Promoting cooperation by preventing exploitation: The role of network structure

A growing body of empirical evidence indicates that social and cooperative behavior can be affected by cognitive and neurological factors, suggesting the existence of state-based decision-making mechanisms that may have emerged by evolution. Motivated by these observations, we propose a simple mechanism of anonymous network interactions identified as a form of generalized reciprocity—a concept organized around the premise “help anyone if helped by someone'—and study its dynamics on random graphs. In the presence of such a mechanism, the evolution of cooperation is related to the dynamics of the levels of investments (i.e., probabilities of cooperation) of the individual nodes engaging in interactions. We demonstrate that the propensity for cooperation is determined by a network centrality measure here referred to as neighborhood importance index and discuss relevant implications to natural and artificial systems. To address the robustness of the state-based strategies to an invasion of defectors, we additionally provide an analysis which redefines the results for the case when a fraction of the nodes behave as unconditional defectors.

Figure 1

Mean-squared error (MSE) comparison of the stochastic and the deterministic models. For the stochastic model, the results are averaged across 100 network realizations. (a) Regular graph. (b) ER graph. (c) BA graph. (d) WS graph. (a–d) $b/c=1.8$. (e–h) Same as (a–d), only $b/c=1.5$. (i–l) Same as (a–d), only $b/c=1.2$. (a–l) A lighter shade indicates lower parameter values. All graphs have 100 nodes and average degree 8. In each run we set the initial values $Y_i\left(0\right)=-4$ and $p_i\left(0\right)=0.05$ for all $i$. We assume synchronous update.

Figure 2

Average steady-state payoff $\langle y^{*}\rangle$ as a function of $b/c$ (solid line), averaged over 100 graph realizations. The regions (in lighter shade) enclose the area between the minimum and the maximum node payoff observed in the network. (a) Regular graph. (b) ER graph. (c) BA graph. (d) WS graph. (a–d) All graphs have 1000 nodes and average degree 8.

Figure 3

(a) A visualization of a small random graph with 10 nodes. The nodes are colored according to the order of switching from strong to weak when decreasing the $b/c$ ratio—a darker color indicates higher switching propensity. The node size is proportional to the index $z_i$. (b) Fraction of strong nodes $\sigma$ as a function of the (decreasing) $b/c$ ratio.

Figure 4

(a) Fraction of unconditional cooperators $\sigma$ as a function of $b/c$ for Regular, ER, BA, and WS graphs. Dashed lines indicate the threshold for full network cooperation according to Eq. (7). (b) Probability density function (pdf) of the index $z$ for the same random graphs. We note that for the Regular graph the pdf is concentrated on a single point (i.e., the index $z$ is the same across all nodes, $z=1$). (a, b) The results are obtained by averaging over 100 different graph realizations. All graphs have 1000 nodes and average degree 8.

Figure 5

Heat map for the fraction of defective instances as a function of $\delta =D/N$ and $b/c$. $D$ out of $N$ nodes are randomly chosen to be unconditional defectors and the results are averaged over 100 realizations. (a) Regular graph. (b) ER graph. (c) BA graph. (d) WS graph. (a–d) All graphs have 100 nodes and average degree 8. Results are averaged across 100 graph realizations.