Source-linked AI summary
Self-organization towards optimally interdependent networks by means of coevolution
Zhen Wang, Attila Szolnoki, Matjaz Perc
TL;DR
The paper asks how beneficial interdependence between networks can emerge rather than being assumed. It coevolves strategy, teaching activity, and external-link eligibility in the prisoner’s dilemma, finding self-organized interdependence, a two-class society, and stronger cooperation under adverse conditions.
Problem
Previous work identified an optimal intermediate interdependence level but did not explain how it could emerge from initially isolated networks.
Method
The model lets players adjust teaching activity after strategy-transfer outcomes and form external links when activity exceeds a threshold, while linked players gain utility without cross-network strategy transfer.
Results
Cooperators can match defectors even for b > 1.3, whereas an earlier isolated-network model failed to observe cooperation for b > 1.18.
Takeaways & Limitations
Interdependence self-organizes so approximately half of the players form external links, producing conditions favorable to resolving the prisoner’s dilemma.
Takeaways & Limitations
The beneficial feedback requires sufficiently notable success rewards and failure punishments to act rapidly enough to affect the social dilemma.
Abstract
from arXiv · showhide
Coevolution between strategy and network structure is established as a means to arrive at optimal conditions for resolving social dilemmas. Yet recent research highlights that the interdependence between networks may be just as important as the structure of an individual network. We therefore introduce coevolution of strategy and network interdependence to study whether it can give rise to elevated levels of cooperation in the prisoner's dilemma game. We show that the interdependence between networks self-organizes so as to yield optimal conditions for the evolution of cooperation. Even under extremely adverse conditions cooperators can prevail where on isolated networks they would perish. This is due to the spontaneous emergence of a two-class society, with only the upper class being allowed to control and take advantage of the interdependence. Spatial patterns reveal that cooperators, once arriving to the upper class, are much more competent than defectors in sustaining compact clusters of followers. Indeed, the asymmetric exploitation of interdependence confers to them a strong evolutionary advantage that may resolve even the toughest of social dilemmas.
1. Introduction
Research on cooperation has increasingly shifted from individual network structure toward multiplex and interdependent networks. The paper addresses how optimal interdependence could emerge from initially isolated networks rather than being assumed.
- Interdependence between networks may be as important for system functioning as the structure of an individual network.
- Evolutionary-game research has established that interaction-network structure strongly influences cooperation, including through small-world, scale-free, bipartite, and coevolving networks.
- Previous studies found that an intermediate level of interdependence best deters defection but generally assumed that interdependence already existed.
- The paper introduces a rule in which players with sufficiently successful strategy transmission may form external links to corresponding players in another network.
- The proposed rule is intended to produce spontaneous optimal interdependence and a two-class society in the prisoner’s dilemma.
- The paper describes the game, presents results, and concludes by discussing their implications.
2. Model
The model places cooperators and defectors on two initially independent square lattices, where strategy transmission and external-link formation coevolve. Utilities include benefits from external links, while strategies still spread only through nearest-neighbor interactions within each lattice.
- Players occupy two independent periodic square lattices, each initially connected to four nearest neighbors and randomly assigned cooperation or defection.
- Players receive prisoner’s dilemma payoffs from interactions with their four same-network neighbors, using temptation T = b and mutual-cooperation reward R = 1.
- Fitness utilities combine local payoff πx with cross-network payoff απx′ when an external link connects corresponding players.
- Strategies transfer only between nearest neighbors on the same lattice, never across external links.
- The Fermi strategy-adoption probability incorporates uncertainty controlled by K.
- Teaching activity starts at wmin = 0.01, changes by ±∆ after strategy-transfer success or failure, and permits an external link only when wx ≥ wth.
- Random sequential simulations vary system size from L = 200 to 400 and run for up to 10^5 Monte Carlo steps to reduce finite-size effects and equilibrate the system.
- Figure 1 maps cooperation and eligible-link fractions against wth and ∆, comparing coupled networks with uncoupled networks across specified temptation values.
3. Results
Coevolution produces a sharp threshold beyond which network interdependence self-organizes and cooperation is strongly promoted. This outcome reflects a two-class society in which upper-class cooperators sustain compact clusters more effectively than defectors.
- A sharp threshold in ∆ promotes cooperation across temptation-to-defect values, with intermediate wth becoming increasingly important as b increases.The required ∆ decreases slightly as b decreases, while the suitable interval of wth narrows as the temptation to defect rises.
- The fraction of cooperators closely correlates with the fraction of players having w ≥ wth, indicating that self-organized interdependence supports elevated cooperation.Approximately half the players being allowed to form external links is identified as the favorable regime.
- For ∆=0.05, no player reaches wth=0.5, whereas for ∆=0.4 the population segregates mainly into w=wmin and w=1 classes.The intermediate class is practically absent, with P(w)<0.1, and only the upper class gains external links.
- Upper-class cooperators build compact follower clusters, while upper-class defectors exploit neighbors and are downgraded by negative feedback.This asymmetric use of interdependence gives cooperators an evolutionary advantage in sustaining cooperation.
- When ∆ is below threshold, segregation and interdependence are weak, leaving cooperators confined to isolated, relatively small regions.Once ∆ exceeds the threshold, cooperators can spread across the network.
- Randomly assigning external links weakens the increase in ρC and shifts it to larger ∆ values compared with corresponding-player links.The comparison indicates that disrupting interdependent network reciprocity impairs the cooperative outcome.
4. Discussion
The coevolutionary rule spontaneously produces an approximately optimal interdependence between initially independent networks, improving cooperation through a two-class structure and cooperative clustering. Its effectiveness depends on corresponding cross-network links and exceeds that of coevolving teaching activity on isolated networks.
- Approximately half of the players form external links, matching the manually identified optimum for deterring defection while emerging spontaneously from initially independent networks.The remaining players are denied participation in activities beyond their host network.
- The rule produces a two-class society in which the upper class controls interdependence and gains increased utility, while the lower class receives no interdependence benefits.The middle class is practically non-existent.
- Cooperators exploit upper-class status more effectively than defectors by forming compact clusters on both networks reinforced through interdependent network reciprocity.Defectors tend to exploit neighbors until they are downgraded, whereas cooperative clusters reinforce themselves.
- External-link coevolution is essential for the segregated society, because coevolving teaching activity alone cannot produce the optimal two-class structure within a network.Randomly chosen rather than corresponding cross-network links significantly impair cooperation.
- Cooperators can match defectors for b > 1.3, whereas isolated-network coevolution failed above b > 1.18 and an isolated square lattice had critical temptation b(K = 0.1) = 1.0357.Building on interdependence is reported as significantly more effective than coevolving teaching activity alone.
- The rule models awarding success and punishing failure and demonstrates that this elementary ingredient can produce healthy interdependence between initially independent populations.The authors identify simultaneous emergence of network structure and interdependence as a direction for further research.