Hierarchical invasion of cooperation in complex networks

The emergence and survival of cooperation is one of the hardest problems still open in science. Several
factors such as the existence of punishment, repeated interactions, topological effects and the
formation of prestige may all contribute to explain the counter-intuitive prevalence of cooperation in
natural and social systems. The characteristics of the interaction networks have been also signaled as
an element favoring the persistence of cooperators. Here we consider the invasion dynamics of
cooperative behaviors in complex topologies. The invasion of a heterogeneous network fully occupied
by defectors is performed starting from nodes with a given number of connections (degree) k0. The
system is then evolved within a Prisoner's Dilemma game and the outcome is analyzed as a function of
k0 and the degree k of the nodes adopting cooperation. Carried out using both numerical and
analytical approaches, our results show that the invasion proceeds following preferentially a
hierarchical order in the nodes from those with higher degree to those with lower degree. However,
the invasion of cooperation will succeed only when the initial cooperators are numerous enough to
form a cluster from which cooperation can spread. This implies that the initial condition has to be a
suitable equilibrium between high degree and high numerosity. These findings have potential
applications to the problem of promoting pro-social behaviors in complex networks.