Coevolution of relationship-driven cooperation under recommendation protocol on multiplex networks

TL;DR

A coevolutionary multiplex network model is introduced that incorporates the concepts of a relationship threshold and a recommendation mechanism to explore how the strength of relationships among agents interacts with their strategy choices within the framework of weak prisoner's dilemma games.

Abstract

While traditional game models often simplify interactions among agents as static, real-world social relationships are inherently dynamic, influenced by both immediate payoffs and alternative information. Motivated by this fact, we introduce a coevolutionary multiplex network model that incorporates the concepts of a relationship threshold and a recommendation mechanism to explore how the strength of relationships among agents interacts with their strategy choices within the framework of weak prisoner's dilemma games. In the relationship layer, the relationship strength between agents varies based on interaction outcomes. In return, the strategy choice of agents in the game layer is influenced by both payoffs and relationship indices, and agents can interact with distant agents through a recommendation mechanism. Simulation of various network topologies reveals that a higher average degree supports cooperation, although increased randomness in interactions may inhibit its formation. Interestingly, a higher threshold value of interaction quality is detrimental, while the applied recommendation protocol can improve global cooperation. The best results are obtained when the relative weight of payoff is minimal and the individual fitness is dominated by the relationship indices gained from the quality of links to neighbors. As a consequence, the changes in the distribution of relationship indices are closely correlated with overall levels of cooperation.

Figures (5)

• Figure 1: Coevolutionary game on multiplex networks. The model consists of a multiplex network, namely the relationship layer and the game layer. The nodes in both layers correspond to the same agents. In the relationship layer, the edge weights represent the strength of the associations between agents, while in the game layer, the edges indicate the interactions or games played between agents. According to the coevolutionary rules, the strength of associations in the relationship layer influences the interactions in the game layer, and the outcomes of these interactions subsequently affect the association strength in the relationship layer. Further details can be found in the main text.
• Figure 2: Cooperation density distribution on $b-m$ parameter plane by using different network types. The variations in cooperation density $f_c$ concerning parameters $b$ and $m$ are depicted for four network structures: (a) HL, (b) SL, (c) XL, and (d) WS small-world networks, under fixed values of parameter $p = 0.9$ and $\gamma = 0.5$.
• Figure 3: Dependence of cooperation level on different parameters within four networks. (a) We investigate the dependence of $f_c$ on $b$ under fixed conditions where $m = 0.5$, $p = 0.9$, and $\gamma = 0.2$. (b) To investigate the effect of the threshold $\boldsymbol{\gamma}$, we set $m = 0.5$, $p = 0.9$, and $b = 1.5$ to observe the variation of $f_c$. (c) The parameters $m = 0.50$, $b = 1.5$, and $\gamma = 0.1$ are employed across four different types of network.
• Figure 4: The mean of the relationship index $\bar{A}$ plotted at different $m$ and $b$ in four networks. Different colors represent different network types, as shown in the legend. The solid line, dotted line, and dashed line correspond to the values of $b = 1.0$, $1.5$, and $2.0$, respectively. $\gamma = 0.1$ is employed for all four network types.
• Figure 5: Probability density distributions of initial and stationary state relationship indices under different $m$ values are presented for the four types of networks. In all cases, $p = 0.9$, $b = 1.5$, and $\gamma = 0.1$ are used. The horizontal axis displays the actual values of the relationship index, while the vertical axis shows their probability density. In all density distribution graphs, the initial normal distribution is represented by a red dashed line, while the steady-state distributions for different $m$ values are indicated by blue, green, and orange lines, as shown in the legend.