  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 concerning on
the competition between to cooperate or to defect, 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 interact 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. Particularly, we show that the transitivity increases eventually
assuming a constant behavior when observed as a function of the number of
contacts of an individual. We investigate the influence of the network
reciprocity on that transitivity increasing regime on the promotion of a
cooperative behavior. The dynamics on small-world networks are compared with
those random regular, and annealed networks, the later typically studied as the
well-mixed approach. We observe that the Snowdrift Game converge to an annealed
scenario as randonness and coordination number increase, whereas the Prisoner's
Dilemma becomes more severe against the cooperative behavior under the regime
of an increasing network reciprocity.
