Evolutionary Dynamics with Self-Interaction Learning in Networked Systems

TL;DR

This paper investigates how self-interaction learning, implemented as degree-correlated self-loops, shapes the evolution of cooperation in networked two-player donation games under weak selection. The authors introduce a self-interaction landscape and derive closed-form conditions for cooperation, $(b/c)^*$, across graph families (regular graphs, stars, hub-hub joined stars, and ceiling fans), showing that appropriately tuned self-interactions can lower the cooperation threshold, especially in high-degree networks, and can prevent spite in spite-favoring regimes. Through extensive simulations on regular, random, and real networks, they show that self-interaction landscapes based on $\ln k$, $(k+1)^{-1}$, or $1-k^{-1}$ substantially reduce the required $b/c$, while $e^{-k}$ offers little help in high-degree networks. The results highlight a practical mechanism—self-reinforcement via self-loops—to promote cooperation in complex networks and suggest directions for extending to higher-order or temporal networks.

Abstract

The evolution of cooperation in networked systems helps to understand the dynamics in social networks, multi-agent systems, and biological species. The self-persistence of individual strategies is common in real-world decision making. The self-replacement of strategies in evolutionary dynamics forms a selection amplifier, allows an agent to insist on its autologous strategy, and helps the networked system to avoid full defection. In this paper, we study the self-interaction learning in the networked evolutionary dynamics. We propose a self-interaction landscape to capture the strength of an agent's self-loop to reproduce the strategy based on local topology. We find that proper self-interaction can reduce the condition for cooperation and help cooperators to prevail in the system. For a system that favors the evolution of spite, the self-interaction can save cooperative agents from being harmed. Our results on random networks further suggest that an appropriate self-interaction landscape can significantly reduce the critical condition for advantageous mutants, especially for large-degree networks.

Figures (10)

• Figure 1: Critical thresholds for cooperation in sparse and dense regular graphs with self-interaction. (a) For sparse regular graphs with $N>2k$ cooperation can always be favored. The network parameters are $(N,k)\in\left\{(50,2),(100,2),(50,3),(100,8)\right\}$. (b) For sense regular graphs with $N<2k$ spite is always favored to win over cooperation. The network parameters are $(N,k)\in\left[(50,30),(100,60),(50,40),(100,80)\right]$. Each transition point of $\ell(k)$ from spite to cooperation is marked with arrows by the corresponding color of the legend. (c) and (d) Phase diagrams of cooperation and spite with $N=50$ and $N=70$, respectively. The yellow and blue areas indicate cooperation and spite respectively. (color online)
• Figure 2: Fixation probabilities and conditions for cooperation in lattices with specific self-interaction function. (a) Topology structures of the hexagonal lattice ($k=3$), the square lattice ($k=4$), and the triangular lattice ($k=6$). The periodic boundaries are not shown here but are considered in the simulations. (b)-(d) Fixation probabilities $N\times(\rho_C-\rho_D)$ as the increase of $b/c$ and the variation of self-loop strength functions. Here we expand $\rho_C-\rho_D$ by $N$ times to unify the scale of the vertical axis. The system sizes for (b), (c), and (d) are $72$, $100$, and $98$ respectively. Each data point marks the ratio of cooperation fixation over $2\times10^6$ independent runs, fitted by linear regression. The vertical lines indicate the theoretical results presented as Eq. \ref{['Eq: bcr regular']}. The grey arrow in each panel is the condition for cooperation without self-interaction, which is $(N-2)/(N/k-2)$. (color online)
• Figure 3: Evolution of cooperation and spite in stars with self-interaction.$\alpha$ and $\beta$ denotes the self-interaction strength of hub and leaves, respectively. (a) Self-interaction only applies to the hub vertex, i.e., $\alpha=0$. The evolution of spite is always favored in this case, which is harmful to the system. (b) Self-interaction only applies to the leaf vertices, i.e., $\beta=0$. The system can always favor the evolution of cooperation. The sizes of stars are $N\in\{5,7,9,11\}$ as indicated in the legend. (c) and (d) Phase diagrams of the cooperation (yellow) and spite (blue) with $N=15$ and $N=25$, respectively. (color online)
• Figure 4: Fixation probabilities and conditions for cooperation in stars. (a) An example of a star. The hub vertex has the greatest degree $N-1$, while each leaf has the degree $1$. (b) The fixation probability for different self-interaction strength of hub and leaves $(\alpha,\beta)\in\{(1.0,2.0),(0.5,2.0),(0.5,0.0), (0.4, 1.0)\}$. The system size is $N=10$. The vertical lines present the theoretical results in Eq. \ref{['eq: thm2 star']}. The condition for cooperation without self-interaction is $\infty$. Each data point is the ratio of cooperation fixation in $2\times10^6$ independent runs. (color online)
• Figure 5: Evolution of cooperation in hub-hub joined stars with self-interaction. $\alpha$ and $\gamma$ indicate the self-interaction strength for leaves and hubs, respectively. (a) The self-interaction only applies to two hub vertices with $\alpha=0$. (b) The self-interaction only applies to the leaf vertices with $\gamma=0$. The system sizes are $N\in\left\{5,7,9,11\right\}$. The hub-hub joined stars always favor cooperation and never favor spite. (c) and (d) Heatmaps of cooperation conditions with $N=15$ and $N=25$, respectively. The red area indicates that the requested condition of cooperation is higher. (color online)

Theorems & Definitions (6)

• Definition 1
• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4