Metric clusters in evolutionary games on scale-free networks

Kaj-Kolja Kleineberg1, ∗

1Computational Social Science, ETH Zurich, Clausiusstrasse 50, CH-8092 Zurich, Switzerland(Dated: August 4, 2017)

The evolution of cooperation in social dilemmas in structured populations has been studied exten-sively in recent years. Whereas many theoretical studies have found that a heterogeneous networkof contacts favors cooperation, the impact of spatial effects in scale-free networks is still not wellunderstood. In addition to being heterogeneous, real contact networks exhibit a high mean localclustering coefficient, which implies the existence of an underlying metric space. Here, we show thatevolutionary dynamics in scale-free networks self-organize into spatial patterns in the underlyingmetric space. The resulting metric clusters of cooperators are able to survive in social dilemmasas their spatial organization shields them from surrounding defectors, similar to spatial selection inEuclidean space. We show that under certain conditions these metric clusters are more efficient thanthe most connected nodes at sustaining cooperation and that heterogeneity does not always favor—but can even hinder—cooperation in social dilemmas. Our findings provide a new perspective tounderstand the emergence of cooperation in evolutionary games in realistic structured populations.

Keywords: Evolutionary game theory, structured populations, emergence of cooperation, scale-free networks,network geometry

I. INTRODUCTION

Cooperation among humans has been found to bequite common in social dilemmas [1, 2], and plays amajor role in the emergence of complex modern soci-eties [3, 4]. Therefore, understanding the underlyingmechanisms that can give rise to and sustain cooperationfrom an evolutionary perspective is key to complement-ing Darwin’s theory of evolution [5–9].

In reality, populations are structured, which meansthat the topology of strategic interactions is given bya network of contacts. In structured populations, in-dividuals interact repeatedly with the same individuals.Thus, as a consequence, cooperators can survive in so-cial dilemmas by forming network clusters. This mech-anism is referred to as network reciprocity [2, 10]. Inthe well-studied case of lattice topologies, the result-ing network clusters unfold in Euclidean space [11–13](spatial selection [14]). Realistic networks of contactsare heterogeneous rather than lattices and often scale-free, which means that their degree distribution follows apower-law with exponent γ ∈ (2, 3), where a lower valueof γ means more heterogeneous networks. Heterogene-ity has been shown to favor cooperation [15–17], andcooperating nodes form a connected (or network) clus-ter [18]. However, the geometric organization of theseconnected clusters—similarly to spatial selection in Eu-clidean space—remains elusive.

Real complex networks, in addition to being hetero-geneous, exhibit a high mean local clustering coeffi-cient [19, 20] (this means that the network contains a highnumber of closed triangles). This is particularly impor-tant because a high clustering coefficient implies the ex-istence of an underlying metric space [21]. We show that

∗ kkleineberg@ethz.ch

evolutionary dynamics on scale-free, highly clustered net-works lead to the formation of patterns in the underly-ing metric space, similar to the aforementioned spatialselection in Euclidean space. Using two empirical net-works, the IPv6 Internet topology and the arXiv collab-oration network, as well as synthetic networks, we showthat spatial patterns play an important role in the evo-lution of cooperation. In fact, under certain conditionsmetric clusters can even be more effective at sustainingcooperation than the most connected nodes (hubs). Asa consequence, heterogeneity does not always favor—butcan even hinder—the evolution of cooperation in socialdilemmas.

II. RESULTS

A. Latent geometry of scale-free networks

Real contact networks are usually heterogeneous, andoften scale-free, as well as highly clustered [20] (we referto a high mean local clustering coefficient, i.e. a largenumber of closed triangles [19]). The effect of scale-freetopologies has attracted a lot of attention, and many the-oretical studies have found that heterogeneous networksof contacts favor cooperation in social dilemmas [15–18],although this behavior has not been confirmed in recentexperiments with human players [22]. Importantly, thehigh local clustering coefficients found in real contact net-works have been proven to imply the existence of a metricspace underlying the observed topology [21]. This meansthat the nodes of a given real complex network can bemapped to coordinates in this metric space such that theprobability that pairs of nodes will be connected in theobserved topology depends only on their distance in themetric space. Specifically, heterogeneous networks can beembedded into hyperbolic space [23–25]. In this repre-

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Figure 1. A: Illustration of the hyperbolic spatial structure (Poincare Disc) underlying a synthetic network generated bythe model described in Methods. The network shown here has N = 2000 nodes, a power-law exponent γ = 2.6, mean degree〈k〉 ≈ 6, and mean local clustering coefficient c = 0.6 (temperature T = 0.3 as in Eq. (6), see Methods section for details). Hubs,i.e. high degree nodes, are placed closer towards the center of the disk (lower radial coordinate). The angular space representsthe similarity between nodes, such that nodes tend to connect to other nodes close to them in this space. The green line showsthe hyperbolic disk of radius R (Eq. (5)) around the green node. We highlight the neighbors of the green node in red. For ahigh c (i.e. low T ), as shown here, the green node is highly likely to connect to other nodes within the disc (green line), and avery unlikely to connect to nodes outside of it (the further apart, the less probable). The high mean local clustering coefficientis then a consequence of the triangle inequality in the metric space. B: Synthetic network generated with the same model butwith mean local clustering coefficient c = 0.25 (temperature T = 0.7). We again show the hyperbolic disk of radius R (Eq. (5))around the green node and highlight its neighbors in red. Note that due to the higher temperature more long-range connectionsare formed, i.e. the green node connects to more nodes outside of the disc as compared to (A), and does not connect to somenode inside of the disc. This effect reduces the mean local clustering coefficient as it induces randomness in the link formationprocess. C: Illustration of evolutionary game dynamics. In structured populations, individuals play with their neighbors in anetwork. In each game, they generate a payoff given by the payoff matrix (Eq. (1)). After each round, they choose a randomneighbor and imitate her strategy with a probability (Fermi-Dirac distribution) that depends on the difference between theirpayoffs.

sentation, each node has a radial and angular coordinate.The radial coordinate abstracts the popularity, and hencethe degree of the node, such that hubs are placed closerto the center of the disk (Fig. 1A). The angular coordi-nate abstracts a similarity space, such that the angulardistance is a measure of the similarity between two nodes,whereby nodes tend to connect to more similar nodes. InFig. 1A and B we show an illustration of the hyperbolicmetric structure underlying two different networks (seeMethods section for further details). In the following, weshow that evolutionary dynamics trigger the formationof stable spatial clusters in the angular dimension on theunderlying hyperbolic space.

B. Evolutionary dynamics and the emergence ofmetric clusters

Let us first consider the prisoner’s dilemma game, inparticular T = 1.2 and S = −0.2 (see Eq. (1) in Meth-ods) on synthetic contact networks generated with themodel described in Methods. This model generates re-alistic topologies based on underlying hyperbolic metricspaces, similar to Fig. 1A. We simulate the evolutionarygame dynamics (see Methods and Fig. 1C) and find thatthe system tends to self-organize into a state in whichgroups which are mainly cooperative are clearly separatefrom groups populated mainly by defectors (see Fig. 2A-

C and Supplementary Video 1) [26, 27]. Similarly tothe case of lattice topologies and spatial selection in Eu-clidean space, we observe the formation of clusters ofcooperators in the angular dimension of the underlyinghyperbolic space. In Fig. 2D we show the evolution ofthe density of cooperators in different bins of the angu-lar coordinate θ. We observe that initially cooperationdecreases (see purple line in Fig. 2E) while, at the sametime, the remaining cooperators become concentrated inclusters in the angular space (Fig. 2A). Cooperation thenincreases again as it spreads in the vicinity of the clusters(Fig. 2B) until the system reaches a stationary state withfluctuations only at the borders of the clusters (Fig. 2Cand Supplementary Video 1).

We quantify the degree to which cooperators and defec-tors cluster in the angular space using the Kolmogorov-Smirnov (KS) statistic [28], which measures the differ-ence between two one-dimensional distributions. TheKS-statistic is defined as the maximum absolute differ-ence between the values of two cumulative distributions.In particular, we define the KS-statistic of the distribu-tion of cooperation density in different angular bins, andthe uniform distribution at time t as ρ(t) (see Methodsfor details). A higher value of ρ(t) thus denotes more pro-nounced clustering of cooperators and defectors respec-tively. In Fig. 2F we show the evolution of ρ(t) for 103

different realizations (blue lines) and their mean (blackline). On average, ρ(t) increases initially and approaches

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A B DC

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Figure 2. A-C: Evolution of the system (see also Supplementary Video 1) for a single realization of the prisoner’s dilemma(T = 1.2 and S = −0.2). Here, we have generated a synthetic network with N = 5000 nodes, power law exponent γ = 2.8,and mean local clustering c ≈ 0.5. Cooperators are marked in blue, whereas red denotes defectors. The system was startedwith randomly selected cooperators (c(0) = 0.5). (A) shows the state of the system after 103, (B) after 104, and (C) after105 generations. D: Density of cooperators (color coded) in different angular bins (shown on the y axis) as a function of time(shown on the x axis). Here, we have divided the angular space θ ∈ [0, 2π) into 20 equidistant bins. E-H: Results for the samesynthetic networks as before but with N = 2 × 104 nodes. E: Evolution of the density of cooperation C for 103 independentrealizations of the system (blue lines) and their average (black line). The purple line corresponds to the realization shown in(A-C). F: Evolution of the KS-statistics ρ for 103 independent realizations of the system (blue lines) and their average (blackline). The purple line corresponds to the realization shown in (A-C). G: Evolution of the distribution of cooperation C observedin the system (colors denote time). H: Evolution of the distribution of ρ observed in the system (colors denote time).

a constant value after approximately 102 ∼ 103 gener-ations. Among different realizations, ρ(t) varies signifi-cantly and we study the evolution of its distribution inFig. 2H. We find that at relatively low times, the dis-tribution shows a peak at ρ ≈ 0.2, which then declines.Eventually (black line), there is a high proportion of real-izations with ρ ≈ 0, which must be the case if the systemapproaches a state with nearly full cooperation or defec-tion. It can also be observed from the evolution of thedistribution of cooperation (Fig. 2G) that the probabil-ity of high cooperation C ≈ 1 increases over time. Incombination, these observations indicate that the evolu-tionary path towards full cooperation includes a phaseof significant clustering of cooperators. The stationarydistribution of ρ (colored lines converge to the black linein Fig. 2H) shows that apart from the aforementionedrealizations, the values of ρ are distributed around themean of ρ ≈ 0.2.

To conclude, evolutionary dynamics on scale-free net-works lead to the formation of stable spatial patterns,which can be observed as metric clusters in the angulardimension of underlying hyperbolic metric spaces. This

behavior is similar to spatial selection in lattice topolo-gies, where cooperators form spatial clusters in Euclideanspace.

C. Metric clusters can be more effective than hubs

Let us now consider two empirical networks, the Inter-net Ipv6 topology, which has N = 5162 nodes, a degreedistribution with power-law exponent γ = 2.1, averagedegree k = 5.2, and a mean local clustering coefficient ofc = 0.22 and the arXiv collaboration network, which has1905 nodes, mean degree 〈k〉 = 4.6, mean local cluster-ing coefficient c = 0.66, and a power-law degree exponentγ = 3.9. To address the question of whether spatial clus-ters or the hubs of a network are more efficient at sustain-ing cooperation, we use the initial conditions as a proxyfor possible control mechanisms [26, 29–31]. Specifically,we distribute the initial cooperators (always c(0) = 0.5)in the system as follows: (i) we randomly assign 50% ofthe nodes as cooperators; (ii) we assign the same numberof cooperators preferentially to the hubs of the system,

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Figure 3. Final density of cooperators (after 2 · 105 updatesteps) averaged over 50 realizations of the system, as a func-tion of the game parameters T and S from Eq. 1. Colorsdenote regions in the parameter space where final coopera-tion exceeds the threshold value of 0.3. This is always thecase in the blue area, only if started with either cooperativehubs or a metric cluster in the green region, only if startedwith cooperative hubs in the yellow area, only if started withmetric clusters in the gray area, and for none of the consid-ered initial conditions in the red region. A: Results for theIPv6 Internet topology. B: Results for the arXiv collabora-tion network. C: Synthetic network with N = 2 · 104 nodes,power-law exponent γ = 2.4, mean degree 〈k〉 ≈ 6, and clus-tering c = 0.5. D: The same as before but for networks withpower-law exponent γ = 2.9 and clustering c = 0.6.

i.e. we select nodes proportional to their degree; (iii)we assign the same number of cooperators into a metriccluster in the similarity space (see Methods). The firststrategy serves as a null model, the second mimics thepotential of the hubs to drive the system towards coop-eration, while the last strategy serves as a proxy for theability of metric clusters of cooperators to survive.

Fig. 3A shows the result for the Internet IPv6 topol-ogy, where we show the regions in the T − S plane, inwhich the degree of final cooperation exceeds an arbitrar-ily chosen threshold value of 0.3. In the blue area, thisis always the case. In the green region, this holds if thesystem began with cooperative hubs or a metric cluster.In the yellow region, the cooperative threshold is onlyexceeded if the system began with cooperative hubs (seeSupplementary Materials for details). This behavior issignificantly different in the case of the arXiv collabora-tion network (Fig. 3B). In contrast to the previous case,there is no region where only initially cooperative hubsallow for sustained cooperation. In the gray region, how-

ever, final cooperation only exceeds the threshold value ifthe system was started with cooperators forming a met-ric cluster. Hence, whereas in the Internet IPv6 topologyhubs can drive the system towards cooperation, in thecase of the arXiv network metric clusters are more effi-cient at sustaining cooperation than the most connectednodes. We observe a similar behavior using syntheticscale-free networks with different mean local clusteringcoefficients and power-law exponents. Fig. 3C shows asimilar behavior to that of the Internet (i.e. the hubsare more efficient than metric clusters), whereby the net-works were generated with a power-law exponent γ = 2.4and mean local clustering coefficient c = 0.5 (here, coop-eration is sustained in none of the cases in the red re-gion). In Fig. 3D we find a behavior similar to the arXiv(i.e. metric clusters are more efficient than the hubs),where we have generated synthetic networks with power-law exponent γ = 2.9 and clustering c = 0.6. To con-clude, in very heterogeneous networks, hubs are efficientat driving the system towards cooperation, whereas inless heterogeneous—but including scale-free—networks,metric clusters are more efficient.

To investigate this effect in detail, let us now con-sider the prisoner’s dilemma game, in particular param-eters S = −0.5 and T = 1.5 in the payoff matrix fromEq. (1), which is widely used as a proxy for real socialdilemma situations. We vary the network topology us-ing the model mentioned earlier. In particular, we tunethe heterogeneity in terms of the power-law exponent γand the mean local clustering coefficient, c, which is ameasure of the strength of the underlying metric struc-ture [32]. We consider the different strategies of allocat-ing the initial cooperators discussed before.

The combination of the initial conditions and the net-work topology yields particularly interesting insights. Ifthe initial cooperators are distributed randomly, finalcooperation is always very low for the chosen parame-ters T and S (Fig. 4A). We find the same result (seeFig. S4 in the Supplementary Materials) if the initial co-operators are assigned into a connected (i.e. unique net-work [18]) cluster (see Methods). However, if the initialcooperators are distributed among the hubs of the sys-tem and the network is sufficiently heterogeneous, theyare able to drive the system to a highly cooperative state(see blue region Fig. 4B, and Supplementary Video 2).Large mean local clustering c, which implies a strongmetric structure, adds to this effect (cf. green region inFig. 4D and Fig. 4E), in agreement with [33]. Impor-tantly, if the network is not sufficiently heterogeneous,but still scale-free, the hubs lose their ability to controlthe system and defection eventually prevails (red regionin Fig 4B, Supplementary Video 3). In contrast, if webegin with the initial cooperators clustered in the met-ric space, this will allow for sustained cooperation evenin scale-free networks, but only if the metric structure issufficiently strong (see Fig. 4C, blue region in Fig. 4D,and Supplementary Video 4). If the network becomestoo heterogeneous, the clusters are no longer sustained

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A B C D

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Figure 4. A-C: Final (after 2 · 105 update steps) density of cooperators (color coded) for the prisoner’s dilemma game(T = 1.5 and S = −0.5) averaged over 50 realizations as a function of the degree distribution power-law exponent γ and meanlocal clustering c. Networks have N = 2 · 104 nodes and a mean degree 〈k〉 ≈ 6. The initial density of cooperators is alwaysc(0) = 0.5. A: Randomly assigned initial cooperators. B: Hubs are assigned as initial cooperators with a probability p ∝ k.C: Initial cooperators are localized in the angular space. D: Regions where the final cooperation exceeds a threshold valueof 0.3 for the cases presented in (A-C). E: Final cooperation as a function of the network heterogeneity for different values ofc starting with preferential hub assignment, representing different cuts through (B). Errorbars denote one standard deviationfrom top to bottom. F: Final cooperation as a function of the network heterogeneity for different values of c starting withcooperators assigned into a metric cluster, representing different cuts through (C). G: Final cooperation density as a functionof the network size for synthetic networks with power-law exponent γ = 2.9, c = 0.6, and mean degree 〈k〉 ≈ 6. H: Finalcooperation for different network sizes (see legend) and the same parameters as before. Initial cooperators (c(0) = 0.5) areassigned into different numbers of disjoint metric clusters, whose number is plotted on the x-axis in the top panel. The bottompanel shows on the x-axis the resulting absolute size of each cluster, given by the number of nodes divided by twice the numberof cooperating clusters.

(see Fig. 4F and Supplementary Video 5).

We also investigate whether network and cluster sizeaffect the ability of metric clusters of cooperators to sur-vive. In Fig. 4G we show that the final cooperation den-sity increases with the system size and saturates to avalue close to C = 0.5. For a fixed network size, coop-eration decreases if we assign cooperators into a largernumber of smaller clusters (see Methods), as shown inFig. 4H (top). However, if plotted as a function of theabsolute size of the individual clusters (which can be cal-culated by dividing the number of nodes by twice thenumber of clusters), the curves that correspond to differ-ent network sizes collapse, see Fig. 4H (bottom). Thissuggests that the survival of a metric cluster of coopera-tors is directly related to its absolute size.

Finally, we can formulate approximatively conditionsfor the survival of cooperating metric clusters. Their sur-vival is favored if they are large enough, i.e. their sizeis nc > 103 (see Fig. 4G), if the mean local clustering ishigh enough, i.e. c > 0.5 (see Fig. 4F), and if the networkis not too heterogeneous, i.e. γ > 2.5.

D. Fraction of intercluster links explains thesurvival of metric clusters

The survival of metric clusters of cooperators can beunderstood as analogous to spatial selection in Euclideanspace in lattice topologies. In this case, clusters of coop-erators survive because they are shielded from surround-ing defectors, such that the interactions between cooper-ators and defectors only occur at the border of the clus-ters. Similarly, in heterogeneous networks, metric clus-ters survive because they are shielded from surroundingdefectors and their spatial organization reduces the num-ber of interactions between cooperating and defecting in-dividuals. For larger clusters, the relative surface area ofthe border in contact with adjacent defectors decreases,which shields them more effectively and hence explainswhy they are more likely to survive (see Fig. 5A). For agiven size, two different mechanisms determine the num-ber of links between spatially clustered cooperators anddefectors. Firstly, the greater the degree of heterogene-ity, the larger the number of hubs, i.e. high degree nodes.

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A B

Figure 5. A: Fraction of links between nodes in a met-ric cluster spanning half of the network to nodes outside ofthe cluster compared to the total number of links in the net-work as a function of the network size for synthetic networkswith power-law exponent γ = 2.9, c = 0.6, and mean degree〈k〉 ≈ 6. The shaded area denotes one standard deviationfrom top to bottom. B: The same fraction of links (colorcoded) for synthetic networks with N = 2 · 104 nodes and〈k〉 ≈ 6 as a function of the power-law exponent γ and meanlocal clustering c.

These nodes are connected to many other nodes, andtherefore form long-range connections in the metric rep-resentation, which are likely to connect cooperators anddefectors. This is the reason why more heterogeneity hin-ders the survival of metric clusters (Fig. 5B). For a fixedlevel of heterogeneity, increasing the mean local cluster-ing coefficient will reduce the temperature T , which re-duces the amount of long-range connections due to ran-domness, cf. Fig. 1A and B. Therefore, a higher degree ofmean local clustering reduces the number of interclusterlinks, which in turn favors the survival of metric clustersas explained before (see Fig. 5B).

III. DISCUSSION

Structured populations play an important role in theevolution of cooperation in social dilemmas. Real con-tact networks are heterogeneous (often scale-free) andexhibit a high mean local clustering coefficient. The lat-ter implies the existence of an underlying geometry [21].Specifically, real heterogeneous networks can be embed-ded into hyperbolic space comprised of a popularity (ra-dial) dimension and a similarity (angular) dimension.

We have shown that this underlying metric space playsan important role in the evolution of cooperation in het-erogeneous contact networks. Specifically, evolutionarydynamics lead to the formation of clusters of cooperatorsin the angular dimension of the underlying metric space,akin to spatial selection in Euclidean space [13, 14]. Thisbehavior can be understood in terms of the fraction of in-tercluster links that determines how well metric clustersof cooperators are shielded from surrounding defectors.Depending on the power-law exponent γ of the degreedistribution and the mean local clustering coefficient c(which is proportional to the strength of the metric struc-

ture), metric clusters can be more efficient at sustain-ing cooperation than the most connected nodes, which isthe case in the arXiv collaboration network. Only whenthe network is very heterogeneous, such as in the caseof the Internet IPv6 topology, are hubs more effectiveat promoting cooperation. We have shown that if coop-erators are clustered in the metric space, heterogeneitycan hinder cooperation in the prisoner’s dilemma. Fi-nally, one could argue that such a configuration is morerealistic than random initial conditions, as for examplethe nodes in the Internet network that correspond tothe same countries are naturally clustered in the metricspace (see [25]), and different countries adopt differentattitudes towards mitigating climate change [34].

Our findings reveal that heterogeneity does not alwaysfavor cooperation in evolutionary games on structuredpopulations, but can even have the opposite effect, thuscomplementing existing studies about the impact of het-erogeneity of realistic contact networks. Furthermore,our framework unifies the description of spatial effectsand the heterogeneity of contact networks. This frame-work can be applied to different games and extended tomultiplex networks, opening promising new lines of re-search.

ACKNOWLEDGMENTS

We thank Stefano Duca for interesting and helpful dis-cussions. We thank Heinrich Nax for feedback on themanuscript. We thank Eoin Jones for proofreading themanuscript. K-K. K. acknowledges support by the ERCGrant “Momentum” (324247).

METHODS

Evolutionary game dynamics

In the evolutionary game dynamics considered here, in-dividuals play strategic games with their contacts where,for instance, they have two strategic choices: they caneither cooperate (C) or defect (D). The payoff of eachtwo-player game is then described by the payoff matrix

M =

C D

C 1 S

D T 0

. (1)

Parameters T and S define different games [6]. T < 1 andS > 0 defines the “harmony” game, T < 1 and S < 0corresponds to the “stag hunt” game, T > 1 and S < 0yields the “prisoner’s dilemma”, and finally for T > 1and S > 0 we obtain the “snowdrift” game.

One round of the game consists of each individual play-ing one game with each of her neighbors in the net-work of contacts. For each game, nodes collect pay-offs given by Eq. 1, which depend on the strategies of

7

the involved players. Here, we consider the evolution ofthe system to be governed by imitation dynamics [35–37](Fig. 1C), reflecting that individuals tend to adopt thestrategy of more successful neighbors. After each roundof the game (synchronous updates) each node i choosesone neighbor j at random and copies her strategy withprobability Pi←j , specified by the Fermi-Dirac distribu-tion [15, 38, 39]

Pi←j =1

1 + e−(πj−πi)/K, (2)

motivated by maximum entropy principles in Glauber-like dynamics [35, 40]. πi and πj measure the payoffs ofnodes i and j, while K denotes the irrationality of theplayers, which we set to 0.5. After all nodes have updatedtheir strategy simultaneously, we reset all payoffs.

In this contribution, games are played only on the giantconnected component (GCC) of the network of contacts.

Complex networks embedded into underlying metricspaces

Metric spaces underlying complex networks providea fundamental explanation of their observed topolo-gies [23, 24]. In the class of models used here, each nodei is mapped into the hyperbolic disk where it is repre-sented by the polar coordinates ri, θi. These coordinatesabstract the popularity and similarity of nodes [24]. Theradial coordinate ri is related to the expected degree ofnode i and therefore abstracts its popularity. More pop-ular nodes are located closer to the center of the disk(lower radial coordinate). The angular distance betweennodes i and j, ∆θij = π − |π − |θi − θj ||, is an abstractmeasure of their similarity. Lower distance implies highersimilarity. The hyperbolic distance [23],

xij = cosh−1 (cosh ri cosh rj − sinh ri sinh rj cos ∆θij) ,

(3)

combines information about both popularity and similar-ity of nodes i and j, such that the connection probabilityfor a given pair of nodes depends only on their hyperbolicdistance.

To generate networks based on hidden hyperbolicspace, we distribute nodes on the hyperbolic disk by as-signing polar coordinates (ri, θi) to each node. In par-ticular, we draw θi from the uniform distribution U[0,2π)and radial coordinates ri from the distribution

ρ(r) =1

2(γ − 1)e

12 (γ−1)(r−R) , (4)

where R denotes the disk radius given by [23]

R = 2 ln

[2TN

k sin T π

(γ − 1

γ − 2

)2], (5)

where N denotes the number of nodes, γ is the power-law exponent of the degree distribution, and T denotesthe temperature. Finally, we connect pairs of nodes iand j with probability p(xij), which depends exclusivelyon the hyperbolic distance xij between nodes i and j.The connection probability is given by the Fermi-Diracdistribution

p(xij) =1

1 + e1

2T(xij−R)

, (6)

where the aforementioned temperature T controls thestrength of the metric structure and the level of meanlocal clustering, c. This is illustrated in Figs. 1A and B.

Finally, given a real network, coordinates of the nodescan be inferred using maximum likelihood estimationtechniques [41, 42]. This enables us to identify the setof coordinates that maximize the probability that theobserved real-world network was generated using the de-scribed model. The inferred hyperbolic maps have provento be very accurate in the case of scale-free, clustered net-works [25, 43, 44].

Mean local clustering and relation to the spatialstructure

The local clustering coefficient of node i is definedas [45]

ci =Number of closed triangles i participates in

ki(ki − 1), (7)

where ki denotes the degree of node i. The maximalnumber of closed triangles a node with degree ki canparticipate in is ki(ki − 1). The mean local clusteringcoefficient of a given network is then the average of ciover all nodes with k > 1 (nodes with k = 1 cannotparticipate in any triangles).

In the framework introduced in the previous section, alow temperature T implies a high mean local clustering,which is the consequence of the triangle inequality in theunderlying metric space (Fig. 1A). A high temperature,however, induces more randomness in the form of long-range connections (see Eq. (6)), which reduces the meanlocal clustering coefficient (Fig. 1B). See [23] for furtherdetails.

KS-statistic

The Kolmogorov-Smirnov (KS) statistic [28], whichwe denote as ρ, is defined as the maximum absolutedifference between the values of two cumulative distri-bution. We are interested in measuring the differencebetween the distribution of cooperators in the angularspace, c(θ), whose cumulative distribution is given by

C(θ) =∫ θ0

dθ′ c(θ′), and the uniform distribution. Then,

8

the KS statistic is given by

ρC = maxθ∈{0,2π}

∣∣∣∣C(θ)− θ

2π

∣∣∣∣ (8)

and analogously

ρD = maxθ∈{0,2π}

∣∣∣∣D(θ)− θ

2π

∣∣∣∣ , (9)

where D(θ) = θ − C(θ) denotes the cumulative distribu-tion of defectors in the angular space. Finally, we define

ρ = cρC + (1− c)ρD , (10)

where c denotes the density of cooperators at the currenttimestep. Note that here we omitted the time depen-dency.

Assignment of initial cooperators

In this contribution, we always start with an initialcooperation density of C(0) = 0.5. However, the distri-bution of initial cooperators in the network can be differ-ent. In particular, we distinguish between the followingprocedures.

Random assignment: Each node is initialized as acooperator with 50% probability and as a defector oth-erwise.

Hubs: We preferentially assign hubs as initial coop-erators. To this end, we assign N/2 cooperators whichwe select proportional to their degree, i.e. pc(k) ∝ k. Ndenotes total number of nodes in the network.

Metric cluster: We sort all nodes by their angularcoordinate θ, and assign the first N/2 nodes as coopera-tors.

Multiple metric clusters: We again sort all nodesby their angular coordinate θ. We now fix a number ofdistinct clusters, nc, and assign the first N/(2nc) nodesas cooperators, the second N/(2nc) nodes as defectors,the third N/(2nc) as cooperators and so on. See Supple-mentary Materials for an explicit example.

Connected cluster: We assign N/2 nodes into a con-nected cluster, or unique network cluster [18]. To thisend, we start from the initial graph and randomly re-move nodes until the size of the giant connected compo-nent (GCC) reaches N/2. The nodes that are now in theGCC are assigned as cooperators in the original graph,and the remaining N/2 nodes are assigned as defectors.

This procedure ensures that the initial cooperators forma unique connected cluster. Note that a network clusterin general is not the same as a metric cluster.

Empirical networks

The arXiv data is taken from [46] and contains co-authorship networks from the free scientific repositoryarXiv. The nodes are authors that are connected if theyhave co-authored a paper. In arXiv, each paper is as-signed to one or more relevant categories. The data onlycovers papers containing the word “networks” in the ti-tle or abstract from different categories up to May 2014.Here, we consider the category “Molecular Networks” (q-bio.MN). The network has approx. 1905 nodes, meandegree 〈k〉 = 4.6, clustering coefficient c = 0.66, and apower-law degree exponent γ = 3.9.

The IPv6 Autonomous Systems (AS) Internet topologywas extracted from the data collected by the Archipelagoactive measurement infrastructure (ARK) developed byCAIDA [47]. The connections in each topology are notphysical but logical, representing AS relationships. AnAS is a part of the Internet infrastructure administratedby a single company or organization. Pairs of ASs peerto exchange traffic. These peering relationships in theAS topology are represented as links between AS nodes.CAIDA’s IPv6 [48] datasets provide regular snapshots ofAS links derived from ongoing traceroute-based IP-leveltopology measurements. The considered topology wasconstructed by merging the AS link snapshots during thefirst 15 days of January 2015, which are provided at [49].The network consists of N = 5162 nodes, has a powerlaw degree distribution with exponent γ = 2.1, averagenode degree k = 5.2, and average clustering c2 = 0.22.

The hyperbolic maps for both datasets have been takenfrom [44].

Supplementary Fig. S2 shows the final density of coop-erators for both networks and for the different allocationstrategies described in the main text.

Data and code availability

To facilitate the application of our framework forfuture work, we enclose the empirical datasets andtheir inferred hyperbolic maps as well as an imple-mentation of the model networks used in this pa-per (figshare.com/articles/DataAndModel zip/4817947).An implementation of the technique to construct hyper-bolic maps for real networks [41, 42] is publicly availableat [50].

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