week ending PRL 116, 128702 (2016) PHYSICAL REVIEW LETTERS 25 MARCH 2016 Linking Individual and Collective Behavior in Adaptive Social Networks Flávio L. Pinheiro,1,2,3 Francisco C. Santos,1,3 and Jorge M. Pacheco2,3,4 1INESC-ID and Instituto Superior Técnico, Universidade de Lisboa, 2744-016 Porto Salvo, Portugal 2Centro de Biologia Molecular e Ambiental da Universidade do Minho, 4710-057 Braga, Portugal 3ATP-group, P-2744-016 Porto Salvo, Portugal 4Departamento de Matemática e Aplicações da Universidade do Minho, 4710-057 Braga, Portugal (Received 11 December 2015; published 24 March 2016) Adaptive social structures are known to promote the evolution of cooperation. However, up to now the characterization of the collective, population-wide dynamics resulting from the self-organization of individual strategies on a coevolving, adaptive network has remained unfeasible. Here we establish a (reversible) link between individual (micro)behavior and collective (macro)behavior for coevolutionary processes. We demonstrate that an adaptive network transforms a two-person social dilemma locally faced by individuals into a collective dynamics that resembles that associated with an N-person coordination game, whose characterization depends sensitively on the relative time scales between the entangled behavioral and network evolutions. In particular, we show that the faster the relative rate of adaptation of the network, the smaller the critical fraction of cooperators required for cooperation to prevail, thus establishing a direct link between network adaptation and the evolution of cooperation. The framework developed here is general and may be readily applied to other dynamical processes occurring on adaptive networks, notably, the spreading of contagious diseases or the diffusion of innovations. DOI: 10.1103/PhysRevLett.116.128702 Complex networks provide a powerful representation of together with resolution of conflicts of interest reflecting the underlying web of social ties that interconnect indi- the process at stake. In particular, as detailed below, when viduals in a given community or population [1–5]. As time conflicts of interest are described by the paradigmatic unfolds, individuals may adjust their behaviors (or their prisoner’s dilemma (PD) of cooperation—which from a social ties), which typically induces changes in the social rational point of view would lead to widespread defection network they are embedded in (or in their individual —network adaptation favors cooperation at a population- behavior). It is thus no surprise that the heterogeneous wide level. Consequently, and despite the fact that, locally, structures we often find in empirical analyses result from every individual engages in a PD game, globally, the game the interplay between at least these two coevolving mech- being played must be a different one. anisms. Naturally, one expects the emerging features of a But which one? Of which type? With which features? complex network to depend on the nature of the inter- These questions remain open and are of fundamental actions between peers. For instance, if what is at stake is the importance given that, often, all we can gather empirically spreading of a disease, healthy individuals ought to break (e.g., from populations of microbes to human societies (secure) the links with infected (healthy) individuals, [14]) are time-series data of aggregate information, without provided they know they are (not) infected [6–8], whereas direct information on individual behavior. An intuitive social dilemmas of cooperation [3,9–11] by considering example of our empirical constraints regards epidemic individuals with different preferences (regarding the main- outbreaks: What one often collects is aggregate statistical tenance of interactions) often lead to more complex information of the community as a whole, instead of decision processes. The time dependence of both social individual information that may allow practitioners to infer structure and individuals’ choices can be conveniently directly from the data the characteristics of the, say, virus described by means of a coevolutionary process [3,4,11] infectiousness [7]. Thus, and similar to many other areas where individual behavior is allowed to evolve at par with of physics, it becomes ubiquitous to establish a (reversible) the structure of the underlying social network [11]. The link between individual and collective behavior in the latter implies that links are possibly rewired depending analysis of coevolutionary processes. on whether individuals are satisfied or not with a given In this Letter, we develop a novel framework able to interaction. A simple implementation of such a model [11] capture the nontrivial link between the nature of indi- —see below for details and Ref. [12] for related approaches vidual (local) interactions and the population-wide (global) —accounts for many of the stylized facts one expects to dynamics observable in adaptive structures. happen when social behavior and social structure coevolve Figure 1 provides an overview of what is known to [13]: adaptation of individual preferences and behaviors, date by means of computer simulations regarding the 0031-9007=16=116(12)=128702(5) 128702-1 © 2016 American Physical Society week ending PRL 116, 128702 (2016) PHYSICAL REVIEW LETTERS 25 MARCH 2016 (a) (b) FIG. 1. Level of cooperation and network heterogeneity under coevolutionary dynamics. (a) We computed numerically the level of cooperation in the domain bounded by 0.0 ≤ λ ≤ 1.0 and 0.0 ≤ τ ≤ 1.0. It is given by the average final fraction of cooperators after 5 × 103 generations (1 generation equals Z discrete strategy time steps, where Z is the population size) and averaged over 105 independent simulations starting from a configuration with equal abundance of strategies. Orange (blue) regions denote parameter values for which cooperators (defectors) dominate. These two regions are separated by a narrow transition line, approximately given by τ λ ≈ a − 1= bλ − 1 (solid black line, a ≈ 0.93, b ≈ 20). (b) Variance of the degree distribution of the equilibrium network as a functionð Þ of λ,ð obtainedÞ along the curve τ λ . Clearly, the stronger the social dilemma, the more heterogeneous the network becomes, in this way allowing for cooperators toð increaseÞ their chances of overturning defectors in the population. Furthermore, network heterogeneity is most pronounced along the line τ λ marking the transition between full cooperation and full defection [11]. Other parameters are β 10.0, Z 103, and average networkð Þ connectivity k 8. ¼ ¼ h i ¼ coevolution of cooperation and network topology [11], she will try to keep the link; if dissatisfied, she will try to when the social dilemma at stake is a PD. The relative time rewire the link to one of her second neighbors, accounting scales of network adaptation and behavioral evolution are for the myopic nature of individuals regarding the entire controlled by the parameter τ: When τ 0, no network social network. Following Ref. [11], when linked individ- adaptation occurs; when τ 1, no behavioral¼ adaptation uals A and B have a conflict regarding rewiring, resolution occurs, which, as shown in Fig.¼ 1, has a profound impact in is fitness driven, the will of A prevailing with probability −1 the emergence of cooperation. One starts from a homo- σ 1 exp −β fA − fB . Different variants of this geneous random network [15], where all individuals have model¼ f haveþ been½ ð consideredÞg[12], leading to qualitatively the same number of links, and let individual behavior and similar results. Naturally, the framework developed here is network structure coevolve at variable rates. Individuals applicable to different choices of link rewiring, as well as engage in a PD in which mutual cooperation provides a to pairwise interactions of different nature, as discussed reward R 1, mutual defection a punishment P 0, below. whereas when¼ a C meets a D, the C gets a sucker’s payoff,¼ As is well known [10,17], in the mean-field approxi- S −λ, while the D gets a temptation, T 1 λ (λ ≥ 0 mation C’s are not evolutionary viable. But when inter- ¼ ¼ þ is the dilemma strength, such that increasing λ implies actions proceed along the links of a social network, C’sdo a stronger attraction into a full-D configuration). Time not necessarily get extinct [3], even when no network proceeds in discrete steps, with the coevolutionary process adaptation takes place (τ 0), as shown at the bottom allowing, in each step, for link rewiring with probability τ part of Fig. 1(a).Mostimportantly,networkadaptation¼ and behavioral update with probability (1 − τ). Behavioral paves the way for cooperation to prevail, even for game update is modeled via the so-called pairwise comparison parameters that would render cooperation unfeasible in rule [16], a birth-death process in which an individual i nonadaptive networks, as one can easily check by follow- with strategy si (here C or D) imitates a neighbor m with ing trajectories of constant λ in Fig. 1(a). The faster the (a different) strategy sm with probability given by the Fermi rate of network adaptation, the more C’s get an evolu- distribution from statistical physics (where the inverse tionary edge over D’s. Moreover, for τ > 0,ourresults temperature β provides here a measure of the strength of suggest that the adaptive nature of the network nicely natural selection): p 1 exp −β f − f −1, with accounts for the heterogeneity observed in realistic social i;m ¼ f þ ½ ð m iÞg fj accounting for the fitness of individual j, associated with networks [1,2], in which a minority of nodes has a much the payoff accumulated over all interactions with their larger degree than the majority. This is shown in Fig. 1(b), neighbors [16]. Network adaptation assumes that C’s(D’s) where we plot the degree variance of the networks seek for C’s to cooperate with (to exploit), while avoiding emerging from coevolutionary simulations, as we move connections with D’s: An individual is satisfied with all his along the critical line τ λ depicted in Fig.