Diversity and critical behavior in prisoner's dilemma game

The prisoner's dilemma (PD) game is a simple model for understanding cooperative patterns in complex systems. Here, we study a PD game problem in scale-free networks containing hierarchically organized modules and controllable shortcuts connecting separated hubs. We find that cooperator clusters exhibit a percolation transition in the parameter space (p,b), where p is the occupation probability of shortcuts and b is the temptation payoff in the PD game. The cluster size distribution follows a power law at the transition point. Such a critical behavior, resulting from the combined effect of stochastic processes in the PD game and the heterogeneity of complex network structure, illustrates diversities arising in social relationships and in forming cooperator groups in real-world systems.