Topological conditions of scale-free networks for cooperation to evolve

Evolutionary game theory is employed to study topological conditions of scale-free networks for the evolution of cooperation. We show that Apollonian Networks (ANs) are perfect scale-free networks, on which cooperation can spread to all individuals, even though there are initially only 3 or 4 hubs occupied by cooperators and all the others by defectors. Local topological features such as degree, clustering coefficient, gradient as well as topology potential are adopted to analyze the advantages of ANs in cooperation enhancement. Furthermore, a degree-skeleton underlying ANs is uncovered for understanding the cooperation diffusion. Constructing this kind degree-skeleton for random scale-free networks promotes cooperation level close to that of Barab\'asi-Albert networks, which gives deeper insights into the origin of the latter on organization and further promotion of cooperation.