Cao et al. (2010) The evolutionary public goods game on scale-free networks with heterogeneous investment.

Cao et al. (2010) The evolutionary public goods game on scale-free networks with heterogeneous investment. Physica A, 389, 1273-1280.

Evolutionary public goods game on a scale-free graph with investments (for cooperators) that correlate with degree.

Player i will invest Ii = N · kβi / Σl kβl if he is a cooperator, where N is the total investment if all players are cooperators, k_i is the degree of player i, and l runs over all players. If β > 0 (<0), large-degree (small-degree) nodes invest more if they are cooperators. When β = 0, all cooperative players have the same investment.

They assume the “single PGG” model – i.e., payoff for x is determined assuming a single PGG centered on x.

Evolutionary dynamics occur using a Fermi-like probability function (as in Poncela et al., 2010). The parameter \kappa is the “environmental noise”, i.e., there is more of a random component to the replicator dynamics.

Main result: It is shown that the cooperation level is remarkably promoted by negative values of β whereas it is highly depressed by positive values of β.

I.e., when small-degree nodes invest more, cooperation is promoted, and it is impaired when large-degree nodes invest more.

More results:

• Hubs: They selected the top highest-degree nodes and fixed them as either C or D during the whole evolution. Having C hubs increased cooperation, having D hubs decreased cooperation. 
• Increase noise decreased cooperation.