Metric clusters in evolutionary games on scale-free networks

TL;DR

This paper reveals that in scale-free networks, evolutionary dynamics form spatial clusters in an underlying metric space, which can promote cooperation by shielding cooperators from defectors, challenging previous assumptions about heterogeneity always favoring cooperation.

Contribution

It demonstrates that scale-free networks exhibit metric clustering that influences cooperation, providing new insights into spatial effects in structured populations.

Findings

Metric clusters of cooperators can survive by shielding from defectors.

Heterogeneity can hinder cooperation under certain conditions.

Spatial organization in scale-free networks influences evolutionary outcomes.

Abstract

The evolution of cooperation in social dilemmas in structured populations has been studied extensively in recent years. Whereas many theoretical studies have found that a heterogeneous network of contacts favors cooperation, the impact of spatial effects in scale-free networks is still not well understood. In addition to being heterogeneous, real contact networks exhibit a high mean local clustering coefficient, which implies the existence of an underlying metric space. Here, we show that evolutionary dynamics in scale-free networks self-organize into spatial patterns in the underlying metric space. The resulting metric clusters of cooperators are able to survive in social dilemmas as their spatial organization shields them from surrounding defectors, similar to spatial selection in Euclidean space. We show that under certain conditions these metric clusters are more efficient than the 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 to understand the emergence of cooperation in evolutionary games in realistic structured populations.

Equations

$$M=\begin{array}[]{c|cc}&\text{C}&\text{D}\\ \hline\cr\text{C}&1&S\\ \text{D}&T&0\end{array}\,.$$

$$P_{i\leftarrow j}=\frac{1}{1+e^{-({\pi }_j-{\pi }_i)/K}},$$

$$\begin{array}{rl}{x}_{ij}& ={\cosh }^{-1}\left(\cosh r_i\cosh r_j-\sinh r_i\sinh r_j\cos \Delta {\theta }_{ij}\right),\end{array}$$

$$\rho (r)=\frac{1}{2}(\gamma -1)e^{\frac{1}{2}(\gamma -1)(r-R)},$$

$$R=2\ln \left[\frac{2TN}{\bar{k}\sin \bar{T}\pi }{\left(\frac{\gamma -1}{\gamma -2}\right)}^2\right],$$

$$p(x_{ij})=\frac{1}{1+e^{\frac{1}{2\bar{T}}(x_{ij}-R)}},$$

$$c_i=\frac{\text{Number of closed triangles }i\text{ participates in}}{k_i(k_i-1)},$$

$${\rho }_C=\underset{\theta \in \{0,2\pi \}}{\max }\left|C(\theta )-\frac{\theta }{2\pi }\right|$$

$${\rho }_D=\underset{\theta \in \{0,2\pi \}}{\max }\left|D(\theta )-\frac{\theta }{2\pi }\right|,$$

$$\bar{\rho }=c{\rho }_C+(1-c){\rho }_D,$$

Full text

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

I Introduction

Cooperation among humans has been found to be quite common in social dilemmas Martin Nowak (2011); Nowak (2006a), and plays a major role in the emergence of complex modern societies Smith and Szathmary (1995); Pennisi (2005). Therefore, understanding the underlying mechanisms that can give rise to and sustain cooperation from an evolutionary perspective is key to complementing Darwin’s theory of evolution Hofbauer and Sigmund (1998); Nowak (2006b); Smith (1982); Axelrod (1984); Weibull (1995).

In reality, populations are structured, which means that the topology of strategic interactions is given by a network of contacts. In structured populations, individuals interact repeatedly with the same individuals. Thus, as a consequence, cooperators can survive in social dilemmas by forming network clusters. This mechanism is referred to as network reciprocity Allen et al. (2017); Nowak (2006a). In the well-studied case of lattice topologies, the resulting network clusters unfold in Euclidean space Grujić et al. (2014); Perc et al. (2013); Rand et al. (2014) (spatial selection Nowak and May (1992)). Realistic networks of contacts are heterogeneous rather than lattices and often scale-free, which means that their degree distribution follows a power-law with exponent $\gamma\in(2,3)$, where a lower value of $\gamma$ means more heterogeneous networks. Heterogeneity has been shown to favor cooperation Santos and Pacheco (2005); Santos et al. (2006); Szolnoki et al. (2008), and cooperating nodes form a connected (or network) cluster Gómez-Gardeñes et al. (2007). However, the geometric organization of these connected clusters—similarly to spatial selection in Euclidean space—remains elusive.

Real complex networks, in addition to being heterogeneous, exhibit a high mean local clustering coefficient Newman (2010a); Newman et al. (2006) (this means that the network contains a high number of closed triangles). This is particularly important because a high clustering coefficient implies the existence of an underlying metric space Krioukov (2016a).

We show that evolutionary dynamics on scale-free, highly clustered networks lead to the formation of patterns in the underlying metric space, similar to the aforementioned spatial selection in Euclidean space. Using two empirical networks, the IPv6 Internet topology and the arXiv collaboration network, as well as synthetic networks, we show that spatial patterns play an important role in the evolution of cooperation. In fact, under certain conditions metric clusters can even be more effective at sustaining cooperation than the most connected nodes (hubs). As a consequence, heterogeneity does not always favor—but can even hinder—the evolution of cooperation in social dilemmas.

II Results

II.1 Latent geometry of scale-free networks

Real contact networks are usually heterogeneous, and often scale-free, as well as highly clustered Newman et al. (2006) (we refer to a high mean local clustering coefficient, i.e. a large number of closed triangles Newman (2010a)). The effect of scale-free topologies has attracted a lot of attention, and many theoretical studies have found that heterogeneous networks of contacts favor cooperation in social dilemmas Santos and Pacheco (2005); Santos et al. (2006); Szolnoki et al. (2008); Gómez-Gardeñes et al. (2007), although this behavior has not been confirmed in recent experiments with human players Gracia-Lazaro et al. (2012). Importantly, the high local clustering coefficients found in real contact networks have been proven to imply the existence of a metric space underlying the observed topology Krioukov (2016a). This means that the nodes of a given real complex network can be mapped to coordinates in this metric space such that the probability that pairs of nodes will be connected in the observed topology depends only on their distance in the metric space. Specifically, heterogeneous networks can be embedded into hyperbolic space Krioukov et al. (2010); Papadopoulos et al. (2012); Boguñá et al. (2010). In this representation, each node has a radial and angular coordinate. The radial coordinate abstracts the popularity, and hence the degree of the node, such that hubs are placed closer to the center of the disk (Fig. 1A). The angular coordinate abstracts a similarity space, such that the angular distance is a measure of the similarity between two nodes, whereby nodes tend to connect to more similar nodes. In Fig. 1A and B we show an illustration of the hyperbolic metric structure underlying two different networks (see Methods section for further details). In the following, we show that evolutionary dynamics trigger the formation of stable spatial clusters in the angular dimension on the underlying hyperbolic space.

II.2 Evolutionary dynamics and the emergence of metric clusters

Let us first consider the prisoner’s dilemma game, in particular $T=1.2$ and $S=-0.2$ (see Eq. (1) in Methods) on synthetic contact networks generated with the model described in Methods. This model generates realistic topologies based on underlying hyperbolic metric spaces, similar to Fig. 1A. We simulate the evolutionary game dynamics (see Methods and Fig. 1C) and find that the system tends to self-organize into a state in which groups which are mainly cooperative are clearly separate from groups populated mainly by defectors (see Fig. 2A-C and Supplementary Video 1) Kleineberg and Helbing (2017); Amato et al. (2017).

Similarly to the case of lattice topologies and spatial selection in Euclidean space, we observe the formation of clusters of cooperators in the angular dimension of the underlying hyperbolic space. In Fig. 2D we show the evolution of the density of cooperators in different bins of the angular coordinate $\theta$. We observe that initially cooperation decreases (see purple line in Fig. 2E) while, at the same time, the remaining cooperators become concentrated in clusters in the angular space (Fig. 2A). Cooperation then increases again as it spreads in the vicinity of the clusters (Fig. 2B) until the system reaches a stationary state with fluctuations only at the borders of the clusters (Fig. 2C and Supplementary Video 1).

We quantify the degree to which cooperators and defectors cluster in the angular space using the Kolmogorov-Smirnov (KS) statistic Zuev et al. (2015), which measures the difference between two one-dimensional distributions. The KS-statistic is defined as the maximum absolute difference between the values of two cumulative distributions.

In particular, we define the KS-statistic of the distribution of cooperation density in different angular bins, and the uniform distribution at time $t$ as $\bar{\rho}(t)$ (see Methods for details). A higher value of $\bar{\rho}(t)$ thus denotes more pronounced clustering of cooperators and defectors respectively. In Fig. 2F we show the evolution of $\bar{\rho}(t)$ for $10^{3}$ different realizations (blue lines) and their mean (black line). On average, $\bar{\rho}(t)$ increases initially and approaches a constant value after approximately $10^{2}\sim 10^{3}$ generations. Among different realizations, $\bar{\rho}(t)$ varies significantly and we study the evolution of its distribution in Fig. 2H. We find that at relatively low times, the distribution shows a peak at $\bar{\rho}\approx 0.2$, which then declines. Eventually (black line), there is a high proportion of realizations with $\bar{\rho}\approx 0$, which must be the case if the system approaches a state with nearly full cooperation or defection. It can also be observed from the evolution of the distribution of cooperation (Fig. 2G) that the probability of high cooperation $C\approx 1$ increases over time. In combination, these observations indicate that the evolutionary path towards full cooperation includes a phase of significant clustering of cooperators. The stationary distribution of $\bar{\rho}$ (colored lines converge to the black line in Fig. 2H) shows that apart from the aforementioned realizations, the values of $\bar{\rho}$ are distributed around the mean of $\bar{\rho}\approx 0.2$.

To conclude, evolutionary dynamics on scale-free networks lead to the formation of stable spatial patterns, which can be observed as metric clusters in the angular dimension of underlying hyperbolic metric spaces. This behavior is similar to spatial selection in lattice topologies, where cooperators form spatial clusters in Euclidean space.

II.3 Metric clusters can be more effective than hubs

Let us now consider two empirical networks, the Internet Ipv6 topology, which has $N=5162$ nodes, a degree distribution with power-law exponent $\gamma=2.1$, average degree $\bar{k}=5.2$, and a mean local clustering coefficient of $\bar{c}=0.22$ and the arXiv collaboration network, which has $1905$ nodes, mean degree $\left<k\right>=4.6$, mean local clustering coefficient $\bar{c}=0.66$, and a power-law degree exponent $\gamma=3.9$. To address the question of whether spatial clusters or the hubs of a network are more efficient at sustaining cooperation, we use the initial conditions as a proxy for possible control mechanisms Liu and Barabási (2016); Helbing (2013, 2012); Kleineberg and Helbing (2017). Specifically, we distribute the initial cooperators (always $c(0)=0.5$) in the system as follows: (i) we randomly assign $50\%$ of the nodes as cooperators; (ii) we assign the same number of cooperators preferentially to the hubs of the system, i.e. we select nodes proportional to their degree; (iii) we assign the same number of cooperators into a metric cluster in the similarity space (see Methods). The first strategy serves as a null model, the second mimics the potential of the hubs to drive the system towards cooperation, while the last strategy serves as a proxy for the ability of metric clusters of cooperators to survive.

Fig. 3A shows the result for the Internet IPv6 topology, where we show the regions in the $T-S$ plane, in which the degree of final cooperation exceeds an arbitrarily chosen threshold value of $0.3$. In the blue area, this is always the case. In the green region, this holds if the system began with cooperative hubs or a metric cluster. In the yellow region, the cooperative threshold is only exceeded if the system began with cooperative hubs (see Supplementary Materials for details). This behavior is significantly different in the case of the arXiv collaboration network (Fig. 3B). In contrast to the previous case, there is no region where only initially cooperative hubs allow for sustained cooperation. In the gray region, however, final cooperation only exceeds the threshold value if the system was started with cooperators forming a metric cluster. Hence, whereas in the Internet IPv6 topology hubs can drive the system towards cooperation, in the case of the arXiv network metric clusters are more efficient at sustaining cooperation than the most connected nodes. We observe a similar behavior using synthetic scale-free networks with different mean local clustering coefficients and power-law exponents. Fig. 3C shows a similar behavior to that of the Internet (i.e. the hubs are more efficient than metric clusters), whereby the networks were generated with a power-law exponent $\gamma=2.4$ and mean local clustering coefficient $\bar{c}=0.5$ (here, cooperation is sustained in none of the cases in the red region). 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 $\gamma=2.9$ and clustering $\bar{c}=0.6$.

To conclude, in very heterogeneous networks, hubs are efficient at driving the system towards cooperation, whereas in less heterogeneous—but including scale-free—networks, metric clusters are more efficient.

To investigate this effect in detail, let us now consider the prisoner’s dilemma game, in particular parameters $S=-0.5$ and $T=1.5$ in the payoff matrix from Eq. (1), which is widely used as a proxy for real social dilemma situations. We vary the network topology using the model mentioned earlier. In particular, we tune the heterogeneity in terms of the power-law exponent $\gamma$ and the mean local clustering coefficient, $\bar{c}$, which is a measure of the strength of the underlying metric structure Krioukov (2016b). We consider the different strategies of allocating the initial cooperators discussed before.

The combination of the initial conditions and the network topology yields particularly interesting insights. If the initial cooperators are distributed randomly, final cooperation is always very low for the chosen parameters $T$ and $S$ (Fig. 4A).

We find the same result (see Fig. S4 in the Supplementary Materials) if the initial cooperators are assigned into a connected (i.e. unique network Gómez-Gardeñes et al. (2007)) cluster (see Methods). However, if the initial cooperators are distributed among the hubs of the system and the network is sufficiently heterogeneous, they are able to drive the system to a highly cooperative state (see blue region Fig. 4B, and Supplementary Video 2).

Large mean local clustering $\bar{c}$, which implies a strong metric structure, adds to this effect (cf. green region in Fig. 4D and Fig. 4E), in agreement with Assenza et al. (2008). Importantly, if the network is not sufficiently heterogeneous, but still scale-free, the hubs lose their ability to control the system and defection eventually prevails (red region in Fig 4B, Supplementary Video 3). In contrast, if we begin with the initial cooperators clustered in the metric space, this will allow for sustained cooperation even in scale-free networks, but only if the metric structure is sufficiently strong (see Fig. 4C, blue region in Fig. 4D, and Supplementary Video 4). If the network becomes too heterogeneous, the clusters are no longer sustained (see Fig. 4F and Supplementary Video 5).

We also investigate whether network and cluster size affect the ability of metric clusters of cooperators to survive. In Fig. 4G we show that the final cooperation density increases with the system size and saturates to a value close to $C=0.5$. For a fixed network size, cooperation decreases if we assign cooperators into a larger number of smaller clusters (see Methods), as shown in Fig. 4H (top). However, if plotted as a function of the absolute size of the individual clusters (which can be calculated by dividing the number of nodes by twice the number of clusters), the curves that correspond to different network sizes collapse, see Fig. 4H (bottom). This suggests that the survival of a metric cluster of cooperators is directly related to its absolute size.

Finally, we can formulate approximatively conditions for the survival of cooperating metric clusters. Their survival is favored if they are large enough, i.e. their size is $n_{c}>10^{3}$ (see Fig. 4G), if the mean local clustering is high enough, i.e. $\bar{c}>0.5$ (see Fig. 4F), and if the network is not too heterogeneous, i.e. $\gamma>2.5$.

II.4 Fraction of intercluster links explains the survival of metric clusters

The survival of metric clusters of cooperators can be understood as analogous to spatial selection in Euclidean space in lattice topologies. In this case, clusters of cooperators survive because they are shielded from surrounding defectors, such that the interactions between cooperators and defectors only occur at the border of the clusters. Similarly, in heterogeneous networks, metric clusters survive because they are shielded from surrounding defectors and their spatial organization reduces the number of interactions between cooperating and defecting individuals. For larger clusters, the relative surface area of the border in contact with adjacent defectors decreases, which shields them more effectively and hence explains why they are more likely to survive (see Fig. 5A). For a given size, two different mechanisms determine the number of links between spatially clustered cooperators and defectors. Firstly, the greater the degree of heterogeneity, the larger the number of hubs, i.e. high degree nodes. These nodes are connected to many other nodes, and therefore form long-range connections in the metric representation, which are likely to connect cooperators and defectors. This is the reason why more heterogeneity hinders the survival of metric clusters (Fig. 5B). For a fixed level of heterogeneity, increasing the mean local clustering coefficient will reduce the temperature $\bar{T}$, which reduces the amount of long-range connections due to randomness, cf. Fig. 1A and B. Therefore, a higher degree of mean local clustering reduces the number of intercluster links, which in turn favors the survival of metric clusters as explained before (see Fig. 5B).

III Discussion

Structured populations play an important role in the evolution of cooperation in social dilemmas. Real contact networks are heterogeneous (often scale-free) and exhibit a high mean local clustering coefficient. The latter implies the existence of an underlying geometry Krioukov (2016a). Specifically, real heterogeneous networks can be embedded into hyperbolic space comprised of a popularity (radial) dimension and a similarity (angular) dimension.

We have shown that this underlying metric space plays an important role in the evolution of cooperation in heterogeneous contact networks. Specifically, evolutionary dynamics lead to the formation of clusters of cooperators in the angular dimension of the underlying metric space, akin to spatial selection in Euclidean space Nowak and May (1992); Rand et al. (2014). This behavior can be understood in terms of the fraction of intercluster links that determines how well metric clusters of cooperators are shielded from surrounding defectors. Depending on the power-law exponent $\gamma$ of the degree distribution and the mean local clustering coefficient $\bar{c}$ (which is proportional to the strength of the metric structure), metric clusters can be more efficient at sustaining cooperation than the most connected nodes, which is the case in the arXiv collaboration network. Only when the network is very heterogeneous, such as in the case of the Internet IPv6 topology, are hubs more effective at promoting cooperation.

We have shown that if cooperators are clustered in the metric space, heterogeneity can hinder cooperation in the prisoner’s dilemma. Finally, one could argue that such a configuration is more realistic than random initial conditions, as for example the nodes in the Internet network that correspond to the same countries are naturally clustered in the metric space (see Boguñá et al. (2010)), and different countries adopt different attitudes towards mitigating climate change Lee et al. (2015).

Our findings reveal that heterogeneity does not always favor cooperation in evolutionary games on structured populations, but can even have the opposite effect, thus complementing existing studies about the impact of heterogeneity of realistic contact networks. Furthermore, our framework unifies the description of spatial effects and the heterogeneity of contact networks. This framework can be applied to different games and extended to multiplex networks, opening promising new lines of research.

Acknowledgements.

We thank Stefano Duca for interesting and helpful discussions. We thank Heinrich Nax for feedback on the manuscript. We thank Eoin Jones for proofreading the manuscript. K-K. K. acknowledges support by the ERC Grant “Momentum” (324247).

Methods

Evolutionary game dynamics

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

[TABLE]

Parameters $T$ and $S$ define different games Nowak (2006b). $T<1$ and $S>0$ defines the “harmony” game, $T<1$ and $S<0$ corresponds to the “stag hunt” game, $T>1$ and $S<0$ yields the “prisoner’s dilemma”, and finally for $T>1$ and $S>0$ we obtain the “snowdrift” game.

One round of the game consists of each individual playing one game with each of her neighbors in the network of contacts. For each game, nodes collect payoffs given by Eq. 1, which depend on the strategies of the involved players. Here, we consider the evolution of the system to be governed by imitation dynamics Szabó and Fáth (2007); Cressman and Tao (2014); Helbing (1996) (Fig. 1C), reflecting that individuals tend to adopt the strategy of more successful neighbors. After each round of the game (synchronous updates) each node $i$ chooses one neighbor $j$ at random and copies her strategy with probability $P_{i\leftarrow j}$, specified by the Fermi-Dirac distribution Santos and Pacheco (2005); Szabó and Tőke (1998); Amaral et al. (2016)

[TABLE]

motivated by maximum entropy principles in Glauber-like dynamics Szabó and Fáth (2007); Perc (2006). $\pi_{i}$ and $\pi_{j}$ measure the payoffs of nodes $i$ and $j$, while $K$ denotes the irrationality of the players, which we set to $0.5$. After all nodes have updated their strategy simultaneously, we reset all payoffs.

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

Complex networks embedded into underlying metric spaces

Metric spaces underlying complex networks provide a fundamental explanation of their observed topologies Krioukov et al. (2010); Papadopoulos et al. (2012). In the class of models used here, each node $i$ is mapped into the hyperbolic disk where it is represented by the polar coordinates $r_{i},\theta_{i}$. These coordinates abstract the popularity and similarity of nodes Papadopoulos et al. (2012). The radial coordinate $r_{i}$ is related to the expected degree of node $i$ and therefore abstracts its popularity. More popular nodes are located closer to the center of the disk (lower radial coordinate). The angular distance between nodes $i$ and $j$, $\Delta\theta_{ij}=\pi-|\pi-|\theta_{i}-\theta_{j}||$, is an abstract measure of their similarity. Lower distance implies higher similarity. The hyperbolic distance Krioukov et al. (2010),

[TABLE]

combines information about both popularity and similarity of nodes $i$ and $j$, such that the connection probability for a given pair of nodes depends only on their hyperbolic distance.

To generate networks based on hidden hyperbolic space, we distribute nodes on the hyperbolic disk by assigning polar coordinates ($r_{i}$, $\theta_{i}$) to each node. In particular, we draw $\theta_{i}$ from the uniform distribution $\mathcal{U}_{[0,2\pi)}$ and radial coordinates $r_{i}$ from the distribution

[TABLE]

where $R$ denotes the disk radius given by Krioukov et al. (2010)

[TABLE]

where $N$ denotes the number of nodes, $\gamma$ is the power-law exponent of the degree distribution, and $T$ denotes the temperature. Finally, we connect pairs of nodes $i$ and $j$ with probability $p(x_{ij})$, which depends exclusively on the hyperbolic distance $x_{ij}$ between nodes $i$ and $j$. The connection probability is given by the Fermi-Dirac distribution

[TABLE]

where the aforementioned temperature $\bar{T}$ controls the strength of the metric structure and the level of mean local clustering, $\bar{c}$. This is illustrated in Figs. 1A and B.

Finally, given a real network, coordinates of the nodes can be inferred using maximum likelihood estimation techniques Papadopoulos et al. (2015a, b). This enables us to identify the set of coordinates that maximize the probability that the observed real-world network was generated using the described model. The inferred hyperbolic maps have proven to be very accurate in the case of scale-free, clustered networks Serrano et al. (2012); Boguñá et al. (2010); Kleineberg et al. (2016).

Mean local clustering and relation to the spatial structure

The local clustering coefficient of node $i$ is defined as Newman (2010b)

[TABLE]

where $k_{i}$ denotes the degree of node $i$. The maximal number of closed triangles a node with degree $k_{i}$ can participate in is $k_{i}(k_{i}-1)$. The mean local clustering coefficient of a given network is then the average of $c_{i}$ over all nodes with $k>1$ (nodes with $k=1$ cannot participate in any triangles).

In the framework introduced in the previous section, a low temperature $\bar{T}$ implies a high mean local clustering, which is the consequence of the triangle inequality in the underlying metric space (Fig. 1A). A high temperature, however, induces more randomness in the form of long-range connections (see Eq. (6)), which reduces the mean local clustering coefficient (Fig. 1B). See Krioukov et al. (2010) for further details.

KS-statistic

The Kolmogorov-Smirnov (KS) statistic Zuev et al. (2015), which we denote as $\bar{\rho}$, is defined as the maximum absolute difference between the values of two cumulative distribution. We are interested in measuring the difference between the distribution of cooperators in the angular space, $c(\theta)$, whose cumulative distribution is given by $C(\theta)=\int_{0}^{\theta}\!\mathrm{d}\theta^{\prime}\,c(\theta^{\prime})$, and the uniform distribution. Then, the KS statistic is given by

[TABLE]

and analogously

[TABLE]

where $D(\theta)=\theta-C(\theta)$ denotes the cumulative distribution of defectors in the angular space. Finally, we define

[TABLE]

where $c$ denotes the density of cooperators at the current timestep. Note that here we omitted the time dependency.

Assignment of initial cooperators

In this contribution, we always start with an initial cooperation density of $C(0)=0.5$. However, the distribution of initial cooperators in the network can be different. In particular, we distinguish between the following procedures.

Random assignment: Each node is initialized as a cooperator with $50\%$ probability and as a defector otherwise.

Hubs: We preferentially assign hubs as initial cooperators. To this end, we assign $N/2$ cooperators which we select proportional to their degree, i.e. $p_{c}(k)\propto k$. $N$ denotes total number of nodes in the network.

Metric cluster: We sort all nodes by their angular coordinate $\theta$, and assign the first $N/2$ nodes as cooperators.

Multiple metric clusters: We again sort all nodes by their angular coordinate $\theta$. We now fix a number of distinct clusters, $n_{c}$, and assign the first $N/(2n_{c})$ nodes as cooperators, the second $N/(2n_{c})$ nodes as defectors, the third $N/(2n_{c})$ as cooperators and so on. See Supplementary Materials for an explicit example.

Connected cluster: We assign $N/2$ nodes into a connected cluster, or unique network cluster Gómez-Gardeñes et al. (2007). To this end, we start from the initial graph and randomly remove nodes until the size of the giant connected component (GCC) reaches $N/2$. The nodes that are now in the GCC 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 form a unique connected cluster. Note that a network cluster in general is not the same as a metric cluster.

Empirical networks

The arXiv data is taken from De Domenico et al. (2015) and contains co-authorship networks from the free scientific repository arXiv. The nodes are authors that are connected if they have co-authored a paper. In arXiv, each paper is assigned to one or more relevant categories. The data only covers papers containing the word “networks” in the title 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, mean degree $\left<k\right>=4.6$, clustering coefficient $\bar{c}=0.66$, and a power-law degree exponent $\gamma=3.9$.

The IPv6 Autonomous Systems (AS) Internet topology was extracted from the data collected by the Archipelago active measurement infrastructure (ARK) developed by CAIDA Claffy et al. (2009). The connections in each topology are not physical but logical, representing AS relationships. An AS is a part of the Internet infrastructure administrated by a single company or organization. Pairs of ASs peer to exchange traffic. These peering relationships in the AS topology are represented as links between AS nodes. CAIDA’s IPv6 as_ (2015) datasets provide regular snapshots of AS links derived from ongoing traceroute-based IP-level topology measurements. The considered topology was constructed by merging the AS link snapshots during the first 15 days of January 2015, which are provided at lin (January 2015). The network consists of $N=5162$ nodes, has a power law degree distribution with exponent $\gamma=2.1$, average node degree $\bar{k}=5.2$, and average clustering $\bar{c}_{2}=0.22$.

The hyperbolic maps for both datasets have been taken from Kleineberg et al. (2016).

Supplementary Fig. S2 shows the final density of cooperators for both networks and for the different allocation strategies described in the main text.

Data and code availability

To facilitate the application of our framework for future work, we enclose the empirical datasets and their inferred hyperbolic maps as well as an implementation of the model networks used in this paper (figshare.com/articles/DataAndModel_zip/4817947). An implementation of the technique to construct hyperbolic maps for real networks Papadopoulos et al. (2015a, b) is publicly available at hyp .