What represents the selection pressure of this game?

When members of a population engage in dyadic interactions reflecting a prisoner's dilemma game, the evolutionary dynamics depends crucially on the population structure, described by means of graphs and networks. Here, we investigate how selection pressure contributes to change the fate of the population. We find that homogeneous networks, in which individuals share a similar number of neighbors, are very sensitive to selection pressure, whereas strongly heterogeneous networks are more resilient to natural selection, dictating an overall robust evolutionary dynamics of coordination. Between these extremes, a whole plethora of behaviors is predicted, showing how selection pressure can change the nature of dilemmas populations effectively face. We further show how the present results for homogeneous networks bridge the existing gap between analytic predictions obtained in the framework of the pair-approximation from very weak selection and simulation results obtained from strong selection.

Complex networks are ubiquitous and known to profoundly affect the processes that take place on them [1–5]. From a theoretical perspective, some of the most complex processes studied to date, occurring on complex networks, are related to behavioral dynamics and decision-making, often described by means of social dilemmas of cooperation [6]. Among these, prisoner's dilemma (PD) provides the most popular metaphor of such dilemmas, given that its only Nash equilibrium is mutual defection, despite mutual cooperation providing higher returns [7]—thus the dilemma. We may also assume a dynamical (evolutionary) approach to game theory [8, 9], where individuals revise their behavior based on the perceived success of other individuals, creating a gradient of selection (GoS) [10] that dictates the evolution of cooperation in time. In this context, such GoS will always favor free-riders irrespective of the fraction of cooperators and of the relative importance of fitness in the evolutionary process, a result that dictates the demise of cooperation in the population [9].

This result that translates the forecast stemming from a game-theoretical analysis based on Nash equilibrium into a population-wide, dynamical setting, assumes that populations are large and well mixed, that is, everyone interacts with everyone else with equal probability [8]. Yet, when members of a population interact along the links of an underlying complex network this scenario is altered, as the assumption of a well-mixed population no longer holds [6, 11–13]. Only recently has it become possible to analyze the population-wide evolutionary dynamics of a game played on an arbitrarily complex network, very much in the same manner that games were analyzed in well-mixed populations [14]. The so-called average gradient of selection (AGoS, see section 2 for details) has unraveled the fundamental changes in evolutionary game dynamics introduced in populations structured along the links of complex networks. While further analysis is required for other classes of dilemmas [6, 15–17], the results obtained from the PD [14] show that, at a population-wide level (what we call macro-dynamics) the effective game at stake can be very different from that in which pairs of individuals engage (what we call micro-dynamics), with direct implications both in the time-evolution and invasion of cooperation, analogous to what occurs in finite well-mixed populations [18]. In particular, homogeneous networks seem to favor the co-existence of strategies, whereas heterogeneous networks, in turn, favor their coordination [14], irrespective of the fact that the game individuals locally perceive and play is a PD