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While worldwide census data provide statistical evidence that firmly link thepopulation density with several indicators of social welfare, the precisemechanisms underlying these observations are largely unknown. Here we study theimpact of population density on the evolution of public cooperation instructured populations and find that the optimal density is uniquely related tothe percolation threshold of the host graph irrespective of its topologicaldetails. We explain our observations by showing that spatial reciprocity peaksin the vicinity of the percolation threshold, when the emergence of a giantcooperative cluster is hindered neither by vacancy nor by invading defectors,thus discovering an intuitive yet universal law that links the populationdensity with social prosperity.

Percolation threshold determines the optimal population density for public cooperation