Evolution of cooperation under social pressure in multiplex networks

In this work, we aim to contribute to the understanding of human prosocial behavior by studying the influence that a particular form of social pressure, “being watched,” has on the evolution of cooperative behavior. We study how cooperation emerges in multiplex complex topologies by analyzing a particular bidirectionally coupled dynamics on top of a two-layer multiplex network (duplex). The coupled dynamics appears between the prisoner's dilemma game in a network and a threshold cascade model in the other. The threshold model is intended to abstract the behavior of a network of vigilant nodes that impose the pressure of being observed altering hence the temptation to defect of the dilemma. Cooperation or defection in the game also affects the state of a node of being vigilant. We analyze these processes on different duplex networks structures and assess the influence of the topology, average degree and correlated multiplexity, on the outcome of cooperation. Interestingly, we find that the social pressure of vigilance may impact cooperation positively or negatively, depending on the duplex structure, specifically the degree correlations between layers is determinant. Our results give further quantitative insights in the promotion of cooperation under social pressure.

Figure 1

Simulation results for an Erdös-Rényi network (upper panels of both subfigures) and Barabási-Albert network (bottom panels of both subfigures) with average degrees $z=4$ (left) and $z=16$ (right). Panel (a) shows the average fraction of cooperators $\langle \rho \rangle$ as functions of the advantage of defectors $b$, and (b) shows the average fraction of cooperators $\langle \rho \rangle$ as functions of the personal threshold $\theta$ of the nodes (horizontal axis) and as functions of the advantage of defectors $b$ (vertical axis). The maximum standard error (SE) of all $\langle \rho \rangle$ values in the figure is 0.048.

Figure 2

Average fraction of cooperators $\langle \rho \rangle$ (top panels) and average fraction of vigilant players $\langle V\rangle$ (down panels) for Barabási-Albert networks with $z=16$ and cost of vigilance $Cv$ 0.25R (left), 0.5R (middle), and 0.75R (right). The higher the cost of vigilance, the lower the average fraction of cooperators $\langle \rho \rangle$ in the regions of $b$ where the population is not fully vigilant. The maximum SE of all $\langle \rho \rangle$ values in the figure is 0.023.

Figure 3

Average fraction of cooperators $\langle \rho \rangle$ for a duplex structure of networks. The combination of network topologies (Erdös-Rényi and Barabási-Albert networks) and average degrees (both layers with the same degree, $z=4$ or $z=16$) are explored. The vigilance network drives the cooperative outcome of the game dynamics. The vigilance network promotes cooperation as in the monoplex scenario (Fig. 1). The maximum SE of all $\langle \rho \rangle$ values in the figure is 0.05.

Figure 4

Average fraction of cooperators $\langle \rho \rangle$ for a duplex structure of networks, where layers have different average degrees. Vigilance and game networks both as Erdös-Rényi (top panels), and vigilance and game networks both as Barabási-Albert (bottom panels). Vigilance networks with $z=4$ and game networks with $z=16$ (left panels), and vigilance networks with $z=16$ and game networks with $z=4$ (right panels). The maximum SE of all $\langle \rho \rangle$ values in the figure is 0.05.

Figure 5

Average fraction of cooperators $\langle \rho \rangle$ for a duplex structure of Barabási-Albert networks. The degree distributions of the layers are maximally positive correlated (a) and maximally negative correlated (b). The average degree of both layers of the duplex is the same and takes the values $z=4$ (left) and $z=16$ (right). The maximum SE of all $\langle \rho \rangle$ values in the figure is 0.044.

Figure 6

Average fraction of cooperators $\langle \rho \rangle$ for a duplex structure of Barabási-Albert networks when actors can give up being vigilant: (a) There is not enough social pressure in the neighborhood or (b) with a probability $p=0.05$. The average degree of both layers of the duplex is the same and takes the values $z=4$ (left) and $z=16$ (right). The maximum SE of all $\langle \rho \rangle$ values in the figure is 0.03.