Diversity-optimized cooperation on complex networks

We propose a strategy for achieving maximum cooperation in evolutionary games on complex networks. Each individual is assigned a weight that is proportional to the power of its degree, where the exponent $\alpha$ is an adjustable parameter that controls the level of diversity among individuals in the network. During the evolution, every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor, and updates its strategy depending on their payoff difference. It is found that there exists an optimal value of $\alpha$, for which the level of cooperation reaches maximum. This phenomenon indicates that, although high-degree individuals play a prominent role in maintaining the cooperation, too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation. Other pertinent quantities such as the payoff, the cooperator density as a function of the degree, and the payoff distribution are also investigated computationally and theoretically. Our results suggest that in order to achieve strong cooperation on a complex network, individuals should learn more frequently from neighbors with higher degrees, but only to a certain extent.

Figure 1

(Color online) Steady-state cooperator density ${\rho }_c$ versus the multiplication factor $r$ for different values of $\alpha$.

Figure 2

(Color online) Steady-state cooperator density ${\rho }_c$ as a function of $\alpha$ for different values of $r$. Apparently ${\rho }_c$ can be optimized by the diversity parameter $\alpha$. The inset is ${\rho }_c$ versus $\alpha$ for $r=3.5$ for the case where normalized payoffs are used.

Figure 3

(Color online) Payoff as a function of degree for different values of $\alpha$ when the system’s cooperation level is stable. The multiplication factor is $r=1.6$. The solid curves are theoretical results from Eq. (9). The dashed line is of unit slope.

Figure 4

(Color online) Cumulative payoff distribution for different values of $\alpha$. The distribution is obtained after the cooperation density becomes stable. The multiplication factor is set to be $r=1.6$. Solid curves are theoretical predictions from Eq. (13).

Figure 5

(Color online) Times series of cooperator density in hubs’ neighborhoods for (a) $\alpha =1$ and (b) $\alpha =6$. (c) Time series of cooperator density ${\rho }_c$ for the entire network for $\alpha =1$ and $\alpha =6$. The network size is 2000 and the average connectivity is $⟨k⟩=4$. For (a) and (b), open circles and open squares indicate evolutions of cooperators around the highest-degree and the second highest-degree node, respectively, where the two hubs are connected and share five neighbors. Initially only the highest-degree individual is a defector and others are cooperators. The multiplication factor is $r=1.2$ and each data point is obtained by averaging over 50 runs.

Figure 6

(Color online) For $r=1.6$, cooperator density ${\rho }_c$ as a function of degree for different values of $\alpha$.

Figure 7

(Color online) For $N=2000$ and $⟨k⟩=4$, the density ${\rho }_c$ as a function of $\alpha$ for different values of $r$ in the case where all cooperators contribute the same cost $c$ per game. Every cooperator contributes a cost $c=1$ in every neighborhood that it plays. The inset is ${\rho }_c$ versus $\alpha$ for $r=4$ by using normalized payoffs.