Temporal Interaction and its Role in the Evolution of Cooperation

TL;DR

The paper investigates how time-varying interactions influence cooperation in spatial public goods games by introducing two temporal participation mechanisms: stochastic activation with probability $p$ and periodic activation with rate $\lambda$ plus a time lag $\epsilon$. Using an $L^2$ lattice with von Neumann neighbourhood, the study shows there is an optimal, intermediate activation probability that maximizes cooperation, and that local synchronization within regions supports dense cooperative clusters while temporal asynchrony impedes cross-structure spread. The results are robust across network topologies and group sizes, with higher clustering and larger group sizes generally promoting cooperation, and noise level $K$ modulating the strength of these effects. These findings offer insights into fostering cooperation in social and information networks and suggest timing-based strategies for coordinating collective action in real-world systems.

Abstract

This research investigates the impact of dynamic, time-varying interactions on cooperative behaviour in social dilemmas. Traditional research has focused on deterministic rules governing pairwise interactions, yet the impact of interaction frequency and synchronization in groups on cooperation remains underexplored. Addressing this gap, our work introduces two temporal interaction mechanisms to model the stochastic or periodic participation of individuals in public goods games, acknowledging real-life variances due to exogenous temporal factors and geographical time differences. We consider that the interaction state significantly influences both game payoff calculations and the strategy updating process, offering new insights into the emergence and sustainability of cooperation. Our results indicate that maximum game participation frequency is suboptimal under a stochastic interaction mechanism. Instead, an intermediate activation probability maximizes cooperation, suggesting a vital balance between interaction frequency and inactivity security. Furthermore, local synchronization of interactions within specific areas is shown to be beneficial, as time differences hinder the spread of cross-structures but promote the formation of dense cooperative clusters with smoother boundaries. We also note that stronger clustering in networks, larger group sizes and lower noise increase cooperation. This research contributes to understanding the role of node-based temporality and probabilistic interactions in social dilemmas, offering insights into fostering cooperation.

Figures (10)

• Figure 1: Schematic of temporal interaction mechanism. (a) Active patterns of five individuals, represented by solid circles of various colours. Agents $x$ and $y$ engage in stochastic interactions, while agents $u$, $v$, and $w$ follow periodic interactions. (b) Illustrates game payoff scenarios: active cooperators and defectors as filled blue and red circles, inactive ones as light blue and red empty circles. In (b-1), only $w$ is active, leading to no game and zero gain. (b-2) represents the typical situation where the active cooperator $y$ contributes to public goods, and inactive ones receive a basic allowance.
• Figure 2: $r-p$ phase diagrams of the spatial public goods game as obtained for different inactive income $\sigma=0.5$ (left column) and $\sigma=1$ (right column). Subfigures (a) and (b) represent stochastic interaction, (c) and (d) represent periodic interaction. The red (blue) line represents the phase transitions between the mixed C+D and homogeneous D (C) states, and the solid (dashed) lines signify the continuous (discontinuous) phase transitions. The inserted colourmap $(^*-1)$ of each panel shows the equilibrium fraction of cooperators with $r$ and $p$. From red to blue, the colour bar indicates that the cooperation level changes from 0 to 1 accordingly. All results are obtained for $K=0.1$ and $t=10^5$.
• Figure 3: The cooperation level $\rho_c$ as a function of activation probability $p$ for various multiplication factors $r$ under a stochastic interaction mechanism. Results are shown for (a) $\sigma=0.5$ and (b) $\sigma=1$. In each subplot, four distinct curves correspond to $r$ values of 3.5, 3.75, 4, and 4.25, respectively.
• Figure 4: Evolutionary dynamics and spatial distributions of strategies under stochastic interaction mechanism. Column (*-1) illustrate the evolution process of strategy proportions over time, where the blue (red) line indicates the fraction of cooperation (defection) that contains active and inactive states, and (*-7) show the strategy fraction at the equilibrium state.
• Figure 5: The cooperation level $\rho_c$ as a function of activation probability $p$ for various multiplication factors $r$ under a periodic interaction mechanism. The results are presented for (a) $\sigma=0.5$ and (b) $\sigma=1$. In each subplot, the four curves correspond to $r=3.5,3.75,4$ and $4.25$, respectively.