Social dilemmas, network reciprocity, and small-world property

We revisit two evolutionary game theory models, namely the Prisoner and the Snowdrift dilemmas, on top of small-world networks. These dynamics on networked populations (individuals occupying nodes of a graph) are mainly concerned with the competition between cooperating or defecting by allowing some process of revision of strategies. Cooperators avoid defectors by forming clusters in a process known as network reciprocity. This defense strategy is based on the fact that any individual interacts only with its nearest neighbors. The minimum cluster, in turn, is formed by a set of three completely connected nodes, and the bulk of these triplets is associated with the transitivity property of a network. We show that the transitivity increases eventually, assuming a constant behavior when observed as a function of the number of contacts an individual has. We investigate the influence of the network reciprocity on that transitivity-increasing regime on promoting cooperative behavior. The dynamics of small-world networks are compared with those of random regular and annealed networks, the latter typically studied as the well-mixed approach. The Snowdrift Game converges to an annealed scenario as randomness and coordination numbers increase. In contrast, the Prisoner’s Dilemma becomes more severe against the cooperative behavior under an increasing network reciprocity regime.