Poncela et al (2011) Cooperation in scale-free networks with limited associative capacities

Ref: Poncela, J., et al. (2011) Cooperation in scale-free networks with limited associative capacities. Physical Review E, 83, 057101.

They consider scale-free (SF) networks (where the probability of finding an individual with k neighbors is P(k) ~ k^{-g}). However, most previous modes of this type have assumed that all individuals interact with all neighbors simultaneously. They propose that it is unlikely that a social hub will interact with all its neighbors in the evolutionary time window, and therefore one should abandon the hypothesis of unlimited associative capacity of individuals.

Method:

• Prisoner’s dilemma (PD) on a SF network. 
• Every node establishes a fixed number k* of interactions randomly chosen among its neighbors. 
• With and without cost for cooperation. 
• Test two updating rules: replicatorlike, and Fermi-like. 
• Fermilike probability function where x is self payoff minus other payoff: 

Results summary:

With a cost-per-cooperation, for both update rules there is an optimal number of interactions k*opt that renders larger values of cooperation than the unlimited scenario.

When no cost per cooperation, cooperation increases with k*, but this growing behavior saturates at values well below the maximum possible value – it is therefore not necessary to exploit the full associative capacity of network nodes to achieve the large levels of cooperation seen in SF networks.

Results detail:

Run for a long time, simulation always went to All-D or All-C. is the probability of reaching all-C.
For low values of b/c (cooperation is costly), the best is when k* = k_max. However, for larger values of b/c, the opposite trend occurs.


Summary: When cooperation is cheap, it’s better to limit associative capacities. But, for too restrictive values (about k* < 10), this is worse than the max values.

For no cooperation costs (P = S = 0, R = 1, T = b), increases with k*.

• the graphs imply that the optimum value k*opt was previously observed due to the compromise for each node between the costs associated to cooperate with all its neighbors and the benefits obtained in those interactions. 
• The curves converge to maximum values to lower values of k*, implying that you don’t need all the connectivity in the SF networks. 

Take home: In a scale-free network, limiting the social interactions increases the chance that cooperation will evolve. When a cooperator doesn’t contribute as much, and the cost of cooperation is high, then it literally loses more, and each receipt of cooperation counts for more. This is a very different mechanism than our own.