Evolution of Cooperation in the Presence of Higher-Order Interactions: from Networks to Hypergraphs
EEvolution of Cooperation in the Presence ofHigher-Order Interactions: from Networks toHypergraphs
Giulio Burgio , Joan T. Matamalas , Sergio Gómez and Alex Arenas * Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26,Tarragona 43007, Spain; E-Mails: [email protected] (G.B.), [email protected] (S.G.) Harvard Medical School & Brigham and Women’s Hospital, 75 Francis St, Boston MA 02115, USA; E-mail:[email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]
Abstract:
Many real systems are strongly characterized by collective cooperative phenomena whoseexistence and properties still need a satisfactory explanation. Coherently with their collective nature, theycall for new and more accurate descriptions going beyond pairwise models, such as graphs, in which allthe interactions are considered as involving only two individuals at a time. Hypergraphs respond to thisneed, providing a mathematical representation of a system allowing from pairs to larger groups. In thiswork, through the use of different hypergraphs, we study how group interactions influence the evolutionof cooperation in a structured population, by analyzing the evolutionary dynamics of the public goodsgame. Here we show that, likewise network reciprocity, group interactions also promote cooperation.More importantly, by means of an invasion analysis in which the conditions for a strategy to survive arestudied, we show how, in heterogeneously-structured populations, reciprocity among players is expectedto grow with the increasing of the order of the interactions. This is due to the heterogeneity of connectionsand, particularly, to the presence of individuals standing out as hubs in the population. Our analysisrepresents a first step towards the study of evolutionary dynamics through higher-order interactions,and gives insights into why cooperation in heterogeneous higher-order structures is enhanced. Lastly,it also gives clues about the co-existence of cooperative and non-cooperative behaviors related to thestructural properties of the interaction patterns.
Keywords: cooperation; evolutionary dynamics; higher-order interactions; hypergraphs.
1. Introduction
In behavioral terms, cooperation is the providing of a benefit to another individual at some cost forthe provider. In well-mixed populations, where all individuals interact with each other, evolutionary gametheory predicts that cooperation cannot survive due to the existence of selfish behaviors [1,2]. Nevertheless,we do observe cooperation in many different real systems, ranging from genomes to human societies [3–6],and not as a marginal phenomenon: rather, it often strongly shapes and makes even possible the existenceof those systems. For example, without cooperative behaviors like the work division and the establishmentand respect of social norms, the humankind could not have developed even the simplest forms of society.Among microbes, some viruses and cells have been observed to be involved in (involuntary) cooperativebehaviors that are essential for their reproduction and diffusion [4,6]. The existence of cooperation inbiological and social systems has led to hypothesize several mechanisms to explain its ubiquity: inclusivefitness, direct and indirect reciprocity, punishment, cultural group selection, etc. (clearly, those requiring1 a r X i v : . [ phy s i c s . s o c - ph ] J un of 24 substantial cognitive demands, apply only to human’s social systems) [7–13]. Among those mechanisms,we focus on the so-called network reciprocity [14–22].Generally speaking, an act of reciprocity takes place when some individual returns a beneficial actreceived by someone else, establishing a fruitful mutual exchange. Therefore, it is a cooperative act. Inparticular, one refers to network reciprocity when the mutual cooperative exchange is sustained by theexistence of a suitable structure of interactions, represented through a network, among the individuals of apopulation. Two conditions lie at the base of network reciprocity: a limited, recurrent set of interactinggame opponents (thus avoiding the anonymity between them); and a local adaptation mechanism throughwhich an individual can change (update) its behavior (strategy) depending on how its neighbors in thenetwork behave and perform. Apart from its fascinating theoretical character, network reciprocity isregarded as one of the most important and interesting of the mechanisms from an application point ofview. Indeed, the easiness of its requirements to be fulfilled, makes it a transverse mechanism applicablenot only to those system made of rational units (e.g., social, economical), but also to those whose units arecapable of only very simple behaviors (e.g., many biological systems) [21].The use of networks to formalize the topology of interactions within a system, however, tacitlyassumes that all the interactions that occur in the system are essentially pairwise. This may not be asatisfactory description for those real systems exhibiting higher-order interactions to which more than twoentities participate together and which are not reducible to combinations of lower-order ones. Examplesof such interactions include triadic [23] and higher-order closures in social networks [24], co-authorshipnetworks in science [25], spatial coexistence relations among species in an ecosystem [26], and trigenicinteractions in genetic networks [27]. In all such cases, representing the interaction patterns through anetwork (that is, approximating those higher-order interactions with pairwise ones) implies the loss ofessential information to understand their dynamics [28].This work tries to bring network reciprocity closer to reality. We investigate the effect of consideringhigher-order interactions, using hypergraphs, on the evolution of cooperation when the population ofindividuals possesses structural correlations. Hypergraphs allow to distinguish the dynamics on a cliqueof m individuals (an m -clique), in which each of the individuals (vertices) interacts separately, in pair, witheach of the other m − m individuals interact all together,as a group, in an intrinsically different way (represented as a hyperedge of cardinality m ). Moreover,hypergraphs, and also other higher-order representations as well [29], solve the lack of uniqueness indefining group interactions from pairwise information only.We show here that the mechanism of network reciprocity can be successfully extended to higher-orderstructures and that, in fact, it even becomes reinforced.
2. Dynamics
To study cooperation in presence of higher-order interactions we need a game definable for anynumber of players (the size of the group) and belonging to the class of social dilemmas. In such games, theequilibrium solution under unilateral changes of strategy (i.e., considering changes of strategy of only oneplayer at a time), called Nash equilibrium, is not the most efficient solution for all the players (i.e., it is notPareto-efficient). In particular, in the type of game we are interested in, the equilibrium solution is the onein which all the players choose to defect, although the Pareto-efficient solution is the one in which they allcooperate, hence the social dilemma. The prototypical m -players dilemma of this kind is the public goodsgame (PGG). Each player, independently, chooses whether to put (cooperating) or not (defecting) a certainamount b (fixed for all players, in our setup) into a public pot. The collected sum is re-scaled by a synergy of 24 factor α ∈ R , α >
1, and equally redistributed among the m players disregarding of their strategy. When n c ≤ m − σ (set to 1 if cooperates, 0 if defects) gets a payoff f ( σ ) given by f ( σ ) = b (cid:20)(cid:16) α m − (cid:17) σ + α ( m − ) m q (cid:21) (1)where q ≡ n c / ( m − ) is the fraction of neighbors that cooperate. The first term accounts for the cost andproportional gain from the contribution to the pot of the player, while the second corresponds to the gainreceived from the cooperating neighbors. Since the second term is always non-negative, the best responsefor a player depends on whether α is smaller or greater than m . The dilemma exists for α < m because, inthis case, according to Equation (1), the best strategy for any player is to defect. However, each of themgets a null payoff, whereas it would be positive if all of them chose to cooperate. We give here a brief introduction to the connectivity structures we use in this work: simplehypergraphs. Given a finite set V of vertices and a family E of subsets { e I } I ⊆ V of V , the couple H = ( V , E ) defines a hypergraph , and each e I is called hyperedge , representing a relation among all the vertices in it. Themaximum and minimum cardinality (degree) of the hyperedges in H are called the rank and co-rank of thehypergraph, respectively: rank ( H ) = m max and co-rank ( H ) = m min . If they both are equal, with value s ∈ N , the hypergraph is said to be s-uniform [30].Another important concept is the 2 -section of H . We can define it as the graph whose vertices arethe vertices of H , and where two distinct vertices form an edge if and only if they belong to, at least, onecommon hyperedge in H [30].Additionally, an hypergraph H is called simple if e I ⊆ e J ⇒ I = J . This means that it has no repeatedhyperedges, and can be obtained from a generic hypergraph ( V , E ) removing from E all the hyperedgeswhich are subsets of at least another hyperedge.Finally, given the family S ( i ) of hyperedges { e J } J ⊆ V containing vertex i , and assuming no repeatedhyperedge, we define the m-degree k ( i ) m of i as the number of hyperedges of degree m in S ( i ) . The generalizeddegree of node i is the number of hyperedges incident on it, given by k ( i ) = m max ∑ m = m min k ( i ) m (2)We construct a hypergraph starting from a certain base network and converting in hyperedges arandomly selected fraction p of its 3-cliques, see Section 6.1 for further details. From now on, we will callthese selected hyperedges as triangles . At the base of our model lie the following microscopic rules, imposed at each time step (round): (i)each player plays with its strategy, either cooperate or defect, within each of the hyperedges incident on it;(ii) each strategist chooses one of its neighbors uniformly at random in the 2-section of the hypergraph,and performs the strategy update. These steps are performed synchronously by all the players.The payoff f ( i ) ( t ) that player i with m -degree k ( i ) m and playing with strategy σ i ( t ) got at the end of atime step t , obeys the following dynamic equation: f ( i ) ( t ) = b m max ∑ m = m min k ( i ) m (cid:20)(cid:16) α m − (cid:17) σ i ( t ) + α ( m − ) m q ( i ) m ( t ) (cid:21) (3) of 24 where q ( i ) m ( t ) is the probability that a neighbor of node i , in a hyperedge of degree m , cooperates at time t .To explore different selection pressure regimes, we consider smoothed imitation by taking, as probabilityfunction for the strategy update, the Fermi distribution function: F ( ∆ f ; β ) = + e − β ∆ f (4)where ∆ f is the difference between the payoff of the randomly chosen neighbor and the payoff of the focalplayer, and parameter β > β , weak and noisy for low values).The global state of a population of size N is defined by the density of cooperators: c ( t ) = N N ∑ i = σ i ( t ) (5)We initialize the system with a fraction c ≡ c ( t = ) of randomly selected cooperators.From Equation (3) it immediately follows that c = c =
3. Results
For every set of hypergraphs we report the effective asymptotic cooperators density c ∞ and theconvergence time t as a function of α and for different values of the conversion fraction p , as found in theperformed Monte Carlo (MC) simulations; see Section 6.2 for further details. To systematically reveal theeffect on the dynamics of converting 3-cliques in triangles, we report the results found for hypergraphsgenerated through the Holme-Kim model (HK) [31] and the Dorogotsev-Mendes model (DM) [32]. Theyboth grow heterogeneous networks having topological properties typical of many real complex systems,and in particular social networks, like power-law degree distributions, the small-world property, and hightransitivity (i.e., high clustering coefficient).Let us call α surv ( p ) the value of α at which, for a given p , the asymptotic cooperator density c ∞ starts to grow above zero, to be compared with the corresponding critical value α cr ( p ) for a well-mixedpopulation. To this purpose, we define r ( p ) ≡ α surv ( p ) α cr ( p ) (6)where α cr ( p ) = m max ∑ m = m min m p m (7)being p m the fraction of hyperedges of cardinality m in the hypergraph. Values of r ( p ) smaller than 1mean some level of structural reciprocity favoring the survival of cooperation. As a particular case, r ( ) measures the network reciprocity.In Figure 1 we show the results for MC simulations with c = β , on hypergraphsstemming from HK networks with N =
500 vertices, m = (cid:104) k (cid:105) ≈ P t =
1. Parameter P t , the probability of making a triad-formation step after a preferential-attachment stepin the HK model, has been set to 1 to get the maximum possible transitivity, thus making more evidentthe effects of increasing the fraction p of converted 3-cliques. We find that, as we increase p from 0 to 1, r ( p ) decreases monotonically from 0.78 to 0.60 when β =
1, and from 0.79 to 0.63 when β = p , the fraction of 3-cliques promoted to triangular hyperedges.We also note how, by decreasing β (weaker selection), the structural reciprocity decreases as well. of 24 c p 0 0.2 0.5 1 c p 0 0.2 0.5 1 rt rt Figure 1.
Asymptotic density of cooperators c (top) and convergence time t (bottom) versus r , forhypergraphs generated from HK networks with N =
500 vertices, m = (cid:104) k (cid:105) ≈ P t =
1, and for β = β = p are specified in the legend, and c = r = Looking at the convergence times in Figure 1, we see the typical peaked shape characterizing thecritical region. Dealing with small finite systems, we expect the convergence time to show a peak nearthe transition, but still to remain finite. Since the maximum number of allowed time steps t max is fixed to2 × , and we average over 100 realizations, we can state that, if the average convergence time is smallerthan 200, then all the trials converged and the final populations are perfectly homogeneous; note that thisis only a sufficient condition. In this case, an effective cooperation density c ∞ other than 0 or 1 representsan average of homogeneous final states. This is the case for r values out of the transition region.Counting how many times the system has not converged n NC in 100 realizations, and knowingthe value of the average convergence time t , the average time t C the system took to converge in theremaining 100 − n NC trials is t C = [ t − n NC ( t max /100 )] / ( − n NC ) . For the previous hypergraphswe find n NC ≤ β = n NC ≤ β = t C are found to be around 1490 and 710, respectively. Being t C well below t max , we expect that thenon-convergence of some realizations does not depend on the value chosen for the latter. This suggeststhat the lack of convergence is not a purely dynamical effect —like at the critical point for an infinitewell-mixed population—, but it is due to the topological constraints of the structure. In fact, we find thepresence of configurations of local states preventing the system to escape from them, acting as topologicaltraps , see Section 4.3. of 24 c p 0 0.2 0.5 1 c p 0 0.2 0.5 1 rt rt Figure 2.
Asymptotic density of cooperators c (top) and convergence time t (bottom) versus r , forhypergraphs generated from DM networks with N =
500 vertices (average degree (cid:104) k (cid:105) =
4) and for β = β = p are specified in the legends, and c = r = We also note that with the increasing of p , the critical region gets wider, whatever is the value of β . InSection 4.2.2, we give an explanation for this based on the degree heterogeneity and the local clustering ofthe structure.Quite similar results are found for hypergraphs stemming from DM networks. In Figure 2 we showthe results obtained for base networks with N =
500 vertices ( (cid:104) k (cid:105) = p = p =
1, we getthe following monotonically decreasing values of r ( p ) : from 0.75 to 0.58 for β =
1, from 0.78 to 0.62 for β = n NC ≤ n NC = t C are found to be around 2660 and 380. As for HK hypergraphs, setting t max = × , we findno significant differences. In Appendix B we report the results obtained also for c = c = r =
4. Mathematical Analysis
The mean-field (MF) approach relies on two approximations: (i) all the vertices have the same m -degree sequence ( (cid:104) k m min (cid:105) , . . . , (cid:104) k m max (cid:105) ) , averaged over the original hypergraph with m -degree sequence of 24 distribution P ( k m min , . . . , k m max ) ; (ii) the local states of different vertices are not correlated. Then, a simplemaster equation for the density of cooperators c ( t ) follows: c ( t + ) = c ( t ) + c ( t ) ( − c ( t )) ( F ( ∆ f ) − F ( − ∆ f )) (8)where ∆ f = (cid:104) f ( c ) (cid:105) − (cid:104) f ( d ) (cid:105) = b m max ∑ m = m min (cid:16) α m − (cid:17) (cid:104) k m (cid:105) (9)Imposing c ( t + ) = c ( t ) we get, besides the absorbing state solutions c ( t ) = c ( t ) = ∀ α , thestationary solution corresponding to F ( ∆ f ) = F ( − ∆ f ) , that is, ∆ f =
0, given by α = α cr ≡ m max ∑ m = m min (cid:104) k m (cid:105) m max ∑ m = m min m − (cid:104) k m (cid:105) = m max ∑ m = m min mp m = (cid:104) m (cid:105) (10)for all c ( ) ∈ (
0, 1 ) , being p m = m − (cid:104) k m (cid:105) m max ∑ m = m min m − (cid:104) k m (cid:105) (11)the fraction of hyperedges of cardinality m or, for a well-mixed population, the probability of playing in agroup of size m , and (cid:104) m (cid:105) their average cardinality. Equation (10) gives the expected abrupt phase transitionbetween full defection and full cooperation, for an infinite well-mixed population, in the presence of eitherone or more allowed group sizes (corresponding to uniform and non-uniform hypergraphs, respectively).To be precise, the shown abrupt transition holds in the limit of an infinite population. Thefinite-size, trivially-structured version of a well-mixed population is an all-connected-to-all structure. Then,considering a m -uniform complete simple hypergraph of N vertices, it can be shown (see Appendix A)that, for β b (cid:29)
1, the transition is at α ≡ α m cr = m ( N − ) / ( N − m ) , greater than α cr = m for any finite valueof N . Additionally, α m cr / α cr grows with m . This finite-size effect slightly raises the structural reciprocityherein reported. The MF prediction is clearly reliable only for highly homogeneous structures. In all the othercases, instead, it serves as a reference point for measuring the structural reciprocity due to the complexheterogeneity of connections. In this section, we study the conditions for the invasion of the structuredpopulation, or for the resistance to be invaded for a player, using a certain strategy in specific configurations.We give a justification to the higher reciprocity level and the enlargement of the transition region, bothfound when increasing p in the used heterogeneous hypergraphs.4.2.1. The Role of Cooperator HubsFor all the analyzed scale-free networks in Figures 1 and 2), we observe the transition moving towardsnotable smaller values of r , whatever is the value of p . This is not the case for homogeneous hypergraphs,like the ones derived from ER networks. We show here that the key difference is the presence of hubs inthe former networks. of 24 We focus on a cooperator hub with m -degree sequence ( k hubm min , . . . , k hubm max ) , surrounded by neighborswith average k -degree sequence ( ¯ k m min , . . . , ¯ k m min ) , where¯ k m = ¯ k m ( { k hubm } ) = ∑ k m P ( k m | k hubm ) k m (12)Imposing the payoff of the hub playing as cooperator is equal to the average payoff of its neighborsplaying as defectors, f ( hub ) = ¯ f ( d ) , we get the threshold value α hubth above which, whenever the hub andany of its defector neighbors match for the strategy update, the survival of the hub is favored. Indicatingwith ¯ q ( d ) = c N / ( N − ) and q ( hub ) = ¯ q ( d ) − ( N − ) the initial ( t =
0) average fraction of cooperatingneighbors, respectively, of the hub’s defector neighbors and of the hub itself, from Equation (3) we find α hubth = m max ∑ m = m min k hub mm max ∑ m = m min (cid:34) k hub m m + m − m (cid:16) q ( hub ) k hub m − ¯ q ( d ) ¯ k m (cid:17)(cid:35) (13)Considering s -uniform hypergraphs and putting ¯ k s = ξ s k hubs , from Equation (13) follows r hub ≡ α hubth α cr = + ( s − )( q ( hub ) − ¯ q ( d ) ξ s ) (14)which is a monotone decreasing function of s and, for ξ s < − ( N c ) − , of c as well (left panel ofFigure 3). Taking as hubs those vertices with degree equal or larger, respectively, to the 97th and the 99thpercentile of the degree distribution, we get, for each value of s , a small interval in which, the valuesof r ( p ) found in the simulations for c = c , as commented in Appendix D. Moreover, we expect the method to work for β large enough, when the imitation process is more deterministic.To evaluate Equation (13) we take k hub s as the average degree of those vertices classified as hubs andthen compute the corresponding average value of ξ s . Setting c = r hub between 0.71 and 0.79 for s =
2, and between 0.55 and 0.65 for s =
3; for DM hypergraphs we get r hub between 0.72 and 0.79 for s =
2, and between 0.56 and 0.65 for s =
3. In Figure 3 (left panel), to visualize r hub versus s , we reported the values taken by the former computed considering ξ s as independent of s and fixed equal to ( ξ + ξ ) /2, taking ξ and ξ as the average of the respective values found in the usedhypergraphs. In fact, only ξ and ξ are accessible in HK and DM hypergraphs. Nevertheless, ξ s onlydecreases by about 0.01 going from s = s =
3, therefore keeping it fixed is a sensible choice.When α > α hubth , a cooperator hub is resilient to the invasion by any defector in its neighborhoodand, by persisting in the cooperative state, gives rise to a cooperative community whose members sustaineach other (for a similar dynamics on networks see [17]). In this way, this community is able to enlargeitself further or, at least, to resist to be invaded by the defectors in the neighborhoods, thanks to the localtopological constraints. Remarkably, in HK and DM networks, by construction, the hubs are frequentlyvery close to each other (first or second neighbors), making it easier for the cooperative community tothrive when more than one of them happens to be in the cooperative state. When, instead, the hub isa defector, its local proliferation is favored whatever is the value of α . However, once a defector hubeasily succeeded in converting all its lower-degree cooperator neighbors in defectors, none of the playersin this defective community provide/receive any benefit playing within it. This prevents the defectivecommunity to cover the entire population, unless α is enough small, outside the critical region. of 24 s r hub c r W Figure 3.
Left . The ratio r hub ≡ α hubth / α cr versus the rank s of a s -uniform hypergraph, for the survival of acooperator hub, for different values of c . The values of r hub are computed by taking as hubs those verticeswith degree equal or larger to the 97th and the 99th percentile of the degree distribution, respectively; theyare shown as the top and bottom lines corresponding to the same value of c . ξ s = ( ξ + ξ ) /2, ∀ s (seemain text for details). Right . The combined probability of Equation (17) that, in a 2-uniform (solid lines)and 3-uniform (dashed lines) DM hypergraph, an isolated defector d survives and w = | Ω ( d ) | = α , for the values of | Ω | reported in the legend, κ d = β = b = σ -strategist is totally surrounded —up to its z th neighborhood, with z large enough— by σ -strategists, with σ (cid:54) = σ , to calculate the joint probabilitythat the σ -strategist survives and some of the σ -strategists adopt strategy σ , as a function of α . Theprobability p i ( t ) that a player of strategy σ i and neighbors Ω ( i ) changes its strategy, reads: p i ( t ) = | Ω ( i ) | ∑ j ∈ Ω ( i ) σ j (cid:54) = σ i F (cid:16) f ( j ) ( t ) − f ( i ) ( t ) (cid:17) (15)In particular, considering a defector d of degree pair ( k ( d ) , k ( d ) ) as the isolated strategist, the probability forit to survive as a defector at time t is equal to 1 − p d ( t ) , in which the difference between the payoff f ( j ) ( t ) of a cooperator neighbor j and its payoff f ( d ) ( t ) is given by: f ( j ) ( t ) − f ( d ) ( t ) = b (cid:20) α (cid:18) k ( j ) − k ( d ) + k ( j ) − k ( d ) − a ( j ) (cid:19) − k ( j ) − k ( j ) (cid:21) (16)where a ( j ) is equal to 1/2 if j is a neighbor through a link and to κ ( j ) d /3 if it is a neighbor through a triangle,where κ ( j ) d is the number of triangles that j shares with vertex d . Similarly, the probability that a cooperatorneighbor j becomes a defector is equal to F (cid:16) f ( d ) ( t ) − f ( j ) ( t ) (cid:17) / | Ω ( j ) | .To get interesting insights it is sufficient to consider all the defector’s neighbors (all cooperators) asequivalent to the average vertex in the neighborhood. Therefore, they have the same degree pair ( k ( c ) , k ( c ) ) ,the same number of neighbors | Ω | , the same κ d and, consequently, the same payoff f ( c ) ( t ) ; it makes sense because the neighbors are selected uniformly at random when updating the strategies. Then, theprobability for d of surviving as a defector and invading all at once w ≤ | Ω ( d ) | neighboring sites reads ( − p d ( t )) · (cid:20) | Ω | F (cid:16) f ( d ) ( t ) − f ( c ) ( t ) (cid:17)(cid:21) w (17)In Figure 3 (right panel) we show this combined probability computed for w = | Ω ( d ) | =
8, differentvalues of | Ω | , κ d = p =
0) or only triangles ( p = | Ω ( d ) | > | Ω | andthrough links otherwise. This is simply understood observing that, thanks to the percolation of 3-cliques(or triangles) present in HK and DM networks (hypergraphs), k ( d ) (cid:12)(cid:12)(cid:12) p = is close to k ( d ) (cid:12)(cid:12)(cid:12) p = ; in particular, k ( i ) (cid:12)(cid:12)(cid:12) p = = k ( i ) (cid:12)(cid:12)(cid:12) p = − ∀ i , for DM structures. Since when d plays within a triangle obtains two timeswhat it gets playing within a link, it is advantageous playing through triangles whenever it possesses asufficient number of neighbors. In Figure 3 we also note that, for | Ω | sufficiently smaller than | Ω ( d ) | , theprobability of proliferating for the defective strategy counter-intuitively slightly increases with α (or r ),even above the mean-field critical point ( r = α may contribute substantiallyto the reported enlargement of the critical region when p is increased. Specifically, on one hand, theincreasing of α always brings the cooperators’ performance closer to the defectors’ in any local instanceof the game, making it easier for cooperators to invade (with certainty, eventually); on the other hand,defectors with high degrees are increasingly able to invade their neighborhoods, containing the diffusionof the cooperative strategy. This competitive dynamics, existing only for heterogeneous structures, is thusexpected to delay the point at which cooperators succeed in fully invading the population, stretching outthe transition, which is almost abrupt in homogeneous random hypergraphs.Next, we will explain why this competition gets fiercer when p is increased, enlarging even more thetransition region. From Equation (16) we can calculate the condition that the degrees of the defector and ofits cooperator neighbors have to fulfill in order that f ( c ) − f ( d ) decreases when α increases, thus raisingthe competition. Taking, as done before, all the defector’s neighbors (all cooperators) as equivalent, byimposing ∂ ( f ( c ) − f ( d ) ) / ∂α <
0, we get | Ω ( d ) | + | Ω ( d ) | (cid:18) k ( d ) + k ( d ) (cid:19) > k ( c ) + k ( c ) (18)where | Ω ( d ) | = k ( d ) + k ( d ) / κ d is the number of neighbors of the isolated defector, and κ d is the numberof triangles that each of the cooperator neighbors shares with it. Since ( | Ω ( d ) | + ) / | Ω ( d ) | ≤ ∂ ( f ( c ) − f ( d ) ) / ∂α >
0. Thus, when in a heterogeneousstructure a defector possesses enough higher degrees with respect to the surrounding cooperators, itssurvival is favored when α is increased, smearing the transition.To get some understanding about how this phenomenon depends on p , i.e., on the order of theinteractions, we refer to hypergraphs that become 3-uniform when p =
1, and 2-uniform when p = like the ones we have used. Then, Equation (18) splits into the following conditions for p = p = k ( d ) k ( c ) > − k ( c ) (19) k ( d ) k ( c ) > − κ d k ( c ) (20)Equating the expressions taken by | Ω ( σ ) | for p = p =
1, the identity k ( σ ) (cid:12)(cid:12)(cid:12) p = = k ( σ ) (cid:12)(cid:12)(cid:12) p = · κ σ /2,to be read as k ( σ ) computed when p =
1, given k ( σ ) when p =
0, follows. We can then rewrite Equation (20)in terms of k ( d ) and k ( c ) and directly compare it with Equation (19). Equation (20) becomes k ( d ) k ( c ) > κ c κ d − k ( c ) (21)where the degrees are calculated for p =
0. It follows that Equation (21) is a weaker condition thanEquation (19), i.e., Equation (19) ⇒ Equation (21, if and only if κ d / κ c > ∂ ( f ( c ) − f ( d ) ) / ∂α < p = p =
0. Expressing κ σ in terms of the local clustering coefficient C ( σ ) and the link-degree k ( σ ) as κ σ = [ C ( σ ) · ( k ( σ ) − )] p = (seeAppendix E for the derivation), we finally getEquation ( ) ⇒ Equation ( ) ⇐⇒ (cid:34) C ( d ) · ( k ( d ) − ) C ( c ) · ( k ( c ) − ) > (cid:35) p = (22)According to the inequalities in Equations (19) and (21), the most interesting case is obtained when k ( d ) > k ( c ) . Then, Equation (22) can be satisfied even when the isolated defector possesses a smallerclustering coefficient than its cooperators neighbors. Specifically, HK and DM models grow networkswhose local clustering coefficient C scales with the 2-degree k as C ( k ) ∼ k − γ , where γ is around 0.8 forHK ( P t =
1) and exactly 1 for DM (see Appendix E). Substituting in Equation (22), one finds that the aboveimplication holds whenever k ( d ) > k ( c ) , given γ ≤ k ( d ) such that, for a given k ( c ) , Equation (21) is satisfied but Equation (19) is not, wefind the condition 32 κ c κ d k ( c ) − < k ( d ) ≤ k ( c ) − p =
0. Therefore, if k ( d ) takes an intermediate value in the senseof Equation (23), with respect to k ( c ) , Equation (21) is satisfied but Equation (19) is not, allowing thecompetitive dynamics to take place for p = p =
0. As shown in Appendix E, suchintermediate neighborhoods actually exist in the structures we made use of. It must be noted that, for k ( d ) > k ( c ) −
1, the competitive dynamics exists for p = f ( c ) − f ( d ) on α , ∂ ( f ( c ) − f ( d ) ) / ∂α < p = p = k ( d ) > k ( c ) (corresponding to κ d / κ c > α (1) th = 2 k A k A + n cA − 1 α (2) th = 2 k cn k cn − 1 − n cA A A AA A α < α (1) th α > α (1) th α < α (2) th α (2) th < α < α (1) th cycle! Figure 4.
Generic bottleneck trap at work in presence of only links. Cooperators are shown in orange.Vertex A is the bridge vertex, k A is its degree and n cA is its number of cooperator neighbors (all consideredwith same degree k cn to get a unique α ( ) th and make clearer the illustration). For α ( ) th < α < α ( ) th , after sometime steps, the configuration is likely to return to the initial state, giving rise to an endless cycle that avoidsboth strategies to spread through the trap. Note that a minimal degree heterogeneity among vertex A andits neighbors must exists to make the trap work. ( p >
0) than 2-way ( p =
0) interactions, in accordance with the observed enlargement of the transitionregion when increasing p . It must be said that our analysis does not exclude the existence of othermechanisms that could further contribute to the observed enlargement. When running a dynamical process on top of a connected structure, there could be microscopicand mesoscopic configurations which operate as topological traps , blocking the system in a certain state.Traps represent here the only mechanism able to give rise to asymptotic heterogeneous populations inwhich cooperators and defectors co-exist. A trap can have any size and complexity, i.e., it can involveany number of vertices and develop over many time steps. A particularly simple class of traps is thebottleneck-type’s, consisting of a sub-hypergraph in which some vertices act as bridges between groupsof vertices of opposite strategy. In rank-3 hypergraphs, they consist of some bridge-vertices gluing linksand/or triangles, as shown in Figure 4.A trap comes into play when the strategies are updated. According to our setup, it is only the 2-sectionof the hypergraph that matters for the strategy update, and the presence of higher-order interactions solelyaffects the ranges of values of α for which a local structure can act as a trap. Then, the effect of a bottlenecktrap is fully specified by the comparison between the chosen value of α and the threshold values α th s,defined as those values at which a cooperator and a defector, each with its m -degree sequence, get the same final payoff. Any trap other than bottleneck-type is specified simultaneously by a set of coupled α th s,one for any link of the 2-section contributing to the trap.Whatever is the complexity of a trap, its effectiveness is strictly related to the value of the parameter β . A trap is able to act if the selection is strong enough, i.e., for high enough values of β . Indeed, β weighsthe proportional contribution of the difference of the payoffs to the update rule. The smaller is β , themore random the strategy update is, being increasingly disentangled from the payoffs resulting fromthe interactions. Accordingly, the MC simulations made for β = β =
1. Furthermore, we see also a slight reduction of structural reciprocitywhen lowering β , meaning that the structures develop some traps able to further anticipate the survival ofcooperation towards slightly lower values of r .A critical value of α can either satisfy or not some of the local conditions for producing a trap. In otherwords, the found effective densities of cooperators can be the result of an average of both homogeneousfinal states (the evolutionary stable states) and/or heterogeneous ones (produced by the traps). It wasalready unveiled in [34] that, for the entire set of 2 × K -distribution P ( k , k (cid:48) , k (cid:48)(cid:48) ) . Clustering coefficient and second-order degree correlations are clearly related, but preservingone does not fully imply preserving the other automatically. Now, since the action of a trap is defined inmultiple time steps (two at least) in which are involved several vertices with overlapping neighborhoods,the degree correlations of order higher than two should be important in the trapping process. Destroyingthose correlations we expect the traps to dissolve. Quite surprisingly, we find that what really matters forthe presence of traps other than the bottleneck-type, is the high value of the local clustering coefficient(see Figure 5). Preserving the P ( k , k (cid:48) , k (cid:48)(cid:48) ) distribution without caring of the local clustering, unravels theinitially present traps. We find this for both DM and HK hypergraphs. On the contrary, the presence ofbottleneck traps is clearly negatively correlated with clustering; they are found to be dominant for lowlyclustered networks (scarce of redundant paths) exhibiting strong degree heterogeneity, in line with [34].
5. Conclusions
Collective and, in particular, cooperative phenomena, in which many individuals take part as a group,are observed in many real systems, among which human society stands out. In this work we investigatedsystematically the effect that group (higher-order) interactions have on the evolution of cooperation instructured populations. We studied the dynamics arising from a population of individuals interactingby playing a public goods game on top of hypergraphs. While the theory we developed is general, thenumerical simulations specifically regarded rank-3 hypergraphs. In this way, we distinguish whether anelement, when a member of a group (clique), interacts separately, in pair, with each of the other members,or interacts simultaneously with all of them.To get the hypergraphs, we made use of an algorithm preserving the pairwise projection of thestructure, which is crucial to satisfy the constraints implicit in the network topology. In fact, although apairwise projection can correctly describe who interacts with whom, it cannot account for different typesof interactions, the temporal order in which the latter occur, and whether interactions involve pairs orlarger groups of units. Additionally, the 2-section imposes the maximum possible rank one could reach bygeneralizing the herein given procedure. The existence of a maximal size for groups is usually related to a rt p 0 1 r t r a p s rt p 0 1 r t r a p s Figure 5.
Convergence time versus r for the hypergraphs generated from two differently rewired versionsof the used DM networks, with t max = × . The inset plots report the number of times the systemhas not converged (i.e., the number of times the system got trapped). Left . Randomization preservingdegree and local clustering: traps are frequent in the critical regions.
Right . Randomization preservingsecond-order degree correlations: no traps left. The values of p are specified in the legend; c = β = saturation restriction due to some unbalanced cost in forming larger groups. This generally depends onthe nature of the considered system, and addressing its form goes beyond the scope of this work. Here, wejust recognize the existence of such a restriction as encoded in the 2-sections of the used hypergraphs, andpreserve it with our generative method.Monte Carlo simulations clearly indicated that heterogeneous structures, like the ones obtained fromHK and DM networks, are able to sustain cooperation for notable ranges of values of the synergy parameterbelow its critical value. This holds for unstructured populations, whatever the conversion fraction, relatedto the order of the interactions. Our findings, therefore, extend the phenomenon of network reciprocity torank-3 hypergraphs.Remarkably, we show a clear evidence that the sustenance of cooperation is stronger when the fractionof 3-way interactions is increased. Indeed, we attained a monotonous decrease of structural reciprocity withthe increasing of conversion fraction, for hypergraphs derived from both HK and DM networks. Morever,the level of cooperation is always larger for the hypergraph. It is key noting that this enhanced reciprocityis exclusively due to the substitution of some closed triads of first-order interactions (3-cliques) with uniquesecond-order interactions (triangles). In other words, given a heterogeneously-structured population,cooperation thrives more easily if individuals interacting separately, in pairs, within a closed triad aremade to interact all together. It should be noticed that, considering the synergy factor as independent ofthe group sizes, we did not account for explicit synergistic or anti-synergistic effects that could increaseor decrease the found levels of reciprocity. The improvement found here, rather, can be referred to as atopological synergistic effect allowed by the evolutionary dynamics.As could be expected from what is known on networks, an important requirement for a significantreciprocity is the heterogeneity of connections; a separate analysis, not addressed in this work, would beneeded for higher-order generalizations of spatial networks. To prove this, we performed an invasionanalysis able to provide a good estimate for the critical point of emergence of cooperation, for bothhypergraphs with rank equal to 2 and to 3. More generally, the analysis indicated that, the higher the orderof the interactions, the stronger the structural reciprocity is. To be more precise, the marked enhancementof reciprocity has been found for heterogeneous structures with a null or slightly negative assortativity, in
12 3 e e e
12 3 e e e
12 3 e e step 3 step 4 Figure 6.
Illustration of a single instance of steps 3 and 4 of the algorithm used to generate rank-3 simplehypergraphs starting from graphs. Note that, removing step 4, we would end up with a simplicial complex. line with what is reported for networks. According to our analysis, we expect a weaker beneficial effect onhypergraphs derived from assortative networks.Furthermore, the invasion analysis allowed us to justify, in comparison with the almost abrupttransition found in homogeneous structures, the enlargement of the transition region observed for theused heterogeneous hypergraphs. Relying on their locally-clustered structure, it also allowed to partiallyexplain why the enlargement grows with the increasing of conversion fraction.Regarding the heterogeneity of strategies in the asymptotic population, we found the presence oftopological traps preventing the system to converge to a uniform state. We characterized those trapsby means of two randomization procedures. In the used clustered structures, we attended the break-upof the present traps whenever the chosen randomization did not preserve the local clustering, thoughstill preserving second-order degree correlations. We show that traps represent the only way to get anon-uniform asymptotic population.Finally, by tuning the selection pressure, we found an improvement towards cooperation in conditionsof stronger selection. This is precisely what we expect from how topological traps work. Remarkably, thisshows up the contribution coming from some traps to enlarge the region where cooperative behaviors cansurvive.
6. Methods
The algorithm to generate rank-3 simple hypergraphs proceeds as follows:1. A simple network g = ( V , E ) is generated through some model;2. From the set of all the 3-cliques in it, a fraction 0 ≤ p ≤ e = { v , v } , e = { v , v } , e = { v , v } ⊂ E are the three edges forming a chosen 3-clique over the subset { v , v , v } ⊂ V ofvertices, then the hyperedge (triangle) e = { v , v , v } is added to E ;4. To obtain a simple hypergraph, after a fraction p of 3-cliques has been converted to triangles, an edgeis removed if it is subset of at least one triangle: the three edges e , e , e , being subsets of e , areremoved from E .This procedure generates a rank-3 simple hypergraph H = ( V , E ) , where V = V and E is the new set ofhyperedges constructed from E through the steps 2–4. The initial network g is the 2-section of H . SeeFigure 6 for an illustration of the procedure.Since the algorithm relies on the amount of 3-cliques present in the base network g , the more clusteredis the latter, the stronger will be the effect of increasing the value of p . In HK ( P t =
1) and DM networks, every link is part of at least a 3-clique. Therefore, by tuning p , they allow us to explore the entire range ofhypergraphs from the 2-uniform ( p =
0) to the 3-uniform ( p = Every simulation starts by picking, uniformly at random, a fraction c ≡ c ( t = ) of cooperators. Foreach fixed set of parameters β , c and p , we run 100 simulations over a generated hypergraph and take theaverage of both the asymptotic cooperators density and the convergence time, i.e., the number of timesteps the process took to stop. We do it for 10 realizations of the same kind of hypergraph (i.e., stemmingfrom the same network model), and then average again over them. Whenever the maximum number oftime steps (set to 2 × ) is reached without converging to one of the absorbing states c = c =
1, thelast value of c is considered. We briefly summarize the two randomization procedures implemented for the characterization oftopological traps. The selected number of randomizing steps was 10 for each network.6.3.1. Preserving Local Clustering CoefficientThrough the following procedure one can randomize a network preserving, besides the degree, alsothe local clustering coefficient of each vertex. At each step, two vertices are uniformly chosen at randomand their local clustering coefficient is computed. Two links, one for each vertex, are randomly selectedand interchanged. If the clustering coefficients of the two vertices remain unchanged, the randomizingstep is accepted; otherwise, another two vertices are drawn and the randomization is tried again.6.3.2. Preserving Second-Order Degree CorrelationsThis procedure allows to obtain a randomized network with exactly the same degree correlationsup to the second-order, i.e., preserving the joint 3 K -distribution P ( k , k (cid:48) , k (cid:48)(cid:48) ) . At each step, two verticesare uniformly chosen at random and the pair is accepted if only the vertices have equal degree. Then,two links, one for each vertex, are peaked at random (an interchange performed now would ensure theconservation of the 2 K -distribution P ( k , k (cid:48) ) ). The degrees of the two vertices at the end of the two selectedlinks are compared and, if they match, the interchange is performed (now conserving also P ( k , k (cid:48) , k (cid:48)(cid:48) ) );elseways, two new initial vertices are chosen and the randomization is tried again. Author Contributions:
Conceptualization, A.A.; methodology, A.A., S.G., J.M. and G.B.; investigation, G.B.;writing–original draft preparation, G.B.; supervision, A.A, J.M. and S.G. All authors have read and agreed to thepublished version of the manuscript.
Funding:
We acknowledge support by Ministerio de Economía y Competitividad (grants PGC2018-094754-B- C21and FIS2015-71929-REDT), Generalitat de Catalunya (grant 2017SGR-896), and Universitat Rovira i Virgili (grant2019PFR-URV-B2-41). A.A. acknowledges also ICREA Academia and the James S. McDonnell Foundation (grant220020325).
Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A. Critical Point for Uniform Complete Simple Hypergraphs
It is possible to find analytically the critical value of α m cr at which c ∞ = m -uniformcomplete simple hypergraph with N vertices, in the limit of strong selection ( β b (cid:29) N the transition becomes discontinuous and α m surv , above which c ∞ > α m cr . The following proposition holds: Proposition A1.
Given a complete m-uniform simple hypergraph (m ≥ ) with N vertices, the critical value α m cr ,in the limit β b (cid:29) , is given by α m cr = m N − N − m (A1) where β is the noise parameter of the Fermi distribution function and b is the fixed contribution a cooperator puts ateach time step. Proof of Proposition A1.
The expression for α m cr follows imposing that the probabilities p d → c and p c → d that, respectively, a defector becomes a cooperator and vice versa, coincide. Being the hypergraph complete,each vertex possesses ( N − m − ) neighbors. Therefore, the payoff f ( σ ) got by any player with strategy σ reads f ( σ ) = b (cid:18) N − m − (cid:19) (cid:20)(cid:16) α m − (cid:17) σ + α m − m q ( σ ) (cid:21) (A2)where q ( σ ) = q ( c ) = c N / ( N − ) − ( N − ) for a cooperator and q ( σ ) = q ( d ) = c N / ( N − ) for adefector. Defining ∆ f ≡ f ( c ) − f ( d ) , we get p d → c = q ( d ) F ( ∆ f ) = c NN − F ( ∆ f ) (A3) p c → d = (cid:16) − q ( c ) (cid:17) ( − F ( ∆ f )) = (cid:18) − c NN − + N − (cid:19) ( − F ( ∆ f )) (A4)where F indicates the Fermi distribution function. By equating p d → c and p c → d , and solving with respectto α , we get α m cr = m N − N − m (cid:18) N − m − (cid:19) − β b ln (cid:18) c − c (cid:19)(cid:18) N − m − (cid:19) (A5)Taking β b (cid:29)
1, Equation (A1) follows.Note that, only for c = β b in this case. Accordingly, the prediction is very precise also for lowvalues of β whenever c does not differ much from 0.5.In the limit N → ∞ , from Equation (A1), we recover the critical value α m cr = m , which holds for aninfinite well-mixed population, in accordance to Equation (10) evaluated for a m -uniform hypergraph. Appendix B. Simulations for Further Values of the Initial Fraction of Cooperators
In Figure A1 we report the results of the MC simulations performed on hypergraphs derived from DMnetworks for some values of the initial fraction of cooperators, c , other than 0.5. The enhanced structuralreciprocity when increasing p is confirmed and, additionally, it gets stronger when c is increased. Appendix C. Simulations for the Erd ˝os-Rényi Model
In Figure A2 we show the results of the MC simulations performed on hypergraphs stemming fromER networks, taking β = c = r =
1, as expectedfor highly homogeneous structures such as ER networks. Accordingly, the denser and less heterogeneouscase with (cid:104) k (cid:105) =
10, shows a slightly lower reciprocity. Due to the low fraction of 3-cliques, the transitionsare nearly overlapped. Moreover, the average convergence time is always below 300 steps, along with theabsence of traps. c p 0 0.2 0.5 1 c p 0 0.2 0.5 1 rt rt Figure A1.
Asymptotic density of cooperators c (top) and convergence time t (bottom) versus r , forhypergraphs generated from DM networks with N =
500 vertices and average degree (cid:104) k (cid:105) =
4. The valuesof p are specified in the legends; c = c = β =
1. The dashed line at r = Appendix D. On the Role of Cooperator Hubs
Equation (13) relies on the comparison between the payoffs of a cooperator hub and any of its defectorneighbors at the initial round. For any r greater than the value r hub provided by Equation (13), wheneverthe cooperator hub and any of its defector neighbors match for the strategy update, it is more probablethat the defector becomes a cooperator than the other way around. For such values of r , the cooperatorhub (or more cooperators hub together, as it is likely to be, by construction, in HK and DM networks) canstart and sustain the formation of a cooperative community, eventually able to expand over the structureor, at least, to resist to be invaded by any neighboring defective community.In Figure A3 we show some temporal evolution of the average fraction of cooperators in theneighborhood of cooperator hubs, ¯ q ( hub ) , and of cooperator hubs’ defector neighbors, ¯ q ( d ) , for simulationsended with c ∞ > α around the respective values of α surv . It is evident how,independently of c , ¯ q ( hub ) and ¯ q ( d ) rapidly reach values around 0.75–0.85 and 0.3–0.35, respectively.Therefore, Equation (13) estimates the conditions for which such temporal evolution (or those ending with c ∞ =
1) are likely to start. The prediction gives a clear qualitative justification to the fact that r decreases(the reciprocity improves) with the increasing of both c and s (the rank of the hypergraph). The equationgives also a good quantitative prediction for c roughly between 0.4 and 0.7. Outside this range, thepredicted values of r are, too low for high values of c , and too high for low values of c . In other words,the structures provide a lower and a higher reciprocity, respectively, than the one expected by the onlycomparison of the payoffs at the first rounds. The point is that the prediction made through Equation (13) c p 0 1 c p 0 1 rt rt Figure A2.
Asymptotic density of cooperation c (top) and convergence time (bottom) versus r , forhypergraphs generated from ER networks with N =
500 vertices, average degree (cid:104) k (cid:105) = (cid:104) k (cid:105) =
10 (right). The values of p are specified in the legends; c = β =
1. The dashed line at r = is mainly based on the average degree disparity among the hubs and their neighbors, while it does notconsider the dependence on c of the probability that any of those hubs and their neighbors select eachother for the strategy update. As long as c takes values close enough to 0.5, the main contribution comesfrom the degree disparity and Equation (13) works well. However, in asymmetric setups corresponding tolow and high values of c , the approximation used in Equation (13) is no longer sufficient and, consequently, α hubth does not give an accurate quantitative estimation of α surv . Appendix E. On the Local Clustering Coefficient
Both HK and DM models grow networks whose local clustering coefficient C scales with the degree k as C ( k ) = C ( ) k − γ . Using the notation k m to indicate the m -degree, the latter expression reads C ( k ) = C ( ) k − γ . As reported in Figure A4 (left panels), vertices in a DM network follow exactly this law, byconstruction, with C ( ) = γ =
1. Vertices in a HK network ( P t = C ( ) ≈ γ ≈ C ( k ) , one can express the average number of 3-cliques (ortriangles), κ ( i ) , that the neighbors of a vertex i share with it, in terms of k only. If κ ( i ) >
1, then someoverlap exists among those 3-cliques. We used this quantity in our invasion analysis and it is strictlyrelated with the clustering coefficient C ( i ) of node i . The number L ( i ) of links among neighbors of vertex i Figure A3.
Instances of the temporal evolution of the average fraction of cooperators ¯ q ( hub ) in theneighborhood of cooperator hubs (orange), and of the average fraction of cooperators ¯ q ( d ) which areneighbors of defector neighbors of cooperator hubs (cyan). First row corresponds to c = c = c = p = α around the respective values of α surv . The hubs are here those vertices with degree above the 98th percentileof the degree distribution. Very similar results are found for HK networks. is exactly the number of 3-cliques incident on i . This number, if the 3-cliques cover the entire neighborhoodof i , is the 3-degree k ( i ) when we set p =
1. Then, on one hand, κ ( i ) = k ( i ) (cid:12)(cid:12)(cid:12) p = k ( i ) (cid:12)(cid:12)(cid:12) p = (A6)on the other, C ( i ) = k ( i ) (cid:12)(cid:12)(cid:12) p = ( k ( i ) ( k ( i ) − )) (cid:12)(cid:12)(cid:12) p = (A7)and the two quantities are related by κ ( i ) = C ( i ) (cid:16) k ( i ) − (cid:17)(cid:12)(cid:12)(cid:12) p = (A8) C = - k C C = - k C k ( d ) = k ( c ) k ( d ) = h ( k ( c ) ) k ( d ) = ( c ) - k ( c ) k ( d ) Figure A4.
Left . Local clustering coefficient C versus degree k for HK (top) and DM networks (bottom). Right . The shaded area indicates, for a given value of k ( c ) , the values of k ( d ) satisfying Equation (A16); h ( x ) = ( ) (cid:104) ( x − ) + (cid:112) ( x − ) + ( ) (cid:105) . In particular, for DM networks, Equation (A8) becomes κ ( i ) = k ( i ) − k ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = (A9)and k ( i ) (cid:12)(cid:12)(cid:12) p = = ( k ( i ) − ) (cid:12)(cid:12)(cid:12) p = (A10)For HK networks ( P t = κ ( i ) = k ( i ) − (cid:16) k ( i ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = (A11)and k ( i ) (cid:12)(cid:12)(cid:12) p = = (cid:16) k ( i ) − (cid:17) (cid:16) k ( i ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) p = (A12)In the end, this shows that the values of the structural parameters chosen for the computation ofEquation (17) in the main text (Figure 3) are exact and nearly compatible with DM and HK hypergraphs.Since the algorithm presented in Section 6, used to generate the hypergraphs, relies on the 2-sections(base networks), Equations (A6)–(A8) can be generalized to any order. Let us suppose that, from a given2-section, we can construct uniform hypergraphs with rank up to m max . To generalize the given algorithm,let us define p m , for m ∈ {
3, . . . , m max } , as the fraction of m -cliques we transform in hyperedges of degree m ; p corresponds to the p used throughout the text. Whenever we take p (cid:96) = ∀ (cid:96) (cid:54) = m , we get a m -uniform hypergraph for p m =
1, and its 2-section for p m =
0. Then, given a vertex i , the followingrelations hold: k ( i ) m (cid:12)(cid:12)(cid:12) p m = = k ( i ) (cid:12)(cid:12)(cid:12) p m = κ ( i ) m m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p m = (A13) κ ( i ) m (cid:12)(cid:12)(cid:12) p m = = (cid:16) k ( i ) − (cid:17) ! ( m − ) ! (cid:16) k ( i ) − m + (cid:17) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p m = C ( i ) m − (cid:12)(cid:12)(cid:12) p m = (A14)where C ( i ) m is the generalized rank- m local clustering coefficient of vertex i , C ( i ) m = N ( i ) m (cid:18) k ( i ) m (cid:19) (A15)with k ( i ) = k ( i ) (cid:12)(cid:12)(cid:12) p m = . This generalized clustering coefficient is defined as the ratio between the number N ( i ) m of m -hyperedges among the neighbors of i , and the maximum possible number of such m -hyperedges.In particular, C is the standard local clustering coefficient defined for graphs. Clearly, if p m + is set to 1, N ( i ) m corresponds to k ( i ) m + . Additionally, κ ( i ) m is the average number of m -cliques (or m -hyperedges) that theneighbors of vertex i share with it. Thanks to the above relations, by just knowing how the rank- m localclustering coefficient depends on the 2-degree k , one is able to compute κ m and k m for any vertex and, inturn, the correct values of ξ m , see Equation (13). From both HK and DM networks, and a large class ofother random models, one can generate uniform hypergraphs with rank not greater than 3.We now make explicit the form taken by the condition in Equation (23) when C ( k ) = C ( ) k − γ and,to get an algebraic solution, we consider DM networks where γ =
1. After some algebra, Equation (23)becomes 34 (cid:16) k ( c ) − (cid:17) + (cid:115)(cid:16) k ( c ) − (cid:17) + (cid:18) (cid:19) < k ( d ) ≤ k ( c ) − p = p =
0. Taking,for example, k ( c ) =
8, Equation (A16) is fulfilled for k ( d ) ∈ [
11, 15 ] . Note also that, for k ( d ) > k ( c ) − p =
0. However,due to the linear dependence of Equation (16) with respect to α , ∂ ( f ( c ) − f ( d ) ) / ∂α < p = p = k ( d ) > k ( c ) , making the competitive dynamics always fiercer inthe former case. Appendix F. Comparing Results from Randomized Networks
We report in Figure A5 the results of the MC simulations performed for networks randomizedpreserving their second order degree correlations but not fully their local clustering coefficient, indeedlowered to around 0.1. We find r ( ) = P t = r ( ) = c c rt rt Figure A5.
Asymptotic density of cooperators c (top) and convergence time t (bottom) versus r for HK (left)and DM (right) networks ( p =
0) with N =
500 vertices, average degree (cid:104) k (cid:105) = (cid:104) k (cid:105) =
6, respectively.The randomization preserves the degree correlations up to the second order, but not the local clusteringcoefficient; c = β =
1. The dashed line at r = to the specific properties of the two models. Since the values of r ( ) are very close to the ones foundwithout randomization, this means, on one hand, that the truly important factor responsible for thestructural reciprocity is the degree heterogeneity, and on the other, that the better performance shown byDM hypergraphs should be mainly ascribed to their specific structure and scarcely to their higher localclustering. References
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