Effect of chaotic agent dynamics on coevolution of cooperation and synchronization
Rohitashwa Chattopadhyay, Shubhadeep Sadhukhan, Sagar Chakraborty
EEffect of chaotic agent dynamics on coevolution of cooperation andsynchronization
Rohitashwa Chattopadhyay, a) Shubhadeep Sadhukhan, b) and Sagar Chakraborty c) Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016,India (Dated: 22 February 2021)
The effect of the chaotic dynamical states of the agents on the coevolution of cooperation and synchronizationin a structured population of the agents remains unexplored. With a view to gaining insights into this problem,we construct a coupled map lattice of the paradigmatic chaotic logistic map by adopting the Watts–Strogatznetwork algorithm. The map models the agent’s chaotic state dynamics. In the model, an agent benefitsby synchronizing with its neighbours and in the process of doing so, it pays a cost. The agents updatetheir strategies (cooperation or defection) by using either a stochastic or a deterministic rule in an attemptto fetch themselves higher payoffs than what they already have. Among some other interesting results, wefind that beyond a critical coupling strength, that increases with the rewiring probability parameter of theWatts–Strogatz model, the coupled map lattice is spatiotemporally synchronized regardless of the rewiringprobability. Moreover, we observe that the population does not desynchronize completely—and hence finitelevel of cooperation is sustained—even when the average degree of the coupled map lattice is very high. Theseresults are at odds with how a population of the non-chaotic Kuramoto oscillators as agents would behave.Our model also brings forth the possibility of the emergence of cooperation through synchronization onto adynamical state that is a periodic orbit attractor.
I. INTRODUCTION
Cooperation plays a pivotal role in the sustenanceof the biological, the social, and the economic systems.An insightful theoretical approach for studying the emer-gence of cooperation in such systems is the evolutionarygame theory . In the framework of the theory, eachindividual or agent in the population adopts a strategythat is based on the payoff it receives by interacting withthe other agents. It is generally observed that an indi-vidual’s self-interest acts as an obstacle in the emergenceof cooperation, e.g., in the stylized game of the prisoner’sdilemma . Thus, one of the main objectives in the theoryis to understand the situations where even though eachagent has a higher incentive to defect, how the globalcooperation-state emerges. To this end, many modelsand mechanisms for the emergence of global cooperationare available in the research literature . The coevo-lutionary processes—the processes that co-occur with theevolution of strategies—are known to promote coopera-tion effectively . Also, certain types of interaction be-tween the agents affect the degree of cooperation presentin the system .The individual agents may have some associateddynamics—e.g., replicator dynamics , best responsedynamics , imitation dynamics , sampling dynamics ,and opinion dynamics —modelling the evolution of theirstates. In a networked population of agents (who can ei-ther cooperate or defect) when arranged so as to have a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] the Kuramoto model realized as far as their coupleddynamics is concerned, the costly interactions present adilemma, termed the evolutionary Kuramoto dilemma :An agent may cooperate by paying the cost in order tohave its state synchronized with the rest agents in thenetwork; or it may defect and thus not suffer any cost,while expecting the other agents’ states to be synchro-nized to its own. Just as the study of emergence of co-operation has a long history in the evolutionary gamedynamics, the emergence of synchronization in complexnonlinear systems has its own very rich literature .The interplay of synchonization and cooperation in theevolutionary game dynamics of the agents is dependenton the topological details of the network employed. Forexample, in Watts–Strogatz (WS) network of the afore-mentioned agents, the population of reaches fully syn-chronization state only for high values rewiring proba-bility and the coupling strength beyond which such astate is attained, decreases with an increase in the ran-dom rewiring probability . Furthermore, in the WS net-work, and also in the Barabási–Albert network and theErdős–Rényi network , the increase in average degree ofthe nodes, desynchronizes the agents’ dynamics .The primary question we ask in this paper is, given anetwork topology, how robust such conclusions regardingthe behaviour of the population of the agents are if theuncoupled agent dynamics is chaotic. In this context, werecall that in the Kuramoto model of the population ofthe agents, the uncoupled agent dynamics is that of auniform oscillator on a circle, i.e., it is a flow on a 1-torussuch that the phase uniformly changes. The Kuramotomodel is synonymous with the homogeneous sinusoidalcoupling between such non-chaotic oscillators, and onceendowed with such coupling, the oscillators may now betermed as the Kuramoto oscillators. It is all too well a r X i v : . [ n li n . AO ] F e b known that globally coupled identical Kuramoto oscil-lators can not show chaotic dynamics and any chaosthat may be there in the globally coupled non-identicalKuramoto oscillators vanishes in the infinite size limitof the population. However, it may be possible to havechaotic state of the population in the thermodynamiclimit if the condition of global coupling between the non-identical Kuramoto oscillators is done away with . Ourquest, however, seeks a non-chaotic synchronized popula-tion state when the individual uncoupled agent dynamicsare identical and chaotic. More specifically, we are inter-ested in the correlation between the co-occurrence of thetemporal changes in the cooperator and the synchroniza-tion levels of such a population; or in other words, in theco-evolution of the cooperation and the synchronization.Since, discrete dynamical systems—often calledmaps—are capable of showing chaotic behaviour evenwith single phase space variable, in this paper it con-veniently suits our purpose to work with the nonlin-ear maps. Specifically, we work with the paradigmaticchaotic logistic map as a toy model. Another reasonbehind using the logistic map is that its phase variable,when scaled by a constant factor of π , is bounded be-tween zero to π just like the phase of the Kuramotooscillator. One extensively studied model for a largenumber of coupled maps is that of the coupled map lat-tice (CML) , that was introduced as a simple tool tostudy the chaotic behaviour of the spatially extended sys-tems. A CML consists of a lattice of maps, each mapbeing on a unique lattice point that is coupled locally toother lattice points via the edges of the lattice. Despitebeing a rather simple construction, the CML has foundextensive applications in modelling a broad spectrum ofsystems , e.g., pattern formation, crystal growth, theJosephson junction arrays, multi-mode lasers, and vor-tex dynamics.The synchronization of the coupled chaotic maps ondifferent network topology and under various settingsis also an intensively studied topic . It should bepointed out that there is a difference between the syn-chronization on to a fixed point in a CML and the syn-chronization among the Kuramoto oscillators: In the caseof the CML, the synchronization on to a fixed point im-plies that all the lattice points have the identical non-oscillatory static state, while the synchronization in anetwork of the Kuramoto oscillators implies a dynamicstate where all the synchronized oscillators’ states changein unison with time. The former phenomenon has a closeresemblance to the phenomenon of the amplitude deathin the coupled limit cycle oscillators like the coupled Stu-art Landau oscillators .As already mentioned, since our aim is to see the ef-fect of chaos in a given network, in this paper we chooseto work with the WS model of generating networks thatcan be interpolations between a regular network and (al-most) the Erdős–Rényi network depending on the valuesof the random rewiring probability. What we find is thatthe conclusions obtained for the behaviour of coopera- tors and defectors in a network vis-a-vis the evolutionaryKuramoto dilemma is so strongly dependent of the un-coupled agent’s dynamics that they are in stark contrastwith what is known to happen when rewiring probabilityor average degree changes in the Kuramoto model. With-out further ado, in what follows, we introduce our modelin Sec. II and the results in Sec III, before concluding inSec. IV. II. MODEL
We consider a population of N agents, each of whichhas its state defined by x in ( n being the time step and i the index for the node) that evolves in accordance withthe chaotic logistic map— x in +1 = 4 x in (cid:0) − x in (cid:1) —whenthe agents are not interacting with each other. Here, i ∈ { , , · · · , N } , n ∈ { , , , · · · , } , and by construc-tion, x in ∈ [0 , ∀ i, n . As mentioned earlier, this is one ofthe simplest and most well-studied map that possesseschaotic solutions. The corresponding CML is definedmathematically as, x in +1 = 4 x in (cid:0) − x in (cid:1) (1 − s in (cid:15) ) + s in (cid:15)k i N (cid:88) j =1 a ij x jn , (1)where k i is the degree of i th lattice site or node, (cid:15) ∈ [0 , denotes the coupling constant quantifying the strengthof the coupling, and a ij ∈ { , } are the elements of theadjacency matrix. This model effectively associates astrategy, s in , to each agent at time n ; s in can either bezero (defection) or unity (cooperation). Note that weare considering a simple but non-trivial linear couplingbetween the agents so that x in ∈ [0 , ∀ i, n even whenthe agents interact.Now the idea is that a cooperator (an agent with co-operation strategy) chooses to interact with the otheragents (as allowed in accordance with the topology of theCML) and endeavours to synchronize with their dynam-ics while incurring the cost of the interactions; whereasthe defectors (agents with defection strategy) do not in-teract at all and hence incur no cost. The cost associatedwith a cooperating agent is the measure of rate of devi-ation from its underlying dynamics. Thus, the cost, c in ,for the i th agent at the n th time step may be defined asas follows: c in := α (cid:12)(cid:12)(cid:8) x in − x in − (1 − x in − ) (cid:9) − (cid:8) x in − + 4 x in − (1 − x in − ) (cid:9)(cid:12)(cid:12) . (2)where, α is a positive real parameter termed the relativecost. One may note that this is exactly in line with thecost defined in the analogous Kuramoto model . Thebenefit, b in that an agent reaps is measured through theextent to which the agent synchronizes with its neigh-bours. We define it as b in := (cid:80) j r ijn a ij (cid:80) j a ij , (3)where r ijn := (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:0) π √− x in (cid:1) + exp (cid:0) π √− x jn (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) , (4)quantifies for the degree or synchronization between apair of agents, say, i th and j th. Note that r ijn is unity(maximum) when x in = x jn . We realize that b in is thelocal version of the global synchronization parameter , r G , defined as, r G := | (cid:80) Ni =1 exp (cid:0) π √− x i (cid:1) | N . (5)The system is said to be completely synchronized when r G = 1 that corresponds to the population state where x in = x jn ∀ i, j ∈ { , , · · · , N } . For smaller values of r G ,the population is away from the synchronized state; r G =0 corresponds to fully unsynchronized or incoherent statethat is realized as a random distribution of the individualagents’ states.Having defined the cost and the benefit, we can nowdefine the payoff (or utility) of i th agent at the n th timestep for this interaction as U in := b in − c in . In our model,we include a rule for strategy update for each agent ateach point of time; the i th agent would choose betweencooperation ( s in = 1 ) or defection ( s in = 0 ) at time step n depending on how it fared against the agents it interactedwith at the time step n . This rule can be implementedeither deterministically or stochastically. In determin-istic strategy update, also known as the unconditionalimitation rule, at each time step the agents at each nodecompare their own payoffs with all their neighbours andadopt the strategy of the most successful agent (withhighest payoff) among them (the agent and its neigh-bours). A popular stochastic strategy exchange rule isthe Fermi rule for strategy exchange . In this rule, attime n with probability [1 + exp {− β ( U jn − U in ) } ] − , the i th agent chooses the strategy of one of the randomlychosen agent, say j th agent, among all the agents it in-teracted with. Here, β denotes the rationality factor.It is quite obvious that under such update rules, aninitial state of the population such that all agents arecooperators does not change with time; same is true forthe initial state where all the agents are defectors. Aninteresting observation concerning the difference betweenthese two invariant states is that while the former statemay or may not simultaneously be a synchronized state,the latter can never be a synchronized state as it is ac-companied with no coupling between the agents. It is alsointeresting to note that sometimes while a completelycoupled set of chaotic oscillators (as in the case of allcooperator state) may not lead to synchronization, occa-sional uncoupling between the oscillators (as in the caseof a mixed cooperator-defector state) may induce syn-chronization . The mixed state is what we are moreinterested here anyway because we want to know whetherglobal cooperation can emerge in our model starting froma mixed state and how the cooperation co-evolves with FIG. 1.
Bifurcation diagram for a globally couplednetwork of cooperators.
We plot state variable ( x i ) of arandomly chosen i th node in an all-to-all coupled network of100 logistic maps after 2000 time steps for varying couplingstrengths ( (cid:15) ) . We find a fixed point for (cid:15) > . . On turningdown the coupling strength from 0.5 to 0, we find that period2, period 4, and higher periods emerge, ultimately leading tochaos. the synchronization. A straightforward useful definition,in this context, is the degree of cooperation in the popu-lation that is just the fraction of cooperators (agents withstrategy s i = 1 )—denoted by C througout the paper—inthe population. Clearly, C ∈ [0 , .Before proceeding further, we pause a bit to ponderupon the meaning of synchronization in the CML. Theindividual uncoupled agent’s dynamics is chaotic. So, ifthe agents’ dynamics synchronize once coupling is turnedon, then a realizable synchronized state should be an at-tractor in the N -dimensional phase space, i.e., the phasespace of the CML. The attractor could either be a homo-geneous fixed point (i.e., x in = x jn ∀ i, j ∈ { , , · · · , N } as n → ∞ ) or possibly one of the non-fixed point at-tractors such as chaotic attractor and stable homoge-nous periodic orbit. For illustration, consider that all theagents are cooperators and there is all-to-all coupling inthe CML—i.e., ∀ n, i, j , s in = 1 and a ij = 1 − δ ij ( δ ij is theKronecker delta) respectively—given by Eq. (1). The re-sulting equation has a trivial homogeneous fixed points,viz., x i = 0 . However, it is not an attractor. The at-tractors depend on the value of the coupling constant. Abifurcation diagram corresponding this particular CML isdepicted in Fig. 1. We find that all the nodes are synchro-nized to a stable homogeneous fixed point x i = x ∗ = 0 . when the coupling strength is greater than . . As welower the coupling strength, the CML undergoes a pe-riod doubling route to chaos potentially followed by morechaotic regions and periodic windows. Thus, the CML’snodes can also synchronize onto the stable periodic orbitsof various periods depending on the value of the couplingconstant (e.g., period- at (cid:15) ≈ . ) and, in principle,even onto a chaotic attractor. In all the cases, r G = 1 at all times (after sufficient transients are ignored) wouldindicate global synchronization but, of course, it cannotdifferentiate the different attractors. III. NUMERICS AND RESULTS
Our model allows for using any network topology forthe CML. We exclusively use the WS model in whichincreasing a particular parameter—the rewiring proba-bility, p ∈ [0 , —one can transition from a regular net-work ( p = 0 ), to small-world networks, and finally to arandom network ( p = 1 ). In WS model, one builds anetwork topology by starting with a undirected ring lat-tice with N nodes, each having k edges ( k/ on each sideof a node). Subsequently, proceeding in a anticlockwisemanner (say), the left nearest neighbour (along the ring)connections of each node are rewired with a probability p to a random node while avoiding self and multiple con-nections. Once the nearest neighbours of all the nodesare exhausted, one again considers every node sequen-tially and randomly rewires the next nearest left neigh-bour (along the ring) links of each node with a probability p . This process is repeated until all the links of the ringlattice one started with have been considered for rewiringand finally we arrive at a network with N nodes and av-erage degree k . In this paper, neighbours of an agentmean all other agents who have a connecting edge withthe agent.We intend to study how the global synchronizationparameter and the degree of cooperation behave in theCML, thus created, as we change its network topologyby tuning the rewiring probability and the degree of thenetwork. Interestingly, we find that our results pertain-ing to such studies are in sharp contrast with the similarstudies where the uncoupled agent dynamics is of theKuramoto oscillator and not a chaotic map.In what follows, unless otherwise specified, we presentour results using, the WS networks with 100 nodes andaverage degrees up to 98. We evolve the system for 2000time steps that are enough to reach the steady state solu-tions. We do check that the final results are qualitativelyrobust against varying network size. All the reportedvalues of the global synchronization parameter and thedegree of cooperation are calculated using the data atthe final time step. Also, they are averaged over 40 inde-pendent realizations achieved using 40 different randominitial conditions and as many realizations of the networkin Eq. (1), and we use (cid:104) r G (cid:105) and (cid:104) C (cid:105) to denote them re-spectively. Similarly, wherever appropriate, we use (cid:104) k (cid:105) for the average degree in the light of aforementioned av-eraging. A. Critical coupling strength for synchronization
The CML is quite an analytically challenging dynami-cal system. So the natural question, that what the mini- . . . (cid:15) h k i . . . . . . h r G i FIG. 2.
Global order parameter with varying couplingstrength and average degree.
We plot the average globalorder parameter ( (cid:104) r G (cid:105) ) for a WS network (details in the text),as a function of the average degree ( (cid:104) k (cid:105) ) and the couplingstrength ( (cid:15) ) . We fix the relative cost ( α ) to 0.01 and rewiringprobability ( p ) to 0.2. The vertical dashed black line cor-responds to the analytically estimated value of the criticalcoupling strength ( (cid:15) crit ) in the limit N, (cid:104) k (cid:105) → ∞ . mum coupling strength should be so that synchronizationis effected, is not easy to answer analytically. Neverthe-less, there is a case where this question can be answeredand the critical coupling strength, (cid:15) crit , beyond whicha stable synchonized state—a homogenous fixed pointattractor—is realized (e.g., (cid:104) r G (cid:105) = 1 ), apparently turnsout to be a lower bound for the other cases studied in thispaper. The case happens to be the limit of N, (cid:104) k (cid:105) → ∞ in the CML where all agents adopt the strategy of coop-eration in the synchronized state.Thus, let’s recall Eq. (1) with all s i = 1 . Since weare looking for a nontrivial homogeneous fixed point, theonly possibility is x i = x ∗ = 0 . ∀ i . We now must lookfor the minimum value of coupling constant (actually, (cid:15) crit ) such that the fixed point is stable. To this end,we perform standard linear stability analysis by putting x in = x ∗ + h in , h in being the infinitesimal perturbation,in Eq. (1), and on expanding to the first order terms, wearrive at, h in +1 =4(1 − x ∗ ) (1 − (cid:15) ) h in + (cid:15) (cid:104) k (cid:105) (cid:80) Nj =1 a ij h jn . (6)Note we have replaced k i with (cid:104) k (cid:105) because we are inter-ested in an ensemble of networks with the same numberof nodes and the same average degree. Subsequently,we use the Fourier expansion for the perturbation— h in = (cid:80) q ˜ h ( q ) n exp (cid:0) √− iq (cid:1) , and substitute it in Eq. (6)to get for every q th Fourier mode, ˜ h ( q ) n +1 ˜ h ( q ) n =4 (1 − x ∗ ) (1 − (cid:15) ) . (7)Here, we have imposed the condition (cid:104) k (cid:105) → ∞ . Evi-dently, the synchronized state is stable if | ˜ h ( q ) n +1 / ˜ h ( q ) n | < or | − x ∗ )(1 − (cid:15) ) | < . Since x ∗ = 0 . and ≤ (cid:15) ≤ , (cid:15) crit = 0 . , i.e., the CML spatiotemporally synchronizesfor any value of coupling constant greater than half.It is worth remarking that since the estimation of (cid:15) crit has not explicitly required any mention of the type ofnetwork, this threshold value of the coupling strengthshould hold good for any network with N, (cid:104) k (cid:105) → ∞ .Furthermore, although analytically quite challenging, inprinciple, such an estimation of (cid:15) crit may be possible forsynchronization onto a homogeneous n -period orbit at-tractor, ( x ∗ , x ∗ , · · · , x n ∗ ) , such that all the nodes syn-chronously fluctuate between these n values for eachagent’s state as time tends to infinity. In fact, numer-ically we do find (refer to Fig. 2), the system synchro-nizes onto a 4-period orbit around (cid:15) ≈ . that is inter-estingly also seen in Fig. 1. The other synchronizationregime—a synchronization onto the homogeneous stablefixed point—is found at much higher coupling strengths.We note that as (cid:104) k (cid:105) increases the analytical estimate of (cid:15) crit = 0 . matches reasonably well with the numerical re-sults. For all practical purpose, (cid:104) k (cid:105) ∼ is large enoughfor the estimate to be considered valid. It may also beobserved that in Fig. (2), (cid:104) r G (cid:105) is not exactly equal tounity beyond (cid:15) crit not only because (cid:104) k (cid:105) is not strictly in-finity, but also because the averaging is over only a finitenumber of realizations. B. Effect of changing rationality
Another important parameter on which the effective-ness of the critical coupling strength, estimated in theimmediately preceding subsection, depends is the strat-egy update rule.It may be recalled that the stochastic Fermi strategyupdate rule, which we adopt in this paper, has a rational-ity factor β that quantifies the degree of rationality of theagents. Its meaning becomes crystal clear when one notesthat an infinitely rational ( β = ∞ ) i th agent definitelychooses the strategy of one of the randomly chosen j thneighbouring agent if the j the agent obtained more pay-off than the i th agent. On the other hand, the infinitelyrational i th agent never chooses the strategy of one ofthe j th neighbouring agent if the j the agent has compar-atively less payoff. Similarly, in case the i th agent is com-pletely irrational ( β = 0 ), it chooses the strategy of therandomly chosen j th neighbouring agent with probabilityequal to one-half. Thus, β is an apt parameter to charac-terize the degree of rationality, because with the increasein its value, the probability of choosing the strategy of theneighbouring agent with higher payoff increases mono-tonically. It is obvious that the stochastic update rulewith β = ∞ is practically equivalent to the determinis-tic rule—the synchronous unconditional imitation—alsoadopted in this paper (see Sec. II)—only difference beingthat in the former an agent only compares its payoff withthat of a randomly chosen neighbour, while in the latteran agent compares its payoff with all its neighbours. In Fig. 3, we depict (cid:104) r G (cid:105) and (cid:104) C (cid:105) as a function of therewiring probability and the coupling constant (the av-erage degree fixed at ), and also as a function of theaverage degree and the coupling constant (the rewiringprobability fixed at . ) for both the the stochastic strat-egy update rule (with β = 1 , , and ) and the de-terministic strategy update rule. The relative cost ( α ) iskept fixed at . . We see no qualitative difference in theresults obtained from stochastic or deterministic strategyupdate rules as far as degree of synchronization’s depen-dence on the rewiring probability, the coupling constant,and the average degree is concerned. Another fact wenote is that, in line with our discussion above, the plotsfor both (cid:104) r G (cid:105) and (cid:104) C (cid:105) found using the stochastic strat-egy update rule tend towards the corresponding ones ob-tained using the deterministic strategy update rule as weincrease the rationality factor.We further observe that, having kept all other param-eters fixed , the degrees of cooperation and synchroniza-tion achieved using the unconditional imitation rule aregenerally higher than in the Fermi strategy update rule.It is consistent with what is known in the literature .However, the average qualitative behaviour of the dy-namics usually remains the same irrespective of which ofthe two update rules is adopted . Furthermore, the di-rect correspondence of the highly cooperative populationand its being spatiotemporally synchronized is best cap-tured in the deterministic strategy update rule because,as expected, less rational players do not play strategicallybut rather randomly choose a strategy leading to a mixedcooperator-defector state. Thus, henceforth, we exclu-sively work with the the unconditional imitation rule thathas the added benefit that being a non-stochastic scheme,it facilitates a sharper boundary between the completelysynchonized ( (cid:104) r G (cid:105) = 1 ) regions and partially synchro-nized ( (cid:104) r G (cid:105) < ) or unsynchronized ( (cid:104) r G (cid:105) = 0 ) regions.Similar relatively sharper boundary is presented by theunconditional imitation rule in the plots for cooperation( (cid:104) C (cid:105) ) as well. C. Effect of changing rewiring probability
In order to gain insight into the effect of the rewiringprobability, p , on the coevolution of the cooperation andthe synchronization, we present in Fig. 4 how they changewith the coupling strength for various representativerewiring probabilities. We observe that the CML reachesa relatively higher level of the global order parameterfor a very narrow range of the coupling strength around (cid:15) ≈ . where one finds that the system synchronizesonto a 4-period orbit. On further increasing the cou-pling strength, the system desynchronizes until respec-tive values of (cid:15) crit are reached and beyond which com-plete synchronization onto the homogeneous fixed pointtakes place. Thus, we see that the CML is completelysynchronized—regardless of the rewiring probability—beyond some (cid:15) = (cid:15) crit that increases with increasing p . . . . p (a) β = 1 (b) β = 100 (c) β = 1000 (d) Deterministic . . . p (e) (f) (g) (h)22650 h k i (i) (j) (k) (l)0 . . . (cid:15) h k i (m) 0 . . . (cid:15) (n) 0 . . . (cid:15) (o) 0 . . . (cid:15) (p) 0 . . . . . . h C i . . . . . . h r G i . . . . . . h C i . . . . . . h r G i FIG. 3.
Order parameter and cooperation for different strategy update rules.
We plot in (a)-(d) and (i)-(l) theaverage global order parameter (cid:104) r G (cid:105) and in (e)-(h) and (m)-(p) the average fraction of cooperators (cid:104) C (cid:105) in the CML of thelogistic maps. We use a WS network of 100 nodes and r G and C are averaged over 40 independent realizations. The relativecost α is kept fixed at 0.01 for all the subplots. The right most subplot, viz., (d), (h), (l) and (p), in each row is obtained withthe deterministic strategy exchange rule. The rest of three plots in every row is obtained using the Fermi strategy update rulehaving rationality factor ( β ) = 1 , , and respectively, from left to right. Subplots (a)-(h) are obtained with the averagedegree ( (cid:104) k (cid:105) ) fixed at while subplots (i)-(p) are obtained for a fixed rewiring probability, p = 0 . . This result is at odds with the case of the Kuramotomodel governed agent dynamics where the system at-tains complete synchronization only for relatively highrewiring probabilities and moreover the threshold cou-pling strength decreases with increasing rewiring proba-bility .Except for the case where the coupling between theagents is very small (i.e., (cid:15) → ), the CML (with anyvalue of p ) reaches full cooperation state whether or notit is synchronized. An understanding of the mechanismleading to this phenomenon reveals the nature of the lo-cal dynamics in the CML. First, consider the case whenthe coupling is very weak ( (cid:15) → ). In this case, theagents may be thought to be evolving chaotically andindependently, and there is no hope of synchronization among them. Consequently, any agent is not synchro-nized with its neighbours, let alone with the rest of thenon-neighbouring agents. This immediately implies thatthe benefit (see Eq. 3) is very small and on average samefor all the agents—whether cooperators or defectors. Thecost (see Eq. 2; α = 0 . ) is also small and but on av-erage same only for the cooperators; the defectors, byconstruction, incur zero cost. Hence, the payoff of thecooperators is smaller than that of the defectors lead-ing to the obvious scenario that under the unconditionalimitation rule, all the agents would adopt the defectionstrategy sooner or later.Next consider the case, (cid:15) > (cid:15) crit , where the parametersof the CML is so arranged that it is driven towards a com-pletely synchronized state which is a homogeneous fixed . . . . . h r G i (a) p = 0 . p = 0 . p = 0 . p = 0 . p = 0 . . . . . . . (cid:15) . . . . . h C i (b) p = 0 . p = 0 . p = 0 . p = 0 . p = 0 . FIG. 4.
Order parameter and cooperation for differ-ent rewiring probabilities.
We plot (a) average globalorder parameter ( (cid:104) r G (cid:105) ) and (b) average cooperation ( (cid:104) C (cid:105) ) as a function of coupling strength, (cid:15) . The average degreeand relative cost are fixed at 6 and 0.01 respectively. Theblue, the orange, the green, the red, and the violet lines cor-respond to the following values of the rewiring probability: p = 0 . , . , . , . , and . respectively. point. In the basin of attraction of such a globally syn-chronized state, consider a close-by neighbouring state.In that neighbouring state, the benefit obtained by anycooperating agent is high (order ) since, by construc-tion, the benefit is a measure of synchronization betweenthe agent and its neighbours. The the cost should below as is evident from its definition and the fact that all x i s are almost same. The payoff of a cooperator, hence,should be high compared to that of a defector who is dy-namically uncoupled from the neighbours. Naturally, un-der the unconditional imitation rule, all the agents wouldadopt the cooperator strategy as the CML’s state rushestowards the stable synchronization state. Analogous ar-gument goes for the region of synchronization onto stablehomogenous period-4 orbit.Lastly, consider the case: (cid:28) (cid:15) < (cid:15) crit (ignoring theregion of synchronization onto period-4 orbit). While theCML is not synchronized, the degree of cooperation in itis unity. This can be attributed to the following mecha-nism: Although the CML is not completely synchronized, the non-zero value of (cid:104) r G (cid:105) suggests that it is possible for afew agents whose local synchronization parameter (whichactually is the benefit)—which measures their dynam-ics’ closeness with their respective neighbours’—to bemuch higher than its global counterpart. So, within suchgroups, the unconditional imitation rule facilitates theagents to adopt the cooperation strategy eventually. Be-ing locally synchronized, such cooperators in the groupshave higher payoff and hence, any defector—who has ahigh probability of being connected with one of the soabundant cooperators—must also switch to the coopera-tion strategy because of the unconditional imitation rule. D. Effect of changing degree
Continuing from the last subsection, we note inFig. 3(p) that, for a given rewiring probability, the co-operation level decreases in the non-synchronized statesof the CML as the average degree increases. Evidently,in a mixed cooperator-defector state of the population,since a defector incurs no cost, a cooperator would switchover to the defector strategy as per the rules of strategyupdate in case one such defector happens to be in theneighbourhood of the cooperator. The chance of such adefector being in the neighbourhood should be more if thedegree of the node having cooperator is higher. Hence,the networks with higher average degree would decreasethe cooperation.The effect on synchronization is very interesting. Incontrary to the case of non-chaotic agent dynamics wherewith a small increase in the degree of the network,the population becomes completely unsynchronized ,the CML under consideration has a more nontrivial be-haviour. Numerics depict in Fig. 5 that as the rewiringprobability increases, (cid:15) crit also increases for relativelysmall average degree. As we increase the average degree,we find (cid:15) crit becomes independent of the rewiring proba-bility and converges towards (cid:15) = (cid:15) crit = 0 . as it should.Note that as the degree increase across the subplots inthe figure (see also Fig. 2), the intensity of the colourstowards diminishes validating an increase in the level ofthe synchronization with (cid:104) k (cid:105) for (cid:15) < . . However, for (cid:15) > . , on average, there is a decrease in the level of thesynchronization with the average degree.The rise in overall synchronization with an increase indegree , for the case (cid:15) < . , can be attributed to the in-crease in the level of synchronization among local clustersof cooperating agents. Of course, the coupling strengthin this region is not strong enough to bring about globalsynchronization. As far as the region (cid:15) > . is con-cerned, the coupling strength is strong enough to bringabout global synchronization (see Fig. 5). As mentionedin the beginning of this subsection, due to the nonco-operative behaviour encouraged by the internodal game,there is a decrease in the number of cooperators with anincrease in the degree of the network. The direct conse-quence of this loss of cooperators is the loss in complete . . . p (a) h k i = 2 (b) h k i = 4 (c) h k i = 6 (d) h k i = 8 . . . (cid:15) . . . p (e) h k i = 10 . . . (cid:15) (f) h k i = 12 . . . (cid:15) (g) h k i = 24 . . . (cid:15) (h) h k i = 36 . . . . . . h r G i FIG. 5.
Order parameter for varying rewiring probability and degree.
We plot the average order parameter ( (cid:104) r G (cid:105) ),for a CML of 100 nodes averaged over 40 independent realizations, as a function of the coupling strength ( (cid:15) ) and the rewiringprobability ( p ). The relative cost α is fixed at 0.01. Subplots (a)-(h) are obtained with average degrees ( (cid:104) k (cid:105) ) 2, 4, 6, 8, 10, 12,24, and 36, respectively. The vertical dashed black lines correspond to the analytically estimated value of the critical couplingstrength, (cid:15) crit , in the limit N, (cid:104) k (cid:105) → ∞ . synchronization as more agents choose not to interactwith their neighbours. Although the number of coopera-tors is reduced, it does not go down to zero. This impliesthat there are still local clusters of cooperators. Theselocal clusters of cooperators, coupled with a high valueof coupling strength, gives a high value of the global or-der parameter in (cid:15) > . region as compared to (cid:15) < . region for the same degree. Intriguingly, the internodalgame—though successful in breaking the complete syn-chronization of the system for (cid:15) > . —is unable to breakthe small clusters of the cooperating agents that existthroughout the region of < (cid:15) < . These clusters seemto be held together by the degree of the network and thedynamics of the agents placed at the node. The degree ofsynchronization never diminishes to zero entirely in ourmodel and, thus, the type of agent dynamics appears tobe crucial for the sustenance of the synchronized statehaving prevalence of the cooperators. E. Effect of introducing delay
Before we discuss and conclude the results obtained inthis paper, we turn on to another aspect of the model:What if there is a delay in updating in strategy by theagents? A delay in updating strategy has the direct ef-fect that the strategic game in the CML has relativelymore intermittent effect on the dynamics of the agents’individual and collective states. It means that in a mixedcooperator-defector state, a cooperator resist from adopt-ing the defector strategy for a longer time, and hence the coupling between the agent and its neighbours re-main turned on for longer time facilitating synchroniza-tion. Thus, all other parameters kept fixed, larger de-lay should mean synchronization—and emergence of co-operation with it—at the smaller values of the couplingstrength.We implement the delay in the unconditional imita-tion strategy update rule in our system by allowing for astrategy update at each time step with a probability of τ ( τ ∈ [1 , ∞ ) ), where τ = 1 corresponds to the usual caseof the strategy update at every time step; the higher thevalue of τ , the more delayed the strategy update is. Fig. 6illustrates that when we incorporate a delay in our strat-egy update, there is a simultaneous rise in the degreesof synchronization and cooperation at a relatively lowercoupling strength. Moreover, the coordinated falls in (cid:104) r G (cid:105) and (cid:104) C (cid:105) , that occur at a higher value of the couplingstrength, are completely halted is the delay is sufficientlyhigh. This result is at odds with the case of non-chaoticnodal dynamics , where the system’s degrees of cooper-ation and synchronization begin to rise from a very lowinitial values at the same coupling strength regardless ofthe delay time; but the configuration with a higher delaytime, loses synchronization at a relatively higher value ofthe coupling strength.Despite the differences in the details between the twosystems—one with chaotic agent dynamics and the otherwith non-chaotic dynamics—we conclude that, in gen-eral, delay aids in the co-emergence of synchronizationand cooperation for both chaotic and non-chaotic agentdynamics.In passing we remark that as long as α is very low,introducing delay is not expected to have much effect onthe degrees of synchronization and cooperation as com-pared to the case when there is no delay in the strategyupdate. In Fig. 4, we can observe that even at a very lowvalues of (cid:15) , at low cost value ( α = 0 . ), the system hascomplete cooperation. Thus, introducing delay cannotincrease the fraction of cooperators any further. How-ever, high α (e.g., α = 1 in Fig. 6) implies a high costof cooperation and we may expect the system to desyn-chronize in the no-delay case ( τ = 1 ) where the systemis otherwise synchronized for low α . It is important tonote that regardless of the high value of α , the cost canbe brought close to zero if the agents do not change theirstates much with each time step. This happens when theagents are (almost) synchronized, i.e., when the couplingstrength is high enough. Therefore, if the cooperatorsare made to resist from adopting the defector strategyby introducing delay, then we can see the system to syn-chronize again at high values of coupling strength despitethe high value of α as seen in Fig. 6. We also note in Fig. 6(cf. Fig 4) that the region of high degree of synchroniza-tion onto 4-period orbit around (cid:15) ≈ . has disappearedfor all values of delay. This is because α is high andthe low value of (cid:15) in this region cannot bring the agents’states close together quickly enough before the strategyupdate occurs. . . . . . . (cid:15) . . . . . . h r G i , h C i τ = 1 τ = 2 τ = 10 FIG. 6.
The effect of delay in strategy update on syn-chronization and cooperation.
We plot the order parame-ter, (cid:104) r G (cid:105) (solid lines) and the associated average cooperation, (cid:104) C (cid:105) (dashed lines) as a function of the coupling strength fordifferent delay times— τ = 1 (red) , (green) , and (black).The fixed relative cost, the average degree, and the rewiringprobability are taken to be 1, 6, and 0.5, respectively. IV. DISCUSSIONS AND CONCLUSIONS
In this paper, we have investigated the effect of chaoticagent dynamics on the coevolution of cooperation andsynchronization in the networks formed using the WS al-gorithm. Specifically, we have placed logistic maps at each node of the network and have formed a CML thatcan show period doubling route to chaos with the de-crease in the coupling strength. Subsequently, we havestudied the effects of the strategy update rule, the ratio-nality of the agents, the coupling strength, the rewiringprobability of the network, and the average degree ofthe network on the coevolution of cooperation and syn-chronization. We have also analytically estimated—andnumerically validated—a lower bound of the critical cou-pling constant beyond which global synchronization on toa common fixed state for all the agents may be affected.An aspect that is unique to our model is the possibil-ity of more exotic synchronized states like chaotic stateand periodic state. An example for the latter—a -periodorbit—has been highlighted in the paper. We have shownthat the CML is completely synchronized, irrespectiveof what the rewiring probability is, beyond critical cou-pling strength that increases with the value of rewiringprobability. When we increase the degree, we observethat the population of the agents have high values ofthe global coupling parameter even for very high averagedegree of the network. Furthermore, any time delay inthe implementation of the strategy update rule inducesenhancement of the coevolution of cooperation and syn-chronization at comparatively lower coupling strengths.These results are quite different from what is known inthe literature of the coevolution of cooperation andsynchronization with non-chaotic agent dynamics stud-ied by placing the Kuramoto oscillators at each node ofthe network. Thus, we have successfully highlighted inthis paper that the interesting phenomenon of the co-evolution of cooperation and synchronization is cruciallydependent on whether the uncoupled agent dynamics ischaotic or not.This paper initiates a set of potentially insightful inves-tigations into the further implications of different chaoticdynamics of the agents on the emergence of cooperationin a population. One such avenue is when the uncou-pled dynamics at each node of the resulting CML canbe modelled using the replicator map that may evenbe chaotic depending of the payoff matrix elements in it.Each node can either be treated as an individual or agroup of individuals (deme); in the latter case, the dy-namics of the CML would model the intergroup dynamicsin a population. Thus, it presents an exciting opportu-nity of studying group-selection models and the intra-demic cooperation within the paradigm of the coevolu-tion of (interdemic) cooperation and synchronization. ACKNOWLEDGMENTS
The authors are thankful to Archan Mukhopadhyayand Samriddhi Shankar Ray for helpful discussions.0
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