ /ν 0.832) [71]. These results demonstrate that the cooperation percolation belongs to the same universality class as the invasion percolation with trapping in the region of 3/2 < b < 2
Summary
Each individual x can follow one of two strategies: cooperation or defection, described by zx =. In order to study the critical behavior regarding the giant cooperator cluster, we employ the normalized size of the largest cooperator cluster s1, the susceptibility χ and the Binder’s fourth-order cumulant U. These quantities are defined as follows [68]: s1. According to the standard finite-size scaling approach [68], the phase transition point can be identified by Binder’s fourth-order cumulant U and there are the following power-law relationships at the critical value: s1 ∼ N −β/ν and χ ∼ N γ /ν. We can use the tool to identify the phase transitions pertaining to the formation of the cooperator clusters and explore the class to which the phase transition belongs
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