Real populations have been shown to be heterogeneous, in which some individuals have many more contacts than others. This fact contrasts with the traditional homogeneous setting used in studies of evolutionary game dynamics. We incorporate heterogeneity in the population by studying games on graphs, in which the variability in connectivity ranges from single-scale graphs, for which heterogeneity is small and associated degree distributions exhibit a Gaussian tale, to scale-free graphs, for which heterogeneity is large with degree distributions exhibiting a power-law behavior. We study the evolution of cooperation, modeled in terms of the most popular dilemmas of cooperation. We show that, for all dilemmas, increasing heterogeneity favors the emergence of cooperation, such that long-term cooperative behavior easily resists short-term noncooperative behavior. Moreover, we show how cooperation depends on the intricate ties between individuals in scale-free populations.complex networks ͉ evolution of cooperation C ooperation has played a key role throughout evolution (1). Self-replicating cells have cooperated to form multicellular organisms throughout evolutionary history (2, 3). Similarly, we know that animals cooperate in families to raise their offspring and in groups to prey and to reduce the risk of predation (4, 5). Cooperation has been conveniently formulated in the framework of evolutionary game theory, which, when combined with games such as the Prisoner's Dilemma, which is used as a metaphor for studying cooperation between unrelated individuals, enables one to investigate how collective cooperative behavior may survive in a world where individual selfish actions produce better short-term results. Analytical solutions for this problem have been obtained when populations are assumed infinite and their interactions are assumed homogeneous such that all individuals are in equivalent positions. Under such assumptions, noncooperative behavior prevails. Such an unfavorable scenario for cooperation in the Prisoner's Dilemma game, together with the difficulty in ranking the actual payoffs in field and experimental work (6, 7), has lead to the adoption of other games (8, 9), such as the Snowdrift game (also known as Hawk-Dove or Chicken), which is more favorable to cooperation, and the Stag-Hunt game (10), and to numerical studies of cooperation in finite, spatially structured populations (11) in which homogeneity is still retained. Such studies of the role of structured populations have attracted considerable attention, originating from fields ranging from sociology to biology, ecology, economics, mathematics, and physics, to name a few (11)(12)(13)(14)(15)(16)(17)(18)(19). More recently, however, compelling evidence has been accumulated that a plethora of biological, social, and technological real-world networks of contacts (NoC) are mostly heterogeneous (20)(21)(22). Indeed, analysis of real-world NoC (20) has provided evidence for the following (heterogeneous) types: (i) single-scale networks, which are charac...

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 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.

Protein function and dynamics are closely related; however, accurate dynamics information is difficult to obtain. Here based on a carefully assembled data set derived from experimental data for proteins in solution, we quantify backbone dynamics properties on the amino-acid level and develop DynaMine-a fast, high-quality predictor of protein backbone dynamics. DynaMine uses only protein sequence information as input and shows great potential in distinguishing regions of different structural organization, such as folded domains, disordered linkers, molten globules and pre-structured binding motifs of different sizes. It also identifies disordered regions within proteins with an accuracy comparable to the most sophisticated existing predictors, without depending on prior disorder knowledge or three-dimensional structural information. DynaMine provides molecular biologists with an important new method that grasps the dynamical characteristics of any protein of interest, as we show here for human p53 and E1A from human adenovirus 5.