Source-linked AI summary
Interdependent network reciprocity in evolutionary games
Zhen Wang, Attila Szolnoki, Matjaz Perc
TL;DR
The paper examines how interconnectedness between networks affects cooperation in public goods games. Using a symmetric utility coupling across two networks, it identifies interdependent network reciprocity, whose success depends on coordinated cooperation on both networks.
Problem
The paper asks how interactions between networks, beyond within-network structure, affect the evolution and survivability of cooperation.
Method
The authors study public goods games on two interconnected square-lattice networks, using a network-symmetric utility function while restricting strategy imitation to players within the same network.
Results
The study identifies spontaneous interdependent network reciprocity, which can produce higher cooperation than traditional single-network reciprocity when cooperative clusters form and remain coordinated across both networks.
Takeaways & Limitations
Interdependence can promote cooperation beyond isolated-network reciprocity, but disrupting coordination on one network causes cooperation to collapse across both.
Abstract
from arXiv · showhide
Besides the structure of interactions within networks, also the interactions between networks are of the outmost importance. We therefore study the outcome of the public goods game on two interdependent networks that are connected by means of a utility function, which determines how payoffs on both networks jointly influence the success of players in each individual network. We show that an unbiased coupling allows the spontaneous emergence of interdependent network reciprocity, which is capable to maintain healthy levels of public cooperation even in extremely adverse conditions. The mechanism, however, requires simultaneous formation of correlated cooperator clusters on both networks. If this does not emerge or if the coordination process is disturbed, network reciprocity fails, resulting in the total collapse of cooperation. Network interdependence can thus be exploited effectively to promote cooperation past the limits imposed by isolated networks, but only if the coordination between the interdependent networks is not disturbed.
Results
Increasing neighbor-payoff weight promotes cooperation, while interdependent network reciprocity requires coordinated cooperative domains and synchronous evolution across networks. When coordination is disrupted, cooperation can collapse; the mechanism also extends to three symmetrically interdependent networks.
- Results: 0.748 to 0.33: the minimally required r/G for cooperator survival drops as α = β increases from 0 to 0.45.The threshold for complete cooperator dominance also drops from r/G = 1.09 to 0.35.
- Results: At fixed β, increasing α elevates ρC without changing the interdependence level, whereas β’s benefit depends strongly on α and r/G.Higher α supports compact cooperative domains, which makes increasing β more effective.
- Results: Cooperator and defector pair densities on the two networks evolve synchronously, with only small quantitative differences.The pair probabilities Pi and Pe measure internal and external configurations, respectively.
- Results: Mixing strategies on one network disrupts synchronization and drives cooperation to extinction on the undisturbed network despite nonzero within-network coupling.Without the disturbance, cooperators can spread even where traditional single-network reciprocity fails.
- Results: Only cooperative domains initially present on both networks survive early defector pressure and spread across both networks.Uniformly random cooperators and clusters present on only one network die out rapidly.
- Results: For three symmetrically interdependent networks, interdependent reciprocity becomes most effective after α surpasses approximately 0.17.Increasing α or β elevates ρC, and spontaneous coordination and synchronization remain possible beyond two networks.
Discussion
The study identifies interdependent network reciprocity under a network-symmetric utility coupling. Cooperation is promoted when corresponding players form compact cooperative domains on both networks, but fails when clustering is absent on either network.
- The study uses a network-symmetric utility definition to examine cooperation on interconnected networks.
- Increasing nearest-neighbor average-payoff relevance improves cooperator survivability, while cross-network payoff coupling produces interdependent network reciprocity.
- Interdependent reciprocity requires cooperative players on both networks to aggregate into corresponding compact domains.
- If cooperative clustering fails on either network, network reciprocity ultimately fails on both networks.
Methods
The model stages public goods games on two square-lattice networks and introduces interdependence through a utility function combining local and cross-network payoffs. Strategy imitation remains within each host lattice, while simulations use random sequential updating.
- Public goods games are staged on L×L square lattices with periodic boundaries and overlapping groups of size G = 5.Players contribute 1 or nothing, the group total is multiplied by r > 1, and the result is shared equally.
- The two networks are not physically connected; interdependence is introduced through a utility function.
- The utility combines individual payoff, host-network neighbor averages, and corresponding-player payoff from the other network.The definition is network-symmetric, with no master-slave relation; α and β satisfy α + β < 1.
- Strategy imitation occurs only between nearest neighbors on the same lattice, never between players on different networks.The adoption probability is determined using the neighboring player's utility.
- Simulations use random sequential updating, with each player given one average strategy-update opportunity per Monte Carlo step.The linear system size ranges from L = 100 to 600, with equilibration requiring up to 10^5 Monte Carlo steps.