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1 Journal of Theoretical Biology 253 (28) Contents lists availale at ScienceDirect Journal of Theoretical Biology journal homepage: Natural selection of cooperation and degree hierarchy in heterogeneous populations Jesús Gómez-Gardeñes a,, Julia Poncela a, Luis Mario Floría a,c, Yamir Moreno a, a Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 59 Zaragoza, Spain Scuola Superiore di Catania, Via S. Paolo 73, Catania, Italy c Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 59 Zaragoza, Spain article info Article history: Received 25 Octoer 27 Received in revised form 8 Feruary 28 Accepted 6 March 28 Availale online 14 March 28 Keywords: Complex networs Evolution of cooperation Prisoner s Dilemma astract One of the current theoretical challenges to the explanatory powers of Evolutionary Theory is the understanding of the oserved evolutionary survival of cooperative ehavior when selfish actions provide higher fitness (reproductive success). In unstructured populations natural selection drives cooperation to extinction. However, when individuals are allowed to interact only with their neighors, specified y a graph of social contacts, cooperation-promoting mechanisms (nown as lattice reciprocity) offer to cooperation the opportunity of evolutionary survival. Recent numerical wors on the evolution of Prisoner s Dilemma in complex networ settings have revealed that graph heterogeneity dramatically enhances the lattice reciprocity. Here we show that in highly heterogeneous populations, under the graph analog of replicator dynamics, the fixation of a strategy in the whole population is in general an impossile event, for there is an asymptotic partition of the population in three susets, two in which fixation of cooperation or defection has een reached and a third one which experiences cycles of invasion y the competing strategies. We show how the dynamical partition correlates with connectivity classes and characterize the temporal fluctuations of the fluctuating set, unveiling the mechanisms stailizing cooperation in macroscopic scale-free structures. & 28 Elsevier Ltd. All rights reserved. 1. Introduction The Prisoner s Dilemma (PD) models situations where cooperation is expensive (Hamilton, 1964; Axelrod and Hamilton, 1981; Nowa and Sigmund, 25; Nowa, 26; Hofauer and Sigmund, 1998, 23). In this two-players game, each individual adopts (independently and simultaneously) one of the two availale strategies, cooperation (C) or defection (D); oth receive R under mutual cooperation and P under mutual defection, while a cooperator receives S when confronted to a defector, which in turn receives T, where T4R4P4S. Under these conditions, defection is uneatale 1 and reaches fixation in a well-mixed population of replicators. However, if individuals only interact with their neighors as dictated y the underlying networ of (social) contacts, several studies (Nowa and May, 1992; Killingac and Doeeli, 1996; Santos and Pacheco, 25; Lieerman et al., 25; Aramson and Kuperman, 21; Gómez-Gardeñes et al., 27; Poncela et al., 27; Ohtsui et al., 26; Eguíluz et al., 25; Santos et al., 26; Szaó and Fáth, 27; Szolnoi Corresponding author. Tel.: ; fax: addresses: yamir@unizar.es, yamir.moreno@gmail.com (Y. Moreno). 1 Strict est response to itself and strict etter response to others than themselves (Nowa, 26). et al., 28; Vuov et al., 28) have reported the asymptotic survival of cooperation for TXR, on different types of networs. Notaly, cooperation even dominates over defection in scale-free (SF) networs where the distriution density of local degree (or connectivity) follows a power law (Santos and Pacheco, 25; Gómez-Gardeñes et al., 27; Poncela et al., 27). In these latter structures, micro-motives and asymptotic macro-ehavior are much more complexly related and its degree heterogeneity offers the opportunity of positive feedac evolutionary mechanisms allowing cooperation to defeat defection, even not eing a est reply to itself (Nash). In well-mixed populations, there is a transition at P ¼ S etween fixation of defection in the PD ðp4sþ and strategies coexistence in the Haws and Doves (HD) game ðposþ. However, in SF networs, it has een recently shown (Gómez-Gardeñes et al., 27) that there is a wide region of P4S where fixation of the uneatale strategy is generically an impossile event, under the updating rule considered here (see elow). In this region, there is an asymptotic partition of the networ into three su-populations (C, D, F). In C (and D), cooperative (and resp. defective) strategy reaches fixation, while in F no fixation is possile and cycles of node invasions follow indefinitely. In this way, the macroscopic average index of cooperation hci has the form hci ¼r c þ r f ht c i, (1) /$ - see front matter & 28 Elsevier Ltd. All rights reserved. doi:1.116/j.jti

2 J. Gómez-Gardeñes et al. / Journal of Theoretical Biology 253 (28) where ðr c ; r d ; r f Þ is the measure of the partition s sets, and ht c i is the average proportion of time spent y the fluctuating supopulation F as cooperators. In this paper, we go one step forward in the characterization of the dynamical organization of cooperation. Capitalizing on a model networ, we analytically argue that the type of partition previously reported (Gómez-Gardeñes et al., 27) also arises generically in a much wider class of heterogeneous networs. Scale-free networs are then further scrutinized to show the continuous variation of the average level of cooperation and the three inds of strategists in a two parameter space covering the PD game. Next, we show how the densities of the supopulations C and F are correlated with the structural division of the networ into degree classes, so unveiling the role of degree heterogeneity in the evolutionary cooperation enhancement. Additionally, the random variales descriing fluctuations of strategies in F are also analyzed numerically. In what follows the payoffs are scaled to the reward for mutual cooperation, R ¼ 1, and punishment for mutual defection is set to P ¼, so there are two free parameters T ¼ 41, and S ¼. The PD corresponds to values of p. The players occupy the vertices of a fixed graph (i.e. connections etween players do not coevolve with strategies) where agents are represented y nodes, and a lin etween nodes indicates that they interact (play). We implement the finite population analogue of replicator dynamics (Santos and Pacheco, 25; Gómez-Gardeñes et al., 27) with synchronous update. 2 In this setting, a player i adopts the strategy of a randomly chosen neighor j with proaility P i!j ¼ ðp j P i Þ, eing P i and P j the payoffs accumulated after playing with all their neighors once. If i ( j ) is the numer of neighors (connectivity or degree) of agent i (j), and D is the maximal possile payoff difference (D ¼ maxf; g), ¼ðmaxf i ; j gdþ 1 is related to the characteristic inverse time scale. This updating rule has the theoretical advantage of leading rigorously, in the well-mixed population limit, to the celerated replicator equation (Gintis, 2). Note that (irrational) imitation of a neighor with a lower payoff is foridden, a feature which is at the root of the existence of the ðc; D; FÞ partition. In an equilirium configuration the proaility of change in one time step is null. For generic irrational values of and, only all-c (fixation of cooperation) and all-d (fixation of defection) are equiliria. However, in networs, the asymptotic state of evolutionary dynamics is often not an equilirium configuration under the aove rules (Gómez-Gardeñes et al., 27). To see that this generically holds for a wide class of heterogeneous networs, we first consider a model networ where we prove the existence of the asymptotic partition (C; D; F), for a macroscopically large set of initial conditions. The ipolar model networ mimics a local environment of a heterogeneous graph, with simplifications that allow analytical insights. It is perhaps the minimal (though general enough) networ model where the partition can e rigorously proved, so illustrating the dynamical organization of cooperation in heterogeneous graphs. 2. A ipolar model networ Let us consider the graph schematized in Fig. 1, composed of the following: (a) A component F of n F nodes with aritrary connections among them. 2 We have checed that the results are qualitatively the same using asynchronous update. C () A node, say node 1, which is connected to all the nodes in F and has no other lins. (c) A component C of n C nodes with aritrary connections among them. (d) A node, say node 2, which is connected to all the nodes in F and C, ut not to node 1. Consider the set of initial conditions defined y: (i) node 1 is a defector, (ii) node 2 is a cooperator, and (iii) all nodes in component C are cooperators. Note that this choice allows 2 n F different initial configurations. We now prove that, provided some sufficient conditions (see elow), this is an invariant set for the evolutionary dynamics. The payoff of a cooperator node i in F is P c i ¼ c i þ 1 þ ð i c i þ 1Þ, where i is the numer of its neighors in F and c i p i is the numer of those that are cooperators. The payoff of node 1 is then P 1 Xð c i þ 1Þ. For the PD game, where o, the inequality P 1 4P c i always holds, so that node 1 will always e a defector. The payoff of a defector node i in F is P d i ¼ð c i þ 1Þ, where c i is the numer of its cooperator neighors in F, while the payoff of node 2 is P 2 ¼ n C þ n F þ n c F ð1 Þ, where nc F pn F is the numer of cooperators in F. Thus, a sufficient condition for P 2 4P d i is n C 4Intðð F þ 1Þ n F Þ, where F (on F ) is the maximal degree in component F, i.e. the maximal numer of lins that a node in F shares within F. With this proviso, node 2 will always e a cooperator, which in turn implies that all the nodes in the component C will remain always cooperators. This argument proves that provided the sufficient condition n C 4Intðð F þ 1Þ n F Þ, (2) holds, the set of initial conditions defined y (i) (iii) is an invariant set: any stochastic trajectory starting in the set remains there. Moreover, as no equilirium configuration is included in this set, one concludes that no trajectory from this set evolves to an equilirium configuration. While nodes in C and node 2 are permanent cooperators, and node 1 is a permanent defector, nodes in F are forced to fluctuate: at every time, a defector in F has a positive proaility to e invaded y the cooperation strategy, and a cooperator in F has a positive proaility of eing invaded y the defection strategy. In other words, every configuration in the set of initial conditions is reachale (in one time step) from any other, thus it is almost sure that it will e reached (ergodicity). Along any stochastic trajectory starting from the set of initial conditions the networ is partitioned into three susets, a set of F 2 1 Fig. 1. Bipolar model networ. Nodes 1 and 2 are connected to all nodes in F. Node 2 is also lined to all nodes in C. Connections inside F and C are aritrary. The colors represent a set of 2 n F different initial configurations. Blue means defector, red stands for cooperator and green means aritrary strategy.

3 298 J. Gómez-Gardeñes et al. / Journal of Theoretical Biology 253 (28) pure cooperator nodes, a set of pure defector nodes and a set of fluctuating individuals. The fluctuations inside the supopulation F reflect the competition for invasion among two non-neighoring hus with fixed opposite strategies in their common neighorhood, a local situation that occurs in heterogeneous networs. It is also a schematic model for the competition for influence of two powerful superstructural institutions lie mass media, political parties, or loies on a target population. Let us now otain some exact results for the simplest choice of topology of connections inside the fluctuating set, namely F ¼. In this case each node in F is only connected to nodes 1 and 2. Note that the sufficient condition for fixation of cooperation at node 2 is, n C 4 n F. Denoting y cðtþ the instantaneous fraction of cooperators in F, the payoffs of nodes 1 and 2 are P 1 ¼ cn F ; P 2 ¼ n C þ cn F þ ð1 cþn F, and the payoffs of a cooperator node and a defector node in F are, respectively, P c ¼ 1 þ ; P d ¼. Then one finds for the (one time-step) proaility P CD of invasion of a cooperator node in F P CD ¼ c ð1þþ=n F, (3) 2D and using the notation A ¼ þðn C Þ=n F and B ¼ 1 þ n C =n F A þ cð1 Þ P DC ¼, (4) 2DB for the proaility of invasion of a defector node in F. Note that A4 due to the non-invadaility of node 2. At time t þ 1, the expected fraction of cooperators is cðt þ 1Þ ¼cðtÞð1 P CD Þþð1 cðtþþp DC, provided n F 1, the fraction of cooperators c in F, evolves according to the differential equation _c ¼ð1 cþp DC cp CD, which after insertion of Eqs. (3) and (4) ecomes _c ¼ f ðcþ A þ A 1 c þ A 2 c 2, (5) where the coefficients are A ¼ A 2DB, A 1 ¼ 1 A þ Bð1 þ Þ=n F, 2DB 1 þ B A 2 ¼. (6) 2DB One can easily chec (A 4 and A 2 o) that there is always one positive root c of f ðcþ, which is the asymptotic value for any initial condition pcðþp1 of Eq. (5). Thus, cooperation is never driven to extinction even for large values of the temptation to defect. Bac to the general case, i.e. aritrary structure of connections in F, it should e emphasized that the sufficient condition (Eq. (2) aove) does not impose ounds on the networ s average connectivity hi, that can tae on aritrarily large values independent of the game parameters. This result differs from the ound on hi reported in Ohtsui et al. (26) and Nowa (26) for different stochastic updating rules in the wea selection limit. 3. The partition ðc; D; FÞ in SF networs Scale-free networs (Newman, 23; Boccaletti et al., 26) are constructed here following the Baraási and Alert (1999) (BA) model. For this purpose, one starts from a fully connected set of m nodes and at each time step a new node is added and lined to m nodes. These m nodes are chosen following the preferential attachment rule, namely, the proaility that node i receives a new lin is proportional to its degree, i = P j j. Avoiding multiple connections and iterating the preferential attachment rule N m times a networ of N nodes and mean degree hi ¼2m is constructed. The degree distriution of the resulting networ is apowerlaw,pðþ / g, with exponent g ¼ 3. It is worth recalling that scale-free networs constructed with the BA method do not show degree degree correlations etween neighoring nodes. In this way, the proaility that a node of degree is connected to another one with degree is independent of, Pð jþ ¼ Pð Þ=hi. The networs used in our simulations have typically N ¼ nodes and hi ¼4. We start the simulations from a networ where agents have the same proaility of adopting either of the two availale strategies: cooperation or defection. Evolutionary dynamics is then iterated over a fixed transient of generations. After this transient we chec whether or not the system has reached a stationary state as given y the fraction, cðtþ, of individuals that are cooperators. We consider that the population is in equilirium when, taen over a time window of 1 3 additional generations, the slope of cðtþ is smaller than 1 2. Once the dynamical equilirium is reached the system is further evolved over 1 4 additional generations, and we identify the agents that act as pure cooperators (defectors), i.e. those individuals that always cooperate (defect). The dynamical patterns of the rest of the agents (fluctuating nodes) are also stored during these 1 4 generations to characterize the fluctuations (cycles of invasion). All the results are averaged over at least 1 3 different realizations of scale-free networs and initial conditions. Fig. 2 shows the numerical estimates of the partition measure ðr c ; r d ; r f Þ as well as the average index of cooperation hci (i.e. the overall fraction of time spent y all the nodes as cooperators averaged over stochastic trajectories on all realizations) for BA networs in the range of parameter values 1oo2:3 and :25o. These results convincingly show the generic existence (as well as its sizeale importance) of the asymptotic partition ðc; D; FÞ for the replicator dynamics (imitation of a neighor of a higher payoff proportional to payoff differences) of the PD game on SF graphs. In fact, though limited to a smaller range of parameter values, the partition is generic for general graphs, as our own results in random Erdös-Renyi graphs corroorate (Gómez-Gardeñes et al., 27; Poncela et al., 27). Thus the question is how degree heterogeneity influences the partition so enhancing the positive feedac mechanisms of networ reciprocity Strategy-degree correlations To investigate in detail the role that degree heterogeneity plays in the replicator dynamics in SF networs, we have measured the proportions r ðþ a ða ¼ c; d; f Þ of each type of individuals inside the class of nodes with degree, for all values of. Note that rc ðþ þ r ðþ þ r ðþ ¼ 1 for each, and denoting the degree distriution f d density y PðÞ (/ g for a SF networ), one has that r a ¼ P PðÞrðÞ a for each a. For definiteness we consider hereafter in this section a value of ¼, i.e., the so-called (Szaó and Fáth, 27) wea PD, at the orderline separating PD and HD. One should not expect, however, qualitative differences when moving from it, as suggested y results in Fig. 2. In Fig. 3 we show r ðþ c (left panel) and r ðþ (right panel) versus f log and the temptation to defect for BA networs. The figure shows four clearly differentiated ranges of values. The first one, 1oo1:7, corresponds to the regime where pure cooperation

4 J. Gómez-Gardeñes et al. / Journal of Theoretical Biology 253 (28) ε ε Fig. 2. Strategists asymptotic partition in BA networs. The panels represent in color-code: (a) the average level of cooperation hci, and (), (c) and (d) the densities r c, r f and r d, respectively, as a function of game parameters and. The sustrate networ is a Baraási-Alert scale-free graph made up of N ¼ nodes and hi ¼4. 2. ρ c ρ f 1. log() Fig. 3. Strategists proportion y degree classes. Color-coded densities of pure cooperators r c and fluctuating individuals r f y degree classes, as a function of, for BA networs and ¼. Note that r d ¼ 1 r c þ r f. More details are given in the main text. dominates the asymptotic ehavior (i.e. :9pr c o1); in this range only a small numer of fluctuating individuals occupy some nodes with low and intermediate degree. In the second range, 1:7oo2, the fluctuating set fully invades the classes with low values of ðp11þ, corresponding to the decrease of r c down to 1%. In the third range, the fluctuating set invades progressively higher -classes as increases, with pure cooperators still predominating in even higher -classes. At around 2:9, only nodes with the highest degree (hus) remain as pure cooperators. For larger values of, the growth of r ðþ at the expense of the d fluctuating individuals is, on the contrary, fairly insensitive to the degree value, as inferred in the right part of the right panel of Fig. 3. The preferential fixation of cooperation at nodes with high degree when cooperation is very expensive can e understood y the following plausile argument: A necessary (though nonsufficient) condition for a node i to e a pure cooperator is that (at a given time t) the numer c i of instantaneous cooperators in its neighorhood (i.e. the payoff of i in the current round) must e greater than the current payoff of any instantaneous defector neighor j, that is, c i 4c j. This condition is clearly favored when the cooperator node i elongs to a high class and its fluctuating neighors j elong to lower classes. 3 Furthermore, the imitation of a successful (high payoff) cooperator y its neighors reinforces its future success (Santos and Pacheco, 26), then favoring the fixation of cooperation in highly connected nodes. On the contrary, the imitation of a successful defector undermines its future success, so that defection cannot tae long-term advantage from degree heterogeneity. 3 Note that the argument is consistent provided that the heterogeneous networ either has not degree degree correlations (so that the neighors of a node of degree have no preferential degrees) or the networ is assortative (i.e. neighors of high degree nodes have preferentially also high degrees). We remind that degree degree correlations are asent in the (undirected) BA networ.

5 3 J. Gómez-Gardeñes et al. / Journal of Theoretical Biology 253 (28) τ c 8x x1 3 4x1 3 2x T c 1 8x1-1 6x x1-1 2x Fig. 4. Hierarchy of cooperation times in the fluctuating set. Permanence times, t c, of the cooperation strategy of a fluctuating node (top) and the fraction of time, T c it cooperates (ottom) as a function of the node s degree and the game parameter for ¼. See main text for details. In the left panel of Fig. 3, for 42, one oserves that at fixed value of, r ðþ c varies rather quicly from to 1 in a small interval of values of centered around some -dependent value ðþ, so that the nodes with degree 4 ðþ are mostly pure cooperators and those with degree o ðþ are mostly fluctuating (see right panel, 2oo2:9). In the asence of degree degree correlations the degree distriution density in the neighorhood of a given node is independent of the node degree, and thus the proportion of cooperators in the neighorhood of a given node is that of the whole networ. This implies that the necessary condition for a pure cooperator i, stated in the previous paragraph, ecomes i 4 j, where j is the fluctuating neighor of i with highest degree, say j. Now, a small increase D maes those pure cooperators i fulfilling ð þ DÞ 4 i 4 to ecome fluctuating, so that D D. With these provisos one concludes that ðþ grows exponentially with, ðþ /expðþ. The linear shape of the right-color line in the ð; log Þ plane at the left panel of Fig. 3, for 42, nicely confirms this prediction, so supporting the validity of the heuristic argument Fluctuations We have noted that the fluctuating supopulation in the ipolar model is such that any fluctuating individual has a positive proaility of changing strategy in one time step, so that the dynamics is ergodic in the set of all configurations compatile with the partition. This is not necessarily the case in a general heterogeneous networ, eing perfectly possile that a fluctuating node at a given time has a null one-time-step proaility of invasion, ut a positive n-time-steps proaility for some n41; thus, ergodicity in the set of configurations compatile with the partition is neither ensured nor discarded. In SF graphs each fluctuating individual is wired to (and then invadale y) a different numer of fluctuating individuals, and (eventually) pure strategists, so that one should expect that the fraction of time T c it spends as cooperator differs widely from node to node. The lower panel of Fig. 4 shows the average fraction of time T ðþ c a fluctuating node of degree spends cooperating. The average of these quantities P PðÞTðÞ c in the supopulation F, defines the parameter ht c i that appears in Eq. (1), i.e. the average individual contriution of fluctuating nodes to the macroscopic index of cooperation hci. To avoid misunderstandings concerning the relative importance of the contriution of connectivity classes to hci, it is important to ear in mind oth, the power-law shape of PðÞ and the left panel of Fig. 3, showing the fraction r ðþ of f fluctuating nodes inside the class of degree. In the extent that T c is a proportion of time, it does not provide information on the time scales of the invasion cycles that fluctuating nodes experience. The random variale t c (cooperation permanence time) is defined as the time spent as cooperator y a fluctuating node in each cycle. For the ipolar networ, when F ¼, the one time step invasion proailities, P CD and P DC (Eqs. (3) and (4)) ecome time independent in the asymptotic regime. Then one can compute the proaility that the cooperation strategy remains for a time t c X1 at a fluctuating node, simply as Pðt c Þ¼P CD ð1 P CD Þ tc 1. In a similar way, the distriution density Pðt d Þ of defection permanence times is otained as Pðt d Þ¼P DC ð1 P DC Þ t d 1. Thus the distriution densities of oth strategies permanence times are exponentially decreasing. For

6 J. Gómez-Gardeñes et al. / Journal of Theoretical Biology 253 (28) example, at ¼, i.e. at the order etween the PD and the HD game, if one further assumes that the relative size mðfþ of the component F is large enough, i.e. mðfþ!1, and mðcþ!, one otains that the stationary solution of Eq. (5) ehaves as c ð þ 1Þ 1 near the limit mðfþ!1. The distriution density Pðt c Þ of the cooperation permanence times of a fluctuating node, as a function of the parameter is thus tc Pðt c Þ¼ð2þ1Þ 1 2 þ 1, (7) 2 þ 2 and the distriution density Pðt d Þ of defection permanence times td. Pðt d Þ¼ð2ðþ1Þ 1Þ 1 2ð þ 1Þ 1 (8) 2ð þ 1Þ For SF networs, one expects that the permanence times at the fluctuating nodes show some correlation with the node s degree. The upper panel of Fig. 4 represents the average permanence time, t ðþ c, that fluctuating nodes of degree remain as cooperators as a function of and, for oservation times of 1 4 generations. We see that cooperation permanence times are strongly correlated with degree: highest t c s occur along the line ðþ of maximal degree in the fluctuating set. As noted efore, the heterogeneity of social contacts in SF networs provides local environments where cooperation has a distinctive selective advantage at high degree nodes. This not only enhances the size of the supopulation where fixation of cooperation occurs, ut also enlarges the average total fraction of time of cooperation in the fluctuating supopulation. Moreover, the picture emerging from the strategists partition in heterogeneous graphs indicates that individuals are effectively organized in such a way that a fraction of pure cooperators is isolated from contacts with the fluctuating population, and thus safe from invasion. This Eden of cooperation, provides a safe source of enefits to those highly connected pure cooperators in the frontier (Gómez-Gardeñes et al., 27), reinforcing the resilience to invasion and providing a stailizing feedac mechanism for the survival of cooperation. Hence, heterogeneity of social contacts enhances the lattice reciprocity, thus preventing the fixation of the defection strategy even when cooperation is expensive. 4. Conclusions In this paper, we have further analyzed the dynamical organization of strategies when the strategists are coupled through heterogeneous topologies. We have presented analytical insights derived from a model networ that mimics the competition for invasion of two highly connected nodes where strategies have reached fixation. Under the hypothesis that the players imitate a neighor with a higher payoff proportionally to the payoff differences, we have shown that fixation of any strategy in the whole population is an impossile event when there is some degree of heterogeneity in the networ of contacts. Furthermore, the strategists partition into three sets for SF networs has een shown to e correlated with structural properties such as the degree of the nodes. Nodes with higher degree are mainly occupied y pure cooperators or fluctuating strategists who spend most of the time cooperating. In this way, the overall dynamical picture that emerges is that of a set of pure cooperators placed on highly degree classes interacting with memers of the fluctuating set that on its turns occupy lower levels in the degree hierarchy of the networ. In all, this provides cooperation with a stailizing mechanism that does not allow defection to e fixated in the whole population. Acnowledgements We than A. 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