Percolation threshold determines the optimal population density for public cooperation

Zhen Wang, Attila Szolnoki, and Matjaž Perc

Phys. Rev. E 85, 037101 – Published 5 March 2012

Abstract

While worldwide census data provide statistical evidence that firmly link the population density with several indicators of social welfare, the precise mechanisms underlying these observations are largely unknown. Here we study the impact of population density on the evolution of public cooperation in structured populations and find that the optimal density is uniquely related to the percolation threshold of the host graph irrespective of its topological details. We explain our observations by showing that spatial reciprocity peaks in the vicinity of the percolation threshold, when the emergence of a giant cooperative cluster is hindered neither by vacancy nor by invading defectors, thus discovering an intuitive yet universal law that links the population density with social prosperity.

Received 27 December 2011

DOI:https://doi.org/10.1103/PhysRevE.85.037101

Authors & Affiliations

Zhen Wang^1,2, Attila Szolnoki³, and Matjaž Perc⁴

¹School of Physics, Nankai University, Tianjin 300071, China
²Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
³Institute for Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
⁴Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

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Vol. 85, Iss. 3 — March 2012

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Images

Figure 1
The peculiar dependence of the fraction of cooperators $f_{C}$ on the normalized synergy factor $r / G$ for different population densities $ρ$ (see legend), as obtained for the square lattice. While decreasing the population density below 1 facilitates the evolution of public cooperation, there exists a lower bound to $ρ$ below which, for sufficiently high values of $r$ , the effect is reversed. This indicates that the population density crucially affects the evolution of public cooperation, but it does so in a nontrivial way.Reuse & Permissions
Figure 2
Fraction of cooperators $f_{C}$ in dependence on the population density $ρ$ for different values of $r / G$ (see legend), as obtained for the square lattice. Inset depicts the corresponding growth of the fraction of active links $a_{l}$ (occupied nearest neighbor sites) as $ρ$ increases. Cooperators go extinct at $r = R_{C 1} = 0.75$ if $ρ = 1$ . The optimal population density where cooperators can dominate even at smaller $r$ is slightly above the percolation threshold, which is $π = 0.59$ .Reuse & Permissions
Figure 3
The optimal population density on the square lattice decreases if strategy imitations are allowed not just between nearest neighbors but also between all the players that are involved in an instance of the public goods game (compare with Fig. 2). This is because the extension of the imitation range effectively reduces the percolation threshold. Presented is the fraction of cooperators $f_{C}$ in dependence on the population density $ρ$ for different values or $r / G$ (see legend). Note that such an extension changes the behavior irrelevantly at $ρ = 1$ , where $R_{C 1} = 0.77$ .Reuse & Permissions
Figure 4
Fraction of cooperators $f_{C}$ in dependence on the population density $ρ$ for different values or $r / G$ (see legend), as obtained for the triangular lattice. Cooperators go extinct at $r = R_{C 1} = 0.65$ if $ρ = 1$ . Like on the square lattice (see Fig. 2), on the triangular lattice, too, $f_{C}$ is independent of $r$ at small values of $ρ$ , although the deviations occur sooner (at lower $ρ$ ) because of the lower percolation threshold. The optimal population density where cooperators can dominate even at smaller $r$ is $ρ_{o} \approx 0.55$ , which is slightly above the percolation threshold ( $π = 0.5$ ).Reuse & Permissions
Figure 5
Specially prepared snapshots of strategy distributions provide evidence that only in the proximity of the percolation threshold (b) cooperators (green) are able to fully percolate. At lower population densities (a) this is prohibited by empty sites (white), while at higher population densities (c) percolation is prohibited by defectors (black). Different shades of green (there are only four for clarity) are used for cooperators who belong to different cooperative clusters, i.e., cooperators who cannot reach each other by means of nearest-neighbor interactions. Note that the latter is the natural reach of imitation on the square lattice used. Population densities are (a) $ρ = 0.4$ , (b) $ρ = 0.71$ , and (c) $ρ = 0.95$ , while $r / G = 0.8$ .Reuse & Permissions

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