Cooperation Prevails When Individuals Adjust Their Social Ties

Citation: Santos FC, Pacheco JM, Lenaerts T ( Cooperation Prevails When Individuals Adjust Their Social Ties Francisco C. Santos 0 1 2 Jorge M. Pacheco 0 1 2 Tom Lenaerts 0 1 2 0 Editor: Luis Amaral, Northwestern University , United States of America 1 1 Computer and Decision Engineering Department, Institut de Recherches Interdisciplinaires et de De veloppements en Intelligence Artificielle, Universite Libre de Bruxelles , Brussels , Belgium , 2 Program for Evolutionary Dynamics, Harvard University , Cambridge, Massachusetts , United States of America, 3 Department of Physics of the Faculty of Science, Center for Theoretical and Computational Physics, University of Lisbon , Lisbon, Portugal, 4 SWITCH Laboratory , Flanders Interuniversity Institute for Biotechnology, Vrije Universiteit Brussel, Brussels, Belgium, 5 Department of Computer Science, Vrije Universiteit Brussel , Brussels , Belgium 2 PLoS Computational Biology 3 www.ploscompbiol.org Conventional evolutionary game theory predicts that natural selection favours the selfish and strong even though cooperative interactions thrive at all levels of organization in living systems. Recent investigations demonstrated that a limiting factor for the evolution of cooperative interactions is the way in which they are organized, cooperators becoming evolutionarily competitive whenever individuals are constrained to interact with few others along the edges of networks with low average connectivity. Despite this insight, the conundrum of cooperation remains since recent empirical data shows that real networks exhibit typically high average connectivity and associated single-to-broadscale heterogeneity. Here, a computational model is constructed in which individuals are able to self-organize both their strategy and their social ties throughout evolution, based exclusively on their self-interest. We show that the entangled evolution of individual strategy and network structure constitutes a key mechanism for the sustainability of cooperation in social networks. For a given average connectivity of the population, there is a critical value for the ratio W between the time scales associated with the evolution of strategy and of structure above which cooperators wipe out defectors. Moreover, the emerging social networks exhibit an overall heterogeneity that accounts very well for the diversity of patterns recently found in acquired data on social networks. Finally, heterogeneity is found to become maximal when W reaches its critical value. These results show that simple topological dynamics reflecting the individual capacity for self-organization of social ties can produce realistic networks of high average connectivity with associated single-to-broad-scale heterogeneity. On the other hand, they show that cooperation cannot evolve as a result of ''social viscosity'' alone in heterogeneous networks with high average connectivity, requiring the additional mechanism of topological co-evolution to ensure the survival of cooperative behaviour. - Conventional evolutionary game theory predicts that natural selection favours the selfish and strong [1], in spite of existing evidence showing that cooperation is more widespread than theory predicts [2]. When cooperation is modelled in terms of the prisoners dilemma [3] (PD), the solution of the replicator dynamics equation in infinite, wellmixed populations [46] dictates the extinction of cooperators by defectors. Cooperators become evolutionarily competitive, however, whenever individuals are constrained to interact with few others along the edges of sparse graphs as recently concluded in two independent studies [7,8]. Both studies place individuals on the nodes of a static graph, and associate their social ties with the vertices linking the nodes such that, throughout evolution, every individual has the possibility of changing her strategy, but not her social ties. In [7] it has been shown that, under strong selection (fitness is determined by the game payoff), heterogeneous graphs lead to a significant increase in the overall survivability of cooperation, modelled in terms of the most popular social dilemmas, played on networks of different degrees of heterogeneity [9]. For the classical PD in which the act of cooperation involves a cost c to the provider, resulting in a benefit b (b . c) for the recipient, a simple relation has been obtained in [8] for a single cooperator to have a chance to survive in a population of defectors, whenever selection is weak (game payoff introduces a small perturbation onto fitness): b/c . z, where z stands for the average number of ties each individual has (z is the average degree of the graph). Both studies show that games on graphs open a window for the emergence of cooperation, showing how social viscosity alone [8] can contribute to the emergence of cooperation. However, recent data shows that realistic networks [1016] exhibit average connectivity values ranging from 2 to 170, with an associated heterogeneity intermediate between single-scale and broad-scale [11], which differs from the connectivity values typically used in previous studies [7,8]. For instance, the network of movie actors exhibits an average connectivity of 30 [17], whereas collaboration networks based on co-authorship of published papers vary from average values of 4 (mathematics), to 9 (physics) up to 15 (biology) [13]. In terms of the simple rule for the evolution of In social networks, some individuals interact with more people and more often than others. In this context, one may wonder: under which conditions are social beings willing to be cooperative? Current models proposed in the context of evolutionary game theory cannot explain cooperation in communities with a high average number of social ties. Santos, Pacheco, and Lenaerts show that when individuals are able to simultaneously alter their behaviour and their social ties, cooperation may prevail. Moreover, the structure of the final networks corresponds to those found in empirical data. Their article concludes that the more individuals interact, the more they must be able to promptly adjust their partnerships for cooperation to thrive. Consequently, to understand the occurrence of cooperative behaviour in realistic settings, both the evolution of the complex network of interactions and the evolution of strategies should be taken into account simultaneously. cooperation for graphs, the reported values of z require benefits to often exceed costs by more than one order of magnitude for a single cooperator to survive [8]. None of the previous results on strong [7] and weak [8] selection on graphs is capable of explaining how cooperation thrives on such social networks. Other mechanisms have to be at work here that allow for the survival of cooperation. In most evolutionary models developed so far, social interactions are fixed from the outset. Such immutable social ties, associated naturally with static graphs, imply (...truncated)