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FCS, JMP, and TL conceived and designed the experiments, performed the experiments, analyzed the data, and wrote the paper.

The authors have declared that no competing interests exist.

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-broad–scale 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

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.

^{2}SYS, a Marie Curie Early Stage Training Site, funded by the European Commission through the Human Resources and Mobility activity. JMP acknowledges financial support from The Science and Technology Foundation—Portugal. TL acknowledges the support of Frederic Rousseau and Joost Schymkowitz from the SWITCH Laboratory of the Flanders Interuniversity Institute for Biotechnology.

Conventional evolutionary game theory predicts that natural selection favours the selfish and strong [

However, recent data shows that realistic networks [

In most evolutionary models developed so far, social interactions are fixed from the outset. Such immutable social ties, associated naturally with static graphs, imply that individuals have no control over the number, frequency, or duration of their ties; they can only evolve their behavioural strategy. A similar observation can be made on studies related to the physical properties of complex networks [

Using a minimal model that combines strategy evolution with topological evolution, and in which the requirements of individual cognitive capacities are very small, we investigate under which conditions cooperation may thrive. Network heterogeneity, which now emerges as a result of an entangled co-evolutionary dynamics, will be shown to play a crucial role in facilitating cooperative behaviour.

Let us consider two types of individuals—cooperators and defectors—who engage in several of the most popular social dilemmas of cooperation (see below). They

Cooperators and defectors interact via the edges of a graph. B (A) is satisfied (dissatisfied), since A (B) is a cooperator (defector). Therefore, A wants to change the link whereas B doesn't. The action taken is contingent on the fitness Π(A) and Π(B) of A and B, respectively. With probability ^{−β[Π(A) − Π(B)]}]^{−1} (where

The social dilemmas of cooperation examined in this work are modelled in terms of symmetric two-player games, where the players can either cooperate or defect upon interaction. When both cooperate, they receive the payoff

Games Studied and Their Parameter Range

Results for the fraction of successful evolutionary runs ending in 100% cooperation for different values of the time-scale ratio ^{3}, _{critical}

PD with

(Upper panel) Fraction of cooperators at end as a function of _{critical }

(Lower panel) Maximum value of the connectivity in population as a function of _{critical}_{critical},

The fact that in our model cooperators and defectors interact via social ties they both decide upon establishes a coupling between individual _{a}_{e}_{e} / τ_{a} ,

The contour plots in ^{3} and _{critical}_{critical}_{critical}

_{critical} ,_{max}_{critical},_{critical},_{critical},_{critical} ,_{critical},_{critical}

For any _{max} = z,

We study the same social dilemmas of cooperation as in

(Top left panel) Cumulative degree distributions (see _{max} = z,

(Contour plots) The amount of heterogeneity, measured in terms of the variance of the degree distribution (see

For large values of the temptation

The pairwise comparison rule [_{e}_{a}_{e} = β_{a} = β

Fraction of cooperators at the end as a function of _{e} = β_{a}^{3} and

Clearly, the smaller the value of _{critical}

Let us further examine the separate role of _{e}_{a}_{max}_{e},_{a}_{c} ,_{e}_{a}_{e}_{c}_{e} ,_{max}_{c}_{c} ,_{e}_{a}_{a}_{e} ,_{c}_{a} ,_{c}_{a},

Fraction of cooperators as a function of _{e}_{a}^{3}. The dilemma corresponds to the diagonal in the PD domain from {_{e}_{a}_{e}_{a},

Giving individuals control over the number and nature of their social ties, based exclusively on their immediate self-interest, leads to the emergence of long-term cooperation even in networks of high connectivity. As a consequence, “network adaptability” is required next to “social viscosity” to evolve cooperation in realistic social networks. For given values of connectivity in the population, and selection pressure, there is a critical value of

The present work assumes that population size remains constant throughout the co-evolutionary process. This is clearly a simplification, and it remains an open problem: the role of a changing population size. Yet, the message conveyed here is powerful and shows that highly interactive, large social networks can exhibit sustained cooperation even when the benefits do not significantly exceed the costs. Finally, the process by which individuals reassess their social ties may change depending on the social context, the “game” at stake, and the species under consideration. In other words, other (perhaps more sophisticated) mechanisms may be envisaged which will certainly build upon the simple model studied here. Yet, the prospect is quite optimistic since, as shown here, simple mechanisms devoid of large information-processing requirements are capable of promoting sustained cooperation.

We place individuals on the nodes (a total of _{E}_{E} / N^{ −1} ∑_{k}k^{2}N_{k}^{2} (where _{k}^{ −1} _{i}_{max} ,_{max}_{max}

Whenever _{e}^{−1}, a structural update event being selected otherwise. A strategy update event is defined in the following way, corresponding to the so-called pairwise comparison rule [^{−β[Π(B) − Π(A)]}]^{ −1}. The value of

We start from a homogeneous random graph [^{3} and average connectivities ^{8} generations, the population has not converged to an absorbing state, we take as the final result the average fraction of cooperators in the population over the last 1,000 generations. Indeed, especially for the SG, the time for reaching an absorbing state may be prohibitively large [_{max}

Discussions with Martin A. Nowak, Sebastian Maurer-Stroh, Hisashi Ohtsuki, Christoph Hauert, Hugues Bersini, and Arne Traulsen are gratefully acknowledged.

prisoner's dilemma

snowdrift game

stag-hunt game