Scale-free networks may not necessarily witness cooperation
SScale-free networks may not necessarily witness cooperation
Deep Nath, Saptarshi Sinha, and Soumen Roy ∗ Department of Physics, Bose Institute, 93/1 Acharya Prafulla Chandra Road, Kolkata 700009, India
Networks with a scale-free degree distribution are widely known to promote cooperation in variousgames. Herein, we demonstrate that this need not necessarily be true. For the very same degreesequence and degree distribution, we present a variety of possible behaviour. We also reevaluate thedependence of cooperation on network clustering and assortativity.
Evolutionary game theory (EGT) has captured the se-rious attention of evolutionary biologists, ecologists, com-puter scientists and statistical physicists over the last fewdecades. This is chiefly due to its potential to effectivelyunderstand the challenge of evolution and maintenanceof cooperation in the Darwinian context. EGT enablesus to investigate various evolutionary processes by know-ing the frequency-dependent steady-state outcome of twoor more interacting populations [1–3]. Prime factors inevolutionary games include the strategy of players andgame rules. EGT has been rather successful in explain-ing the cooperative behavior of living organisms frommicroscopic to macroscopic. Cooperation between liv-ing organisms may flourish irrespective of the presenceof free-riders [4–7]. EGT has been studied primarily onfour types of games: prisoner’s dilemma (PD), harmony,snowdrift and coordination, which differ in their payoffvalues and steady states [2]. In the last few decades,much research has been carried out on the maintenanceof cooperation in various games. Among these, PD is sig-nificant because defection would be the natural tendencyin PD [8].Apart from game rules, the underlying structure of thepopulation also plays an essential role in the outcome ofthe game [9–13]. The underlying graph topology impartsspatial restrictions on the interactions between players.These spatial restrictions may act in favour of coopera-tion. In PD games played on homogeneous populationstructures, it is difficult to maintain cooperation [3, 14].On the other hand, cooperation could thrive in heteroge-neous populations. Thus, the outcome of a game dependson the structure of the population, types of payoffs andsundry factors like mobility [15].Networks [16, 17] have been found to be useful infields as diverse as mutagenesis and phage resistance [18],image-processing and non-invasive diagnostics [19], in-frastructure [20] and optogenetics [21, 22]. While degreeis only one of the many metrics in networks, it has re-ceived perhaps the most emphasis in network literature[23]. Graphs with power-law degree distributions havebeen generally alluded to as “scale-free networks” in lit-erature [16, 24–26]. Heterogeneity in scale-free networkscan be better understood through measures such as theS-metric [27, 28]. As well-known, the mechanism of gen- ∗ [email protected] eration [24, 29] can imprint its signature on the structureof the network [27].It has been reported earlier that scale-free networkspossess an inherent tendency to promote cooperation[30]. The underlying intuition seems to be that when co-operators are hubs, they can survive in a population byaccumulating higher payoffs as compared to their defect-ing neighbours [31, 32]. It has also been thought that fac-tors like clustering could influence this outcome [33, 34],although at higher mutation rates even highly clusterednetworks may not witness cooperation [35]. Herein, wedemonstrate that these need not necessarily be true. In-deed, for the very same degree sequence and degree dis-tribution, we demonstrate that scale-free networks maydisplay a rich diversity in behaviour with regard to co-operation.Let G ( V , E ) denote a graph, where V and E denote theset of nodes and edges respectively. |V| = N and |E| denotes the number of nodes and edges respectively in G ( V , E ). Henceforth, we often refer to G ( V , E ) as G . s ( G ) = (cid:88) E ij ∈E k i k j (1)Here, i and j are the end nodes of the edge E ij ∈ E . Thedegree of node i and j is denoted by k i and k j respec-tively. If K denotes the degree sequence of G , let G ( K )denote the set of graphs with degree sequence K . We candifferentiate between these graphs through S ( G ) = s ( G ) s max (2)Here, s max = max { s ( G ) : G ∈ G ( K ) } , whence 0
1. Only a completely disconnected graph has S = 0 and is therefore excluded herein. If graphs havingdifferent values of S ( G ) possess the same degree sequence– their degree distribution is obviously identical. Herein, S ( G ) is used to represent different graphs with identicaldegree sequence and hence identical degree distribution[27]. Henceforth, we mostly refer to S ( G ) simply as S .Herein, we simulate evolutionary PD game in hetero-geneous populations. The population structure has beenconsidered to be a BA network which possesses a power-law degree distribution. For each ensemble, initially aBA network, G BA , is generated. From G BA , a set of scale-free networks , {G SF } is obtained by repeated degree pre-serving double-edge swaps . We can hardly overemphasisethat ∀G ∈ {G SF } have the same degree sequence andnaturally their degree distribution is identical to that of a r X i v : . [ phy s i c s . s o c - ph ] J a n G BA . ∀G ∈ {G SF } would obviously possess a value of S different from S ( G BA ). It should be noted that no nodeor edge is removed or added during rewiring by degreepreserving double-edge swaps.We can easily obtain the value of max ( s ) in {G SF } .However it is not possible to achieve an arbitrarily spec-ified low value of S, ∀G ∈ {G SF } . The minimum obtain-able value of S would depend on N , E and edge density of G BA among other factors. Here, N = 1024 and we havebeen able to generate graphs with S as low as S = 0 . S at a given N is computationally inhibitive [27].Each node in G represents a player and the populationstructure greatly influences the interactions between theplayers. At the start of each ensemble, the populationis randomly divided into an equal number of coopera-tors, C , and defectors, D . Thus, the initial fraction ofcooperators, f C i = 0 . R . If two defectors interact with each other, theywill earn punishment, P . On the other hand, interac-tion between C and D will lead to sucker’s payoff, S ,for C and temptation, T , for D . Herein, these payoffvalues are considered to be R = 1 .
0, 1 . < T ≤ . P = 0 . S = 0 . i , interacts with a randomlychosen neighbor, its payoff is π ij . Generally π ij (cid:54) = π ji .The value of π ij would be R , T , P or S . The accumulatedpayoff of i is Π i = (cid:80) j π ij . After payoff determination,individuals will update their strategy synchronously. LetΠ i and Π j denote the accumulated payoffs of i and j respectively. i will imitate the strategy of j with a prob-ability, P i → j = Π j − Π i ( T − S ) × max ( k i , k j ) Θ Π j > Π i (3)Here Θ Π j > Π i = 1 for Π j > Π i and zero otherwise. Thiscondition indicates that individuals will try to maximizetheir payoff and i will imitate j ’s strategy only if Π j > Π i .10 generations of transient time have been considered ineach ensemble. The final fraction of cooperators, f C , isaveraged over 10 generations. For each network, N =1024 and average degree, (cid:104) k (cid:105) = 4.Fig. 1 presents a plot of the fraction of cooperators, f C ,against temptation, T , at different values of S . Highervalues of T favor defection and result in a decrease of f C .We also observe that the dependence of f C on S is highlynon-monotonic. Further, the maintenance of cooperationis high only in and around S = 0 .
4. This demonstratesthat the maintenance of cooperation in scale-free net-works is not decided by the degree distribution alone.The complex variation of f C with S at different val-ues of T is demonstrated in Fig. 2. We observe that atlow values of T and S , cooperation is well-maintained. FIG. 1. Fraction of cooperators, f C , versus temptation, T ,at different values of S-metric, S . Results are for f C i = 0 . N = 1024, (cid:104) k (cid:105) = 4, E N = 1600 ensembles. Cooperation ishigh for S = 0 . T . However, for higher andlower values of S cooperation is not maintained well. Thestandard error is smaller than the size of the data points. However, at higher values of T , the maintenance of co-operation is higher in and around S = [0 . , . FIG. 2. f C versus S at various values of T for scale-free(SF) networks. Red indicates the maintenance of cooperationand blue its absence. Cooperation depends on both S and T . Results are for f C i = 0 . N = 1024, (cid:104) k (cid:105) = 4, E N =1500 ensembles. Cooperation is largely well-maintained or ill-maintained respectively at lower and higher values of T and S .We observe that f C is higher in and around S = [0 . , . In Fig. 3 we examine the behaviour of f C with respectto S at various values of T . We again observe that f C ishigher in and around S = [0 . , . S represent different structures ofscale-free networks. In all of these structures, the con-nectivity between the hubs is different. This difference inconnectivity between the hubs can be easily understoodthrough another topological metric measuring degree-degree correlation in networks, i.e. assortativity, r . Aswell-known, − ≤ r ≤ +1 [36].The variation of r with S is studied in Fig. 4(a) and FIG. 3. Fraction of cooperators, f C , versus S , at various val-ues of T . While only scale-free graphs have been consideredhere – all of them clearly do not promote cooperation. At allvalues of T , f C is higher in and around S = [0 . , . f C i = 0 . N = 1024, (cid:104) k (cid:105) = 4, and E N = 1000. The standarderror is smaller than the size of the data points.FIG. 4. Assortativity, r , versus S for (a) the original graph, G and (b) cooperator graph, G C , and defector graph, G D . T = 1 . f C i = 0 . N = 1024, (cid:104) k (cid:105) = 4, and E N = 1000.The standard error is smaller than the size of the data points. is observed to be consistent with reported literature [37].Graphs with positive and negative values of S are termedassortative and disassortative respectively. In assorta-tive graphs, nodes with higher degree are predominantlyconnected to each other. In disassortative graphs, nodeswith higher degree are predominantly connected to nodeswith lower degree. It has been presumed that hubs areresponsible for the maintenance of cooperation in hetero-geneous population structures. The underlying thoughtseems to be that when the hubs are cooperators they canacquire higher payoffs [32]. Herein, graphs having dif-ferent values of S possess the same degree sequence bydefinition. It can then be expected that f C should notdepend on S . However, from Figs. 1, 2 and 3, it can beeasily observed that f C strongly depends on S .Also, it has been postulated earlier that assortativityamong cooperators could work in favor of cooperation [38]. When hubs act as cooperators they can accumulatehigher payoffs. Hence, cooperation can be maintainedin a population. Also, assortativity between the hubsshould operate in favor of cooperation as they can acquirehigher payoffs as well. Therefore, cooperation should bemaintained in assortative graphs which possess highervalues of S . However, in disassortative graphs coopera-tion might not be maintained. Since r varies linearly with S , it would be expected that f C would possess a lineardependence on S . It is evident from Fig. 3 that for higherand lower values of S , cooperation is not maintained wellenough. A suitable region for the maintenance of coop-eration lies somewhere between highly assortative andhighly disassortative graphs. Hence, we can concludethat networks with scale-free degree distribution do notalways promote cooperation. Also, hubs and assortativ-ity between them might not really be responsible for themaintenance of cooperation.The association between assortativity among coopera-tors and the maintenance of cooperation has been ex-plored earlier [38]. Assortativity among cooperatorscould act in favor of cooperation as cooperators should beable to accumulate higher payoffs. Assortativity amongcooperators, r C , can perhaps be differently scrutinisedthrough the “cooperator graph” , G C , instead of the orig-inal graph, G [13]. Similarly, the “defector graph” , G D ,may be useful to understand the assortativity betweendefectors, r D . We can construct G C and G D from theoriginal graph, G [13]. G C and G D are solely graphs of co-operators and defectors respectively among themselves. G C is obtained by removing every defector and each of itsconnections from G . Similarly G D is obtained by pruningall cooperators and their connections from G . G C and G D respectively capture the connectivity among cooperatorsand defectors themselves in G , but not between any coop-erator and defector. In Fig. 4(b) we study the variationof r versus S for G C and G D . In contrast to the linearbehaviour observed in Fig. 4(a) for the full graph, G , weobserve a non-linear variation in G C and G D . We haveobserved earlier in Fig. 3 that the maintenance of co-operation is higher in and around S = [0 . , .
45] for G .However, Fig. 4(b) for G C and G D demonstrates that thevalue of r C is enhanced at higher values of S . G C cap-tures purely the connections between cooperators only,while, f C is calculated for the full graph, G . Therefore, f C may not be really correlated with r C . Also defectiondominates at S = 0 .
99, while r D is higher at S = 0 .
9. Insummary, the role of assortativity in the maintenance ofcooperation in a population needs far larger scrutiny inorder to arrive at a suitable conclusion.In Fig. 6(a), we study the average clustering coeffi-cient, (cid:104)C(cid:105) , ∀G ∈ {G SF } at different values of S . (cid:104)C(cid:105) maynot prominently depend on S as N → ∞ . Neither hasany such dependence of (cid:104)C(cid:105) on S been widely reportedin literature. We have observed earlier that Figs. 1, 2and 3 exhibit a strong dependence of f C on S . A naturalquestion is whether and how f C would depend on (cid:104)C(cid:105) ,especially if the dependence of (cid:104)C(cid:105) on S is minimal. Pre- FIG. 5. (a) Fraction of hubs acting as cooperators, f H C ,or defectors, f H D , and, (b) H r = f H C /f H D ; versus S at T = 1 .
31. All graphs possess the same degree sequence andtherefore the same number of hubs and an identical degreedistribution. H r peaks at S = 0 .
4, alike f C in Fig. 3. We alsoobserve that hubs are mostly defectors at higher S . Resultsare for (cid:104) k (cid:105) = 4, E N = 1000 ensembles. The standard error issmaller than the size of the data points.FIG. 6. Average clustering coefficient, (cid:104)C(cid:105) , versus S at T =1 .
31. Results are for (a) N = 512, and, (b) N = 1024 nodes. (cid:104)C(cid:105) may not depend prominently on S as N → ∞ . Neitherhas any such dependence been widely reported in literature.Figs. 1, 2 and 3 exhibit a strong dependence of f C on S . Anatural question is whether and how f C would depend on (cid:104)C(cid:105) ,especially if the dependence of (cid:104)C(cid:105) on S is minimal. Resultsare for (cid:104) k (cid:105) = 4 and E N = 1200. The standard error is smallerthan the size of the data points. vious studies have observed that cooperation increaseswith an increase in network clustering [33, 34]. Of course,it is also known that at higher mutation rates, evenhighly clustered networks may not witness cooperation[35]. However, it must also be duly noted that, while thedegree distribution remained unchanged in Refs. [33, 34]– the degree sequence likely changed. Herein, we havestrictly retained the degree sequence throughout.We now address the importance of hubs in a graph bystudying the variation of the number of hubs and theirclustering coefficient with S . It has been claimed that hubs mainly act as cooperators in a scale-free networkand play an important role in maintaining cooperation[31]. ∀G ∈ {G SF } possess identical degree sequence.Therefore, ∀G ∈ {G SF } can be expected to possess anidentical number of hubs. Let k sd denote the standarddeviation of the degree distribution of G . Herein, we con-sider nodes with degree greater than (cid:104) k (cid:105) + k sd as hubs.Let us denote all hubs by H and those which act as co-operators and defectors by H C and H D respectively. The number of these hubs can then be denoted by N H , N H C and N H D respectively. The respective fraction of suchhubs are denoted as f H , f H C and f H D . The value of f H does not depend on the value of S but is decided by K , as aforementioned. In Fig. 5(a), we study the vari-ation of f H C and f H D with S . We observe that as S increases, f H C gradually starts declining but f H D rises. f H C is higher at lower values of S and responsible forthe overall maintenance of cooperation in G . However,as S increases – hubs start adopting defection. There-fore, irrespective of the presence of hubs – cooperationis not maintained at higher values of S . We also study H r = N H C / N H D = f H C /f H D versus S in Fig. 5 (b). H r is highest at S = 0 .
4, where cooperation is also highestas already observed in Fig. 3. Hubs seem to play an im-portant role in maintaining cooperation, when they arecooperators. However, whether they act as cooperatorsor defectors would depend on the topology of the graph.
FIG. 7. (a) Fraction of cooperators, f C i ,C , possessing C i = 0and C i = (0 , f C i , with C i = 0 and C i = (0 , (cid:104)C(cid:105) H C ; versus S at T = 1 . f C ,C peaks at S = 0 . f C ,C rather than f C (0 , ,C decides f C as seenin (b). (cid:104)C(cid:105) H C increases monotonically with S in (c). Resultsare for f C i = 0 . N = 1024, (cid:104) k (cid:105) = 4, E N = 1000. Thestandard error is smaller than the size of the data points. We also study the clustering coefficient, C i , of node, i ,at different values of S . We denote the total number ofnodes in the network possessing C i = 0 and 0 < C i ≤ N C and N C (0 , respectively. The fraction of nodes inthe network possessing C i = 0 and 0 < C i ≤ f C = N C / N and f C (0 , = N C (0 , / N respectively.These numbers and fractions obviously include both co-operators and defectors. We now specifically denote the number of cooperators in the network possessing C i = 0and 0 < C i ≤ N C ,C and N C (0 , ,C respectively. The fraction of such nodes can then be respectively denotedby f C ,C = N C ,C / N and f C [0 , ,C = N C (0 , ,C / N .Fig. 7(a) exhibits the variation of f C ,C and f C (0 , ,C versus S at T = 1 .
31. We observe that the position ofthe peak for f C ,C mirrors that of f C as observed in Fig.3 earlier. Fig. 7(b) records the variation of f C and f C (0 , versus S . Clearly, f C is far influential as compared to f C (0 , in deciding f C . We have represented hubs actingas cooperators by H C . (cid:104)C(cid:105) H C denotes their average clus-tering coefficient. Fig. 7(c) demonstrates that (cid:104)C(cid:105) H C increases monotonically with S .The variation of (cid:104)C(cid:105) H C with respect to S in Fig. 7(c)is in remarkable contrast to the variation of H C versus S as observed in Fig. 5. As aforementioned, it has beenreported earlier that the average clustering coefficient ofa network is considered to work in favour of cooperation.However, we observe that the average clustering coeffi-cient of hubs may not really promote cooperation. As S increases, (cid:104)C(cid:105) H C increases monotonically, while the main-tenance of cooperation progressively decreases. Indeed at S = 0 . (cid:104)C(cid:105) H C is at its highest yet the maintenance ofcooperation is minimal.In order to gain a better understanding into the main-tenance of cooperation, we take recourse to toy networks.In all toy networks considered herein; R = 1, T = 1 . P = 0, S = 0 [30]. Let i and j be two randomly chosenneighbors in the population. Let A and B denote thestrategy of i and j respectively. This strategy can be ei-ther cooperation or defection. Let k i denote the degreeof i , and, k j of j . If k i C and k i D be the number of C and D in the neighborhood of i , then k i C + k i D = k i .Similarly, k j C + k j D = k j . The accumulated payoff of i isΠ i = (cid:88) j π ij = k i C ( π C − A ) + k i D ( π D − A ) (4)Obviously A can be either C or D . π C − C = R (reward), π C − D = T (temptation), π D − C = S (sucker’s payoff)and π D − D = P (punishment).The accumulated payoff ofan arbitrarily chosen neighbor, j , of node, i , isΠ j = k j C ( π C − B ) + k j D ( π D − B ) (5)Individual, i , would upgrade to the strategy of j with aprobability P ( i → j ) as shown in Eqn. 3. Similarly, j canalso imitate the strategy of i , with probability, P j → i = Π i − Π j ( T −S ) × max ( k i ,k j ) Θ Π i > Π j .The star graph in Fig. 8(a) has one hub and five leaves.Suppose the hub, i , is a defector and the leaves are co-operators. Let j be any arbitrarily chosen neighbor of i . Then, A = D , B = C , k i C = 5, k i D = 0, k j C = 0, k j D = 1. The accumulated payoff of i and j is Π i = 5 . j = 0. Since Π i > Π j , i will not imitate the strategy ofits neighbor j . However, j will imitate the strategy of i with the probability P ( j → i ) = 1. Hubs play a signifi-cant role in the maintenance of cooperation. If the hubis a cooperator, it will acquire a higher payoff and gainan evolutionary advantage over its neighbors. However,maintenance of cooperation becomes fragile if the hub isa defector. FIG. 8. Blue and red denote cooperators and defectors respec-tively. (a) Star graph with ( N , E ) = (6 , N , E ) = (10 ,
13) with S b >S C and r b >r C .A direct link between two hubs, i and k , makes this networkvulnerable to defection. (b) Defection is likely to dominate if i is a defector. i can turn k into a defector. (c) Cooperationis possible as k will never adopt defection. From Fig. 8(a) we observe that the strategy of thehub would dominate the outcome. Cooperation wouldbe maintained in a star graph when the hub is itself acooperator. However, the presence of many cooperatorhubs in a network is not enough in itself for maintainingcooperation. There are two hubs in both the toy networksin Fig. 8(b) and 8(c). Fig. 8(b) indicates that if hub i is a defector, k might adopt defection with a probability, P ( k → i ). From Eqn. 3, we have P ( k → i ) = 0 . i and k . Dueto this connection, a defector hub can easily affect itsneighboring cooperator hub. On the other hand, in Fig.8 (c), we observe that if a defector hub and a cooperatorhub are directly connected to each other, the defectorhub would not be able to affect the cooperator hub.In summary, a significant body of study in literaturestates that scale-free networks can facilitate coopera-tion. Herein, we examine the prisoner’s dilemma gameon scale-free networks. We demonstrate that identicalpower-law degree distributions and indeed even an iden-tical power-law degree sequence may exhibit remarkablydifferent outcomes with regard to cooperation.Our resultsindicate the maintenance of cooperation could be higherin SF networks within a narrow range of S .We review the correlation of assortativity among co-operators and maintenance of cooperation. For this weborrow the notion of “cooperator graph” , G C , and “defec-tor graph” , G D [13]. We measure assortativity betweencooperators, r C , through the help of G C . We observe thatthe maintenance of cooperation does not always arise asa direct consequence of the assortativity between them.We also study the average clustering coefficient of thenetwork at different values of S . It has been reportedthat clustering directly influences the maintenance of co-operation in a network. However, we observe that forscale-free graphs with identical degree sequence, cooper-ation may not really depend on clustering.In addition, we evaluate the role of hubs towards themaintenance of cooperation. In a heterogeneous popula-tion, cooperator hubs play a crucial role in accumulat-ing higher payoffs. ∀G ∈ G SF , cooperation does not de- pend merely on the number of hubs, but rather on thosehubs which are cooperators. However, whether the hubsbecome cooperators or defectors would depend on thetopology of the network. It appears that hubs are morelikely to be directly connected to each other in graphsassociated with higher values of S . 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