Role of population density and increasing neighborhood in the evolution of cooperation on diluted lattices

We investigate the evolution of cooperative behaviors with increasing neighborhood size on diluted lattices. For three typical pairwise game models which include prisoner's dilemma, snowdrift and stag hunt games, all numerical results indicate that cooperation can persist or emerge around the optimal population density which is dictated by the percolation threshold on the square lattice. Meanwhile, the neighborhood size determines the interaction ranges of focal players and then dominates the percolation threshold, and extensive numerical simulations demonstrate that the intermediate neighborhood size is the most beneficial to the evolution of cooperation in the current lattice setup. The current findings can help to deeply understand the sustenance and emergence of collective cooperation in many natural, social and economic systems.