Szolnoki & Perc (2009) Resolving social dilemmas on evolving random networks

Szolnoki, A., & Perc, M. (2009). Resolving social dilemmas on evolving random networks. Eur Phys Lett, 86, 30007.

The evolution on networks is increasingly often accompanied by the evolution of networks.

Here, we aim to show that simple coevolutionary rules affecting the interaction network may lead not just to heterogeneous states promoting the cooperative strategy, but also to new dynamical processes that positively affect the evolution of cooperation.

Start with a random interaction network and introduce a coevolutionary rule entailing both deletions of existing and additions of new links between players.

While existing links are deleted whenever a player adopts a new strategy or its degree exceeds a threshold value, new links are added after each given number τ of game iterations (this is a time scale parameter).

The random topology and heterogeneity of the network are largely preserved, so that the resolution of social dilemmas is due the Red Queen mechanism and group selection, the appearance of which depend on τ.

Methods:

• Consider payoff matrices for PD, snowdrift, and stag hunt games. 
• Players start either C or D with equal prob, on random network with avg connectivity 4. 
• Strategy evolution:
• Randomly selected player x acquires payoff p_x by playing with all players. One randomly selected neighbor or x, y, also gets payoff p_y. 
• If p_x > p_y, y takes x’s strategy with probability (p_x – p_y)/bk_q, where k_qis the largest of the two degrees of both players. 
• Network updating:
• Whenever player x adopts a new strategy, all its links are deleted except the one with the donor of the new strategy. 
• All individuals form a new link with a randomly chosen player with whom they are not already connected every τ Monte Carlo steps. 
• When k_x reaches k_max (“aging”), it is replaced with a new player with the same strategy and one randomly selected link. 
• If a player accidentally gets totally detached, he is relinked with a random single connection. 

Results:

• For τ = 1, cooperation is highly constrained in the PD game. In the snowdrift game, cooperation either dominates or has a mixed frequency, with some oscillatory states. 
• For τ = 500, there are no mixed states, and cooperation does very well in many circumstances, even in the PD game. 
• In the snowdrift game, there is a steady mixed state for τ = 1. As τ increases, cooperators dominate. As it continues to increase, there is a step-wise increase with periods of stable cooperator frequency (of the same length as τ). This allows the emergence of homogenous and virtually isolated groups of players. These groups remain inactive until they are reconnected. They argue that this is a group selection effect. 

NOTE: The same model is more thoroughly analyzed just for the PD game in:

Szolnoki, A. & Perc, M. (2009). Emergence of multilevel selection in the prisoner’s dilemma game on coevolving random networks. New J. Phys, 11, 093033.