Coevolution of relationship and interaction in cooperative dynamical multiplex networks

TL;DR

This work addresses how evolving social relationships co-influence cooperative behavior in multiplex networks playing the weak prisoner's dilemma. It introduces a two-layer model with a relationship layer of edge weights $W(i,j)$ and an interaction layer, where fitness combines payoff $\Pi_i$ with a relationship-index $A_i=\sum_j W(i,j)$ via $f_i=\Pi_i+mA_i$, and strategy updates follow a Fermi rule with $K=0.1$. Across honeycomb, square, hexagonal lattices and Watts–Strogatz/Newman–Watts networks, the study shows that higher average degree in the relationship layer can promote cooperation in regular graphs, while randomness can hinder it in harsher regimes; a stronger coupling $m$ between relationship index and fitness generally reduces cooperation by dampening the direct payoff-feedback. The distribution of the relationship index $A_i$ correlates with the overall cooperation level, exhibiting topology-dependent skewness and kurtosis that reflect how relationships polarize under evolving strategies. These findings illuminate how structural and relational coevolution shapes cooperative outcomes and offer insights for designing systems with robust cooperative dynamics.

Abstract

While actors in a population can interact with anyone else freely, social relations significantly influence our inclination towards particular individuals. The consequence of such interactions, however, may also form the intensity of our relations established earlier. These dynamical processes are captured via a coevolutionary model staged in multiplex networks with two distinct layers. In a so-called relationship layer the weights of edges among players may change in time as a consequence of games played in the alternative interaction layer. As an reasonable assumption, bilateral cooperation confirms while mutual defection weakens these weight factors. Importantly, the fitness of a player, which basically determines the success of a strategy imitation, depends not only on the payoff collected from interactions, but also on the individual relationship index calculated from the mentioned weight factors of related edges. Within the framework of weak prisoner's dilemma situation we explore the potential outcomes of the mentioned coevolutionary process where we assume different topologies for relationship layer. We find that higher average degree of the relationship graph is more beneficial to maintain cooperation in regular graphs, but the randomness of links could be a decisive factor in harsh situations. Surprisingly, a stronger coupling between relationship index and fitness discourage the evolution of cooperation by weakening the direct consequence of a strategy change. To complete our study we also monitor how the distribution of relationship index vary and detect a strong relation between its polarization and the general cooperation level.

Figures (6)

• Figure 1: Coevolutionary game on multiplex networks. The multiplex network consists of two layers, termed as "relationship layer", and "interaction layer". Each node within both layers corresponds to the same agent. In the relationship layer, weighted edges denote intensity of the association between agents. Edges in the alternative layer denote proper interactions among agents in the from of cooperation or defection. According to the coevolutionary rule, a link in the relationship layer will affect whether the game occurs in the interaction layer, and the outcomes of this game also effects the sate of the link in the relationship layer (color online).
• Figure 2: Cooperation heatmaps for four networks. The $f_C$ density of cooperators is shown on the $b-m$ parameter plane at $p=0.9$ for honeycomb lattice (a), square lattice (b), hexagonal lattice (c) and WS small-world graph (d). The meaning of colors is explained in the right-hand side legend for each panel. It is generally valid that the cooperation level decays as we simultaneously increase the temptation value and the relative strength of relationship index in the fitness function. The stationary value of $f_C$ was averaged over 500 MC steps after 5500 relaxation steps in a graph containing $N=2500$ nodes (color online).
• Figure 3: Cooperation level in dependence on parameter $b$ and $p$ for four networks. Panel (a): $f_c$ as a function of $b$ at fixed $m=0.50$ and $p=0.90$ values. Panel (b): $f_c$ as a function of $p$ at fixed $b=1.50$ and $m=0.50$ (b). Curves with different colors represent different topology of relationship layer as indicated in the legend: HL (honeycomb), SL (square), XL (hexagonal lattice), and WS (small-world).
• Figure 4: Clustering of cooperators when using square lattice topology in relationship layer. On the upper row we use $b=1.5$ and $m=0.5$ for all cases at different $p$ values, which are $p=0.5$ (a), 0.75 (b), and 1 (c). On the lower row we apply $p=0.9$ and $m=0.5$ at various temptation levels. They are $b=1.3$ (d), 1.5 (e), and 1.7 (f). Cooperators are marked by blue while defectors are denoted by white color in a lattice containing 2500 players. The upper low indicates that we need a large $p$ value to spatial reciprocity work. For moderate $p$ values large cooperator islands cannot form. The lower row illustrates that the cooperation level remains reasonable even at $b = 1.5$ and the major decay of cooperation happens for larger $b$ values. Below this parameter condition cooperators can maintain their tight formation but above this the change is significant. This particular behavior, exclusively characterizes square lattice, separates other cases observed for alternative topologies where the temptation value has less critical role on the cooperation level.
• Figure 5: The mean of relationship index $(\bar{A})$ plot in dependence on $m$ and $b$ for four networks. We demonstrate the influence of parameter $m$ on the mean of the relationship index for four distinct network structures at $p=0.9$. Different colors indicate different network type as indicated in the legend. They are honeycomb (HL), square lattice (SL), hexagonal lattice (XL) and small-world graph (WS). Solid, dash-dot, and dashed lines respectively correspond to $b=1.0$, 1.5, and 2.0 values.