Adaptive Payoff-driven Interaction in Networked Snowdrift Games

TL;DR

An adaptive network model in the snowdrift game is introduced to examine dynamic levels of cooperation and network topology, involving the potential for both the termination of existing connections and the establishment of new ones, and reveals that adaptive networks are particularly effective in promoting a robust evolution toward states of either pure cooperation or complete defection.

Abstract

In social dilemmas, most interactions are transient and susceptible to restructuring, leading to continuous changes in social networks over time. Typically, agents assess the rewards of their current interactions and adjust their connections to optimize outcomes. In this paper, we introduce an adaptive network model in the snowdrift game to examine dynamic levels of cooperation and network topology, involving the potential for both the termination of existing connections and the establishment of new ones. In particular, we define the agent's asymmetric disassociation tendency toward their neighbors, which fundamentally determines the probability of edge dismantlement. The mechanism allows agents to selectively sever and rewire their connections to alternative individuals to refine partnerships. Our findings reveal that adaptive networks are particularly effective in promoting a robust evolution toward states of either pure cooperation or complete defection, especially under conditions of extreme cost-benefit ratios, as compared to static network models. Moreover, the dynamic restructuring of connections and the distribution of network degrees among agents are closely linked to the levels of cooperation in stationary states. Specifically, cooperators tend to seek broader neighborhoods when confronted with the invasion of multiple defectors.

Figures (6)

• Figure 1: A schematic representation of the adaptive mechanism: evolution of network state from $t$ to $t+1$. Here, cooperative nodes are visually denoted by the color blue, whereas defectors are represented in red. We partition the evolutionary process into two distinct phases: the strategy updating and the network updating, elucidating its underlying principles and mechanisms. During the strategy updating, agents adjust their strategies based on the accrued payoffs $\pi_{ij}$, while in the network updating, agents rewire their connections driven by the disassociation tendency $D_{ij}$. In the network updating phase, gray dashed lines indicate the connections to be severed, while red lines represent newly formed connections following the adjustment of adverse neighbor relationships.
• Figure 2: Time evolution of $f_c$ for various $r$ values under different initial network configurations. The evolution of $f_c$ is examined for various values of $r$ with respect to the number of evolution iterations depicted for initial network structures of RG (a) and WS (b), where solid and dashed lines correspond to $k=4$ and $k=10$ respectively. Different colors in the curves indicate distinct values of $r$ as illustrated in the legend. Across all initial topologies, cooperation flourishes predominantly at lower values of $r$, with evolution stabilizing at approximately 200 steps.
• Figure 3: The $f_c$ is investigated as a function of $r$ ranging from 0 to 1.0 for both adaptive and static networks. It is explored how $f_c$ varies with $r$ under a consistent initial network structure. Essentially, the initial topological structure of the adaptive network mirrors that of the static network, yet the adaptive network undergoes experiences co-evolution according to the pre-defined adaptive rules. Significant variability in $f_c$ between different $r$ values and network topologies is observed, with distinct trends emerging at lower and higher $r$ values, which highlights the enhanced adaptability of dynamic networks to foster cooperation under varying environmental constraints compared to rigid static networks.
• Figure 4: The proportions of C-C, C-D, and D-D edge types vary across different networks and $r$ values. The upper row of the network structures depicts RG for all cases across various $r$ values: $r = 0.2$ (a), 0.4 (b), and 0.6 (c). Consequently, the lower row applies WS with the same values for $r$ as the upper row: $r = 0.2$ (d), 0.4 (e), and 0.6 (f). The solid lines represent an average degree of the network of $k=4$, while the dashed lines represent $k=10$. Blue, orange, and red represent the C-C, C-D, and D-D edge types, respectively. Upon contrasting the two sets of sub-figures, the initial network structures exhibit an insignificant influence on edge types. The first column indicates that, under a scenario dominated by cooperation, the C-C type edges increase rapidly, while the proportion of D-D type edges decreases faster than that of C-D type edges. Results from the second column suggest that when the ratios of cooperation and defection are similar, the proportion of C-D type edges remains almost constant, with a slight increase in C-C edges and a corresponding decrease in D-D edges. The third column implies that, under a predominance of defectors, there is an increase in the proportion of D-D edges, which become nearly equal to C-D type edges, and noticeably, networks with $k=4$ and $k=10$ exhibit significant differences.
• Figure 5: Degree distributions in stationary RG with varying average degrees of connectivity. It displays the degree distributions for nodes in stationary state networks, differentiated by the average degrees of 4 and 10 within RG. Specifically, (a) presents a network with an initial average degree of $4$ and a cooperation rate of $0.2$, while (b) presents a network with an initial average degree of $10$ and the same cooperation rate. Moving to (c) and (d), the cooperation rate remains constant at $0.4$, but the initial average degree changes to $4$ and $10$, respectively. Similarly, (e) and (f) maintain a cooperation rate of $0.6$, with initial average degrees of $4$ and $10$, respectively. Each panel delineates the degree distributions for cooperators (blue) and defectors (red), along with all agents (gray).