Source-linked AI summary
Optimal interdependence between networks for the evolution of cooperation
Zhen Wang, Attila Szolnoki, Matjaz Perc
TL;DR
The paper examines how much interdependence between networks best promotes cooperation in evolutionary games. It finds that cooperation is optimized by an intermediate fraction of sufficiently strong links, through heterogeneity and asymmetric strategy flow around externally linked players.
Problem
The paper asks how much interdependence between networks is needed to promote cooperation in evolutionary games.
Method
The study models spatial prisoner’s dilemma and snowdrift games on two interconnected lattices, varying the fraction and strength of external links.
Results
An intermediate fraction of sufficiently strong links optimally promotes cooperation, while externally linked players exhibit higher cooperation through asymmetric strategy flow.
Takeaways & Limitations
Optimal interdependence arises from heterogeneity created by enhanced utility for externally linked players and the resulting asymmetric strategy flow.
Takeaways & Limitations
The games studied are not intended to model a particular real-life situation, and future work could examine more complex topologies and other games.
Abstract
from arXiv · showhide
Recent research has identified interactions between networks as crucial for the outcome of evolutionary games taking place on them. While the consensus is that interdependence does promote cooperation by means of organizational complexity and enhanced reciprocity that is out of reach on isolated networks, we here address the question just how much interdependence there should be. Intuitively, one might assume the more the better. However, we show that in fact only an intermediate density of sufficiently strong interactions between networks warrants an optimal resolution of social dilemmas. This is due to an intricate interplay between the heterogeneity that causes an asymmetric strategy flow because of the additional links between the networks, and the independent formation of cooperative patterns on each individual network. Presented results are robust to variations of the strategy updating rule, the topology of interdependent networks, and the governing social dilemma, thus suggesting a high degree of universality.
Results
Cooperation is promoted most effectively by sufficiently strong interdependence involving an intermediate fraction of players, because externally linked players become cooperative leaders and mutually connected networks support cooperative pattern formation.
- ρ ≈0.5 yields an optimal cooperation outcome across temptation-to-defect values, provided coupling strength α is sufficiently large.Beyond α = 0.7, cooperation may fade slightly at low or moderate temptation, but sufficiently strong coupling remains the prevailing condition.
- Externally linked distinguished players cooperate more than ordinary players because higher payoffs create an asymmetric strategy flow that makes them community leaders.Followers adopt strategies from these higher-payoff players, selecting cooperation around them.
- Reducing distinguished players’ teaching activity to w = 0.05 removes their cooperation advantage and eliminates the interdependence-driven promotion of cooperation.When ordinary players instead receive w = 0.05, the leadership of distinguished players and cooperation promotion are strengthened.
- The optimal mechanism requires distinguished players in both networks; unilateral links provide at most a marginal improvement, whereas bilateral links strongly promote cooperation.Independent cooperative pattern formation on each network and heterogeneity on both networks are necessary for mutual amplification.
- If distinguished players are too rare, increasing α cannot sustain cooperation because their influence cannot percolate through the population.At high temptation to defect, cooperation disappears below a density threshold even for large coupling strength.
- The intermediate-ρ optimum and qualitatively similar α dependence persist under best-takes-over and proportional-imitation updating, different topologies, and the snowdrift game.The same qualitative pattern appears on triangular and square lattices and across the tested social dilemmas.
Discussion
The study finds that cooperation is best promoted by bilateral interdependence involving an intermediate fraction of sufficiently influential links. This optimality is attributed to asymmetric strategy flow and mutually supporting cooperative patterns across networks, while the model’s real-world scope remains limited.
- Discussion: Optimal cooperation requires an intermediate fraction of links that significantly affect players’ utilities across the two networks.The authors attribute this to heterogeneity between externally linked and unlinked players, producing asymmetric strategy flow and influential cooperative hubs.
- Discussion: Externally linked players can become influential leaders and strong cooperative hubs because enhanced utility creates heterogeneity between players.The resulting asymmetric strategy flow supports the emergence of cooperative hubs in each network.
- Discussion: Independent compact cooperative patterns on both networks mutually support one another through the links connecting corresponding players.The mechanism works best when interdependence is bilateral.
- Discussion: The models are not intended to represent a particular real-life situation, although they capture aspects of situations where individuals differ in their opportunities or preferences for external connections.The authors identify more complex topologies, other games, coevolution, and growth as directions for future study.
Methods
The study simulates evolutionary games on two interdependent lattices, assigning strategies and external links randomly before calculating payoffs from local interactions. Strategy updating then uses utilities that include potential external-link contributions and a Fermi-based adoption process.
- Methods: The game is staged on two L × L square lattices, with players initially assigned cooperation or defection at equal probability.A randomly selected fraction ρ of players on each lattice may form external links with corresponding players in the other lattice.
- Methods: Payoffs follow standard social-dilemma rules, with the prisoner’s dilemma using T = b, R = 1, and P = S = 0.The temptation parameter satisfies 1 < b ≤2.
- Methods: Interdependence changes player utilities but does not transfer strategies between networks.Each player’s utility includes games with nearest neighbors and any additional external-link contribution.
- Methods: The Fermi strategy-adoption process uses K = 0.1 to quantify uncertainty from decision errors and imperfect information.A player compares utility with a randomly selected same-network neighbor before attempting strategy adoption.