Hierarchical invasion of cooperation in complex networks

Daniele Vilone1,2, Valerio Capraro3,4 and José J. Ramasco5
1LABSS (Laboratory of Agent Based Social Simulation), Institute of Cognitive Science and Technology, National Research Council (CNR), Via Palestro 32, 00185 Rome, Italy.
2Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matematicas, Universidad Carlos III de Madrid, 28911 Leganes, Spain.
3Center for Mathematics and Computer Science (CWI), 1098 XG, Amsterdam, The Netherlands.
4Business School, Middlesex University London, NW44BT, London, United Kingdom.
5Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Palma de Mallorca, Spain.

(February 2018)

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.