Distance dependent competitive interactions in a frustrated network of mobile agents
11 Distance dependent competitive interactions in afrustrated network of mobile agents
Sayantan Nag Chowdhury, Soumen Majhi, and Dibakar Ghosh
Abstract —Diverse collective dynamics emerge in dynamical systems interacting on top of complex network architectures. Along thisline of research, temporal network has come out to be one of the most promising network platforms to investigate. Especially, suchnetwork with spatially moving agents has been established to be capable of modelling a number of practical instances. In this paper,we examine the dynamical outcomes of moving agents interacting based upon their physical proximity. For this, we particularlyemphasize on the impact of competing interactions among the agents depending on their physical distance. We specifically assumeattractive coupling between agents which are staying apart from each other, whereas we adopt repulsive interaction for agents that aresufficiently close in space. With this set-up, we consider two types of coupling configurations, symmetry-breaking andsymmetry-preserving couplings. We encounter variants of collective dynamics ranging from synchronization, inhomogeneous smalloscillation to cluster state and extreme events while changing the attractive and repulsive coupling strengths. We have been able tomap all these dynamical behaviors in the coupling parameter space. Complete synchronization being the most desired state inabsence of repulsive coupling, we present an analytical study for this scenario that agrees well with the numerical results.
Index Terms —Time varying networks, Mobile agents, Extreme events, Synchronization, Inhomogeneous small oscillation. (cid:70)
NTRODUCTION C OMPLEX network [1], [2] is the unifying paradigm behindmany scientific disciplines ranging from physics, mathemat-ics to computer science and biology. Besides the study related tolocal and global statistical properties of complex networks [3], re-searchers are equally interested in paying attention to the dynamicsof their interacting units [4]. Among those collective dynamics, awidely studied phenomenon is that of synchronization [5], [6], [7],[8], [9], [10], [11], [12] of coupled oscillators arranged over theconstituents of complex networks. Although, in recent times, therehave been attempts in examining synchronizability of time-varyingnetworks with different underlying structural arrangements, inmost of these studies of synchronization, the nodes of the complexnetworks are spatially static. But, in real world networks, theindividuals (nodes) move around and exchange information withinclose proximity. Such nodes in the contextual literature are wellknown as mobile agents [13], [14], [15], [16], [17], [18], [19], [20],[21] referring to those nodes whose movement highly influencethe systems’ collective dynamics. But, in general, the states ofthose oscillators situated on top of those agents do not affect theagents’ mobility. The studies on the process of synchronization inmobile agents [22], [23], [24], [25], [26], [27], [28] have, so far,primarily concentrated on those particular cases when the mobileagents interact with each other only when they are closed enoughin space. But, in all of these studies, the network architecture,i.e., how the oscillators are connected with each other, dependsonly on the attractive interaction. So, a basic query arises that howis the flow of information gets affected when different types ofinteractions are present in the movement. For instance, an attemptto answer this question can be made while incorporating repulsivecoupling with the attractive one in such a networked system. Such • S. Nag Chowdhury, S. Majhi and D. Ghosh are with the Physics andApplied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road,Kolkata 700108, India.E-mail: [email protected] (D. Ghosh) co-existing dynamical interactions can be observed in a variety ofsystems and have been studied previously in different network set-ups yielding diverse collective states [29], [30], [31], [32], [33].Synergy of attractive-repulsive coupling can lead to severalinteresting dynamical behaviors. One of such example is theemergence of solitary states in multiplex networks [34] possessingpositive inter-layer and negative intra-layer interactions. Earlier,Maistrenko et al. [35] found a variety of stationary states innetworks of globally coupled identical oscillators with attractiveand repulsive interactions. Sathiyadevi et al. [36], [37] founddiverse chimera states due to the introduction of nonlocal repul-sive coupling together with an attractive coupling in a networkof coupled oscillators. Also, interplay between conformists andcontrarians leads to traveling wave along with the stationary states[38]. The effect of competitive interactions on the synchronizationmanifold among globally coupled identical Van der Pol oscillatorsthrough their velocities is rigorously addressed in ref. [39]. Even,the presence of a tiny fraction of repulsive couplings is found toenhance the synchronization of nonidentical dynamical units thatare attractively coupled in a small-world network [40]. Originationof dragon-king-like extreme events as a result of the coexistence ofexcitatory and inhibitory chemical synaptic couplings is reportedin the ref. [41]. Besides emergent dynamical patterns, co-presenceof attractive and repulsive couplings carry high significance toseveral biological [42], physical [43], ecological [44] and evensocial scenarios [45].Recently, Nag Chowdhury et al. [46] proposed an universal − π rule to identify the bipartiteness of networks from theanti-phase synchronization states. This article describes such asystem of attractively coupled oscillators with negative links as frustrated networks [47]. Such frustrated systems under the in-fluence of attractive-repulsive co-existing interactions may induceunanticipated phenomenon, like extreme events [48]. Most of theexisting articles [41], [49], [50], [51], [52] on extreme events indynamical systems and networks define such an event as rare a r X i v : . [ n li n . C D ] A ug event. Mathematically, a rare event is defined on a probabilityspace χ equipped with a σ -algebra B of events such that P ( A ) issmall, where P is the probability measure designed to quantify theprobability of the occurrence of any event A ∈ B . But, extremeevents can be rare events [53], [54], [55], or they can often befrequent in time and space [56], [57], [58], [59]. Examples ofsuch intermittent frequent extreme events include population sizeof Paris [60], the bubbling of share market [61] before a crash andmany more.Along with its small occurrences, extreme event has anotherthreat in the form of large impact. Financial and commoditymarket crashes, tsunamis, hurricanes, floods, epidemic diseasespread, global warming-related changes in climate and weather,warfare and related forms of violent conflict, asteroid impacts,solar flares, acts of terrorism, industrial accidents, 8.0+ Richtermagnitude earthquake are few examples of such large impactevents. In dynamical systems, researchers are recently startingto measure the impact of such events in terms of the amount ofvariation from the central tendency of that observable [41], [49],[51], [62], [63], [64]. So, if ψ is the observable defined on thestate space U to R , then the extreme events are members of theset { u ∈ U : ψ ( u ) > H S } , where H S = m + dσ with d is thenon-zero integer. Here, m and σ are respectively the mean valueand the standard deviation of all the peak values in a time seriesof u . But, if H S is too large, there will be very few values tomodel the tail of the distribution correctly as the variance is likelyto be large due to only very extreme observations remaining. Onthe other hand, a low threshold value of H S will include too manyvalues giving a high bias. Recently, m +8 σ is found to appropriateindicating extreme event threshold for Weibull distribution [65].Inspired by these observations, we consider a frustrated net-work of limit cycle oscillators, where each node of the networkis a mobile agent. Earlier all studies [22], [23], [25], [27], [28],[65], [66] related to mobile agents, considered only limited lo-cal interactions. Instead, we consider here a global network ofmobile agents with co-existing switching interactions. Dependingsolely on the relative distance between the agents, attractive orrepulsive interaction is activated. Specifically, we choose attractiveinteraction between those agents which are staying apart fromeach other, while we consider repulsive coupling for the agentsthat are sufficiently close. We consider two types of couplingschemes, namely symmetry-breaking and symmetry-preservingcouplings. Under this set up, we explore the effect of couplingstrengths which reveal quite interesting phenomena like clustersynchronization [67], complete synchronization , inhomogeneoussmall oscillation , extreme events etc. We also analytically derivethe critical coupling strength for achieving complete synchroniza-tion using time-average Laplacian matrix in absence of repulsiveinteraction and numerically verify the results. ATHEMATICAL FRAMEWORK
We consider N mobile agents which are moving independentlyon a two-dimensional ( D) physical plane S = [ − g, g ] × [ − g, g ] . Initially, the agents are randomly distributed on S .The i -th mobile agent can move in any direction with velocity (cid:0) v cos θ i ( t ) , v sin θ i ( t )) (any kind of collision among the agentsare not allowed), where v is the uniform moving velocity and θ i is drawn randomly from [0 , π ] . Higher values of velocity v implies that the agents move the whole physical space and whichincreases the possibilities of interactions. Thus, if (cid:0) p i ( t ) , q i ( t ) (cid:1) Fig. 1. Schematic diagram at a particular time: We consider N = 3 mobile agents in the two-dimensional XY-plane, where g = 10 and α = 10 . The red and black lines represent repulsive and attractive links,respectively. is the position of the i -th agent at time t , then motion updatingprocess will be maintained by the following relation, p i ( t + 1) = p i ( t ) + v cos (cid:0) θ i ( t ) (cid:1) ,q i ( t + 1) = q i ( t ) + v sin (cid:0) θ i ( t ) (cid:1) . (1)If at some time t , p i ( t ) , q i ( t ) exceed | g | , then we re-generate anew θ i ( t ) in [0 , π ] such that − g ≤ p i ( t ) , q i ( t ) ≤ g . We definethe distance between any two agents by the standard Euclideanmetric as d ij ( t ) = (cid:114)(cid:16) p i ( t ) − p j ( t ) (cid:17) + (cid:16) q i ( t ) − q j ( t ) (cid:17) . (2)Next, we write the the dynamical equations characterizing thetime-evolution of each agent ( i = 1 , , · · · , N ) in the networkedsystem as follows, ˙ x i = F ( x i ) + N (cid:88) j =1 { k A B ij + k R C ij } H ( x i , x j ) , (3)where x i ∈ R n is the state variable and F ( x i ) is the correspond-ing vector field of the i -th agent. Here H ( x i , x j ) is chosen as thediffusive type coupling function, k A > and k R < respectivelycorrespond to the attractive and repulsive coupling strengths.Moreover, B and C are the distance dependent adjacency matricesassociated to the attractive and repulsive interactions such that B ij = (cid:40) , if d ij ( t ) > α , if d ij ( t ) ≤ α, and C ij = (cid:40) , if d ij ( t ) > α , if d ij ( t ) ≤ α, where α is the parameter lying within [0 , √ g ] . Here √ g is themaximum possible distance between any two agents lying in the -dimensional plane [ − g, g ] × [ − g, g ] . At a particular instance, theagents whose Euclidean distance is less than α , repel to each other Fig. 2. Variation of degree distribution of a randomly chosen i -th agent out of N = 100 mobile agents: (a) attractive neighbor and (b) repulsiveneighbor. Each i -th agent is connected with remaining ( N − agents whenever α ∈ [0 , √ g ] . Note that the numbers of attractive and repulsiveneighbors both are widely spread. The simulation is carried out for t = 3 × . (c, d) Variation of degree of both matrices with respect to time t :Just like the upper panel, we find that the neighbors of any agent are rapidly changing depending on the agent’s random movement. Here, we take g = 10 and α = 10 , and the uniform moving velocity v = 2 . . and repulsive interactions arise between them. Attractive couplingoccurs between the agents when the distance between them isgreater than α .To illustrate this, we draw a schematic diagram with N = 3 mobile agents in the -dimensional plane [ − g, g ] × [ − g, g ] inFig. (1). At a particular time t , we plot the position of these threeagents in Fig. (1) where g = 10 and α = 10 are considered. Forthese choices of α and g , the adjacency matrices correspondingto attractive and repulsive interactions at that specific time instantlook like B = and C = Now, we consider fix physical plane with g = 10 and fix theparameter α = 10 . Each agents moving within the physical plane [ − g, g ] × [ − g, g ] using the rule given in (1) with fixed velocity v =2 . . Now we see the variation of degree distributions of the time-varying network. For this, we calculate the number of attractiveand repulsive neighbors depending on the critical distance α . InFig. 2, the degree distributions of moving agents for attractiveand repulsive interactions are shown. We randomly select an agentwhich is moving with uniform velocity v = 2 . . We then find thatthe number of attractive neighbors are varying within the interval [5 , (cf. Fig. 2(a)) and the number of repulsive neighbors ofthat agent is varying within [7 , (cf. Fig. 2(a)). At any particulartime-step, the sum of the number of attractive neighbors and thenumber of repulsive neighbors is N − for any i -th agent. Thisscenario is well expressed by the lower panel of Figs. 2(c, d).Now, on top of each mobile agent, a two-dimensional Stuart-Landau (SL) oscillator is placed where the state dynamics of each limit cycle oscillator is represented by F ( x i ) = (cid:2) − (cid:0) x i + y i (cid:1)(cid:3) x i − ω i y i (cid:2) − (cid:0) x i + y i (cid:1)(cid:3) y i + ω i x i , (4)where ω i = ω = 3 . is the identical intrinsic frequency for each i = 1 , , · · · , N .We consider two types of diffusive interactions, namelysymmetry-breaking coupling H ( x i , x j ) = ( x j − x i , T andsymmetry-preserving coupling H ( x i , x j ) = ( x j − x i , y j − y i ) T .The interesting and important part of this distance dependent cou-pling functions is for α = 0 , the Eq. (3) becomes globally couplednetwork with attractive coupling only. In this case, movement ofthe agents does not effect the attractive adjacency matrix B andthe network becomes static. But, for α > , depending upon thevalues of attractive and repulsive coupling strengths k A and k R ,two different cases can be implemented:1) k A (cid:54) = 0 and k R = 0 : In this case, the mobile agentswill interact attractively when the relative distance d ij isgreater than α and remain disconnected with each otherwhen they are closed to each other and relative distance d ij is less than equal to α . There is no repulsion amongthe agents. Here, synchronization is the most desired statefor a suitable value of k A . k A (cid:54) = 0 and k R (cid:54) = 0 : In this case, interplay betweenattractive and repulsive interactions is considered. Duringthe spatial movement of the agents, they experience bothtypes of interaction among them. Here, different states may arise, like inhomogeneous small oscillation, syn-chronization, extreme event and other intermittent states.In the following section, we will explore the dynamics of thetime-varying network (3) with two types of diffusive couplingsand under above mentioned distinct coupling conditions. Our mainemphasis will be to identify the parameter space by varying thetwo coupling strengths k A and k R with fixed values of othernetwork parameters g = α = 10 . and v = 2 . related to themovement of the agents. ESULTS
For numerical simulations, we integrate Eq. (3) using fifth orderRunge-Kutta-Fehlberg method with a integration time step δt =0 . . At each integrating time step, θ i ( t ) and thus v i ( t ) is updatedaccording to rule given in (1) for each i -th mobile agent ( i =1 , , · · · , N ). Hence, both the matrices B and C change at eachintegrating time step leading to fast switching approximation [68].All simulations are done for fixed initial conditions x i (0) = ( − i iN ,y i (0) = ( − i iN , (5)unless stated otherwise . In this Sec. (3), the symmetry-breakingcoupling H ( x i , x j ) = ( x j − x i , T is considered. For the chosen values of the parameters as mentioned above, wefind different dynamical behaviors of the network system (3). Wefirst look at the collective dynamics of the network whenever thereis no repulsion among the agents and only attractive couplingis activated, i.e., when k A (cid:54) = 0 and k R = 0 . In this scenario,the agents with relative distance d ij ( t ) ≤ α = 10 remaindisconnected with each other and agents will interact with eachother if their distance is greater than α .We next intend to analytically derive the critical interac-tion strength k critical for which synchrony appears, based onthe approach of constructing the time-average Laplacian matrix G = [ g ij ] . For the sake of simplicity, let us start with N = 2 mobile agents. Then, depending on their relative distance, twopossible Laplacian matrices can be observed.1) Whenever d ij > α , then the agents interact with one an-other so that the corresponding Laplacian matrix becomes G A = (cid:18) − − (cid:19) .2) If d ij ≤ α , then the Laplacian matrix due to absence ofindividuals’ interaction takes the form G = (cid:18) (cid:19) .Since G is a null matrix so in this case G reduces to G = pG A , where p is the probability of interaction between the twoagents. Since, we are considering a planar space S of area g sq.units, then p = 1 − Area of the interactionArea of S becomes p = 1 − πα g .
1. Inhomogeneous small oscillation depends crucially on the initial condi-tions, and the basin of attraction of this state is small. If the initial conditionsare not chosen suitably from the basin of attraction, then the trajectoriesmay converge to a different periodic attractor. However, other observeddynamical states can be reached from any random initial conditions startingfrom [ − , × [ − , . We next calculate the average Laplacian matrix G for N = 3 agents. For instance, depending on the spatial positions of thesemobile agents for a fixed value of α , eight possible configurationshave been encountered for N = 3 . These possible cases aredescribed below:1) All the agents interact with each other and thus G A = − − − − − − .2) There are three cases in which two out of the threeagents interact with each other and the third one remainsisolated. The corresponding Laplacian matrices are G = − − , G = −
10 0 0 − and G = − − , where G ij is theLaplacian matrix when i -th and j -th agents areinteracting with each other, and the other third agentremains isolated. Also note that, G + G + G = G A .3) It is also possible that two of the agents lie withinthe relative distance α so that they are not interactingwith each other, but the third agent stays far away andinteracts with both of them. The associated Laplaciansare as follows G = − − − − , G = − − − − and G = −
10 1 − − − ,where G k is the Laplacian matrix when the k -th agentinteracts with the other two agents, but those twoagents do not interact with each other. Here note that G + G + G = 2 G A .4) Majority of the agents, i.e., all three oscillators situatedwithin the relative distance α , hence no one can establisha connection with the others. So, Laplacian matrix canbe indicated by G = .Then the time-average matrix for N = 3 becomes, G = p G + p G + p G + p G + p G + p G + p G + p A G A . Here p ’s stand for the probabilities associatedto the respective network configurations for which we have p = p = p = p a (say), and p = p = p = p ab (say).Then G = (2 p a + p ab + p A ) G A and hence G = pG A , where p = 2 p a + p ab + p A is the probability of interaction betweenany two agents. The scenarios corresponding to N ≥ can besimilarly tackled, all of which yields the time-average matrix as G = pG A , where G A is the Laplacian matrix of order N × N inthe form, G A = N − − − · · · · · · · · · − − N − − · · · · · · · · · − − − N − · · · · · · · · · − · · · · · · · · · · · · · · · . · · · · · ·· · · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · ·− − − · · · · · · · · · N − ,so that G is basically a rescaled all-to-all global Laplacian matrix.As per our described model, the links among the agents getrewired at each integrating time step, so the convergence of allthe attractors to a single attractor [69] can be assured if the time-average matrix supports synchronization [17]. As stated in theRef. [17], if there exists a constant T such that G ( t ) satisfies T (cid:82) t + Tt G ( τ ) dτ = G , the time-average of G ( t ) and the systemof coupled oscillators given by ˙ x i = F ( x i ) − k (cid:80) Nj =1 g ij H ( x j ) , i = 1 , , ..., N (6)with fixed interacting Laplacian matrix G = [ g ij ] possesses astable synchronization manifold, then there exists (cid:15) ∗ > suchthat for all fixed (cid:15) satisfying < (cid:15) < (cid:15) ∗ , the coupled systemaccording to the time-varying Laplacian G ( t(cid:15) ) defined by ˙ x i = F ( x i ) − k (cid:80) Nj =1 g ij ( t(cid:15) ) H ( x j ) , i = 1 , , ..., N (7)also sustains a stable synchrony manifold under sufficiently fastswitching sequence between the network configurations.Then, the stability of the synchronized state can be investigatedby the eigen values of G . For a time-independent coupling matrix,a necessary condition for synchronization is that the masterstability function (MSF) be strictly negative in each transversedirection [70]. With our choice of parameter ω and the couplingfunction H ( x i , x j ) = ( x j − x i , T , the SL-system belongs toclass-I MSF [70], i.e., the synchronization manifold is stable in theinterval of type [ β , β ]. The N eigen values of G are λ = 0 and λ j = pN , j = 2 , , ..., N . Then the critical range of k A to attaincomplete synchrony is obtained by the inequality β ≤ k A λ j , j = 2 , , ..., N which implies, k A ≥ β pN = k critical . (8) Fig. 3. Synchrony region: The red curve is the derived critical curve plot- ted using the relation (8). The blue dots are the stable synchronization point ( N, k A ) at which E < − . The results are accumulated for independent numerical realizations. The green shaded region is thedesynchronization region, with E ≥ − . To verify the relation (8), we plot N − k A parameter spacein Fig. 3 where the red solid curve is our analytically found curve k A = β pN for β (cid:39) . . This curve agrees well with ournumerically found synchronization region (blue shaded region). In order to numerically study the emergence of synchrony, wedefine the synchronization error as E = (cid:28) (cid:80) Ni,j =1( i (cid:54) = j ) √ ( x j − x i ) +( y j − y i ) N − (cid:29) t , (9)where (cid:104)·(cid:105) t represents time average obtained over a long timeinterval (taken as × time steps here) after initial transientof × time steps. In our work, complete synchronizationcorresponds to the state when E becomes less than − accounting independent network realizations. Note that, wefind a small fluctuations in the relation (8) for finding k critical .These fluctuations may occur due to various reasons, includinginsufficient realizations, approximate choice of the value of β ,or for the effect of the uniform moving velocity v . Even, thissmall fluctuation may be due to our choice of p . Although, forstatic agent assumption or v = 0 , our chosen p = 1 − πα g works absolutely correctly. But when the agents are moving, theagents do not necessarily maintain this exact relation (8). In fact,in deriving the relation (8), we use the negativity of the maximumLyapunov exponent in the transverse direction of the synchroniza-tion manifold, which is a necessary condition, not a sufficient one.All these lead to small fluctuations in k critical compared to ourderived critical synchronization curve k critical = β pN . Notably,this density dependent threshold for the emergence of synchronyis found to be relevant and is of practical interest, particularlyin the studies of the bacterial infection, biofilm formation andbioluminescence where quorum-sensing transition is observed inindirectly coupled systems [71], [72].This result is quite important due to the fact that if werandomly distribute N agents in the plane S and then set v = 0 (i.e., when the network becomes static), complete synchronizationis not guaranteed depending on their initial placements. As theconnected agents may exhibit complete synchronization depend-ing on suitable coupling strength k A , but the disconnected agentsremain isolated and as a result of that they maintain isolatedtrajectories. But, the mobility of each agent allows them to interacteven if they are long-distant, under sufficiently long simulations.These occasional interactions induce complete synchronizationamong them for appropriate critical coupling strength.Such occasional minimal interaction is found to be benefi-cial for the emergence of complete synchronization, instead ofcontinuous-time coupling, from the point of view of optimalinteractional cost [73], [74]. This confined interaction may beuseful in case of robotic networks and wireless communicationsystems, where transmission signals are turned on if the agentsare lying sufficiently close to each other and where we will haveto inspect if synchronization can still occur. To analyze the effect of only attractive coupling, one requiresto turn-off the repulsive coupling k R . But this is not the onlypossible way of setting on repulsion free interaction. Instead, onemay realize the scenario by setting α = 0 . This specific choice of α transforms the repulsive matrix C to null matrix, and attractivematrix in the form Fig. 4. Cluster Synchronization: The oscillators maintain a small oscillation around (a) . for the set U and (b) − . for the set V . All theoscillators of each set U and V converge to a single attractor. (c) The time-series of x i of all N = 100 oscillators reveal that trajectories convergeto two distinct trajectories. (c) These final converged trajectories look like oscillation death states, but the upper panel (a, b) already reveal theexistence of small oscillation. (d) Two clusters are clearly observed. Here, the attractive coupling strength k A = 1 . . B = · · · · · · · · ·
11 0 1 · · · · · · · · ·
11 1 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Note that the matrix B now becomes a static, i. e., a time-independent matrix. Under this set-up, one may expect completesynchronization, but for higher values of k A , we can not obtainconvergence of those attractors into one attractor. Instead ofthat, we find incomplete synchronization in the form of clustersynchronization for k A ≥ . . We define U = { x i | i is odd } and V = { x i | i is even } . After the initial transient, the x i oftrajectories of the sets U and V converge respectively to twodifferent trajectories as shown in Figs. 4(a, b) for k A = 1 . .Both the sets U and V contain oscillators. The global networkevolves into two equipotent subsets of oscillators in which mem-bers of the same cluster are synchronized to the same trajectory,but members of different clusters, oscillating with different smallamplitudes. Notice that the oscillators that start with negative x i asper relation (5) end up with positive x i and vice-versa. Figure 4(c)portrays the time series of x i for i = 1 , , · · · , . Although,Fig. 4(c) reflects oscillation death like scenario, but Figs. 4(a,b) reveal a small oscillation around . for the set U and − . for the set V . Figure 4(d) depicts snapshot of theposition of the oscillators at a particular time t . All N = 100 oscillators are symmetrically distributed around the origin, theunstable stationary point of the system (3).Such partial synchronization may show up in swarms of un-manned autonomous vehicles, power grids and swarms of animals[75]. This interesting phenomenon is also hold for random initial conditions choosing from [ − , × [ − , . But, in that case,the cardinality of U and V vary significantly in each realization.We also find that, if we increase k A by keeping fixed the initialconditions as per the relation (5), still we do not observe com-plete synchronization for k A ≥ . . They still show clustersynchronization for suitable k A . However, for smaller suitable k A , the trajectories exhibit complete synchronization convergingto a single periodic attractor. For N = 100 oscillators, completesynchronization can be obtained for k A < . . Now, we consider α = 10 along with k A = 1 . and k R = − . ,so that both the couplings can play a decisive role. Under thischoice of parameters, we observe the time-series of each oscillatorexhibits chaotic behavior. Again, those x i of N = 100 oscillatorsdistribute equally on both sides of y i = 0 . Similarly, y i of thoseoscillators are distributed equally on both sides of x i = 0 . One ofsuch scenario is shown in Fig. 5(a). It is observed that each of thesetwo groups is not synchronized. They maintain their chaotic smalloscillations. The simulations are done using initial conditions ofrelation (5).Recently this type of oscillations is reported by Dixit et al.[76] for two coupled SL-oscillators under the influence of dynamicinteraction. But, our study is quite different from them. Our net-work consists of mobile agents, and their relative distance d ij ( t ) actually decides the mutual interaction type, which is completelydifferent from the study [76]. In fact, we found this state for largenetworks, which help to deny the finite-size effect. Moreover, each Fig. 5. (a) Inhomogeneous small oscillation: variation of x i ( t ) of all the oscillators for N = 100 . (b) Complete synchronization: variation of x i ( t ) of N = 51 oscillators. Other parameters are k A = 1 . and k R = − . . The initial conditions are chosen as per the relation (5). (c) Attractor switching:Without any loss of generality, the -th oscillator is chosen randomly among N = 100 oscillators. The time-series depicts the occasional switchingof the attractor. (d) The chaotic attractor: A single attractor is shown here to illustrate the attractor switching. Other parameters are k A = 1 . and k R = − . . split group in our case consists more than -oscillators and thosetrajectories do not lead to any cluster synchronization.To study the effect of network size N , a detailed numericalstudy is perceived. We find that there exists a N critical , beyondwhich network shows such inhomogeneous small oscillation. Nu-merically, we find this N critical is . For N ≥ , such chaoticsmall oscillation is observed. For N < , the attractors areconverged to a single attractor and the trajectories are exhibitingperiodic behavior. In Fig. 5(b), we plot the x i ’s of all N = 51 oscillators, which are oscillating synchronously. Fig. 6. On-off intermittency: The red dashed horizontal line is theextreme event indicating threshold H S = m + 8 σ , where m is thesample mean and σ is the standard deviation of the whole data. Theerror dynamics, E shows irregular and rare switching from the zerovalue to non-zero values. The parameters are N = 100 , k A = 1 . , α = g = 10 . , v = 2 . and k R = − . . To scrutinize the effect of k R , we fix the attractive coupling k A = 1 . , and slowly increase the absolute magnitude of k R .We observe that with increment of k R , the amplitude of theoscillators increase. After a certain value of k R , the attractorsof one group, which are initially separated by the axes (SeeFig. 5(a)), are crossing the axes frequently. The trajectory of i -th oscillator crosses the axes and joins the other group forsometimes, before it comes back to its original group. To illustratethis, we choose randomly an agent from N = 100 oscillators,and plot its time series in Fig. 5(c). The corresponding attractoris also shown in Fig. 5(d) to demonstrate the chaotic switchingbehavior of the attractor. Interestingly, further slight incrementof k R leads to complete synchronization. At k R = − . , all N = 100 oscillators oscillate synchronously and re-gain theirisolated periodic dynamics. They oscillate in the limit cycle regimefrom [ − , . Further decrement of k R (for fixed k A = 1 . ) destabilizes thesynchronization manifold and after a certain value of k R , theerror dynamics becomes intermittent. Specifically, for most of thetime the synchronization error E (cf. Eq. (9)) remains near zero(lies within the interval [10 − , − ] ), but occasionally this E becomes non-zero. For a narrow range of k R with fixed k A = 1 . ,we find this intermittent error trajectories exhibiting extremelylarge values. To reveal this issue, we plot variation of synchroniza-tion error E in Fig. 6, which is accumulated over a sufficientlylong time interval. For this entire data, we calculate the samplemean m = 0 . and the standard deviation σ = 0 . . Wethen define a extreme event threshold H S = m + 8 σ (cid:39) . ,which is plotted over the error dynamics with horizontal dashedline in Fig. 6. In this particular accumulated data of E , we find data points having value of E , which is greater than H S .Thus, the probability of { E > H S } is . . However,this probability varies with each realization, and for our choice of H S = m + 8 σ , Cantelli’s inequality [77], [78] yields the upperbound for each realization as P ( E ≥ H S ) ≤ . (10)In the existing literatures [79], [80], this type of intermittencyis known as on-off intermittency . This type of intermittent error,the trajectories experiencing non-uniform, uncertain extensiveexpeditions from the synchronous manifold and can be consideredas an appropriate candidate for extreme events. Earlier, localinstability of synchronization manifold due to on-off intermittencycausing extreme events in a pair of coupled chaotic electroniccircuits is reported [81]. Also, Nag Chowdhury et al. [65] founda similar mechanism for the generation of extreme events in amobile network of chaotic oscillators. Fig. 7. Non-Gaussian distribution of E n : Local maxima E n of E isaccumulated and histogram of them is plotted in semilog scale. Thishistogram resemblances with L-shaped PDF exhibiting non-Gaussiandistribution. The red dashed vertical line is the significant height H S = m + 8 σ . The parameters are N = 100 , k A = 1 . , α = g = 10 , v = 2 . and k R = − . . To further resolve whether the spikes in Fig. 6 qualify forextreme events or not, we determine the local maximum values E n of E and plot the histogram for the event sizes E n in Fig.7. Note that the length of the collected time series is sufficientenough so that due to statistical regularity, inclusion of new sampledoes not affect the structure of the histogram. Pisarchik et al. [82]pointed out a special characteristic of extreme events, which is theL-shaped probability density function (PDF) of E n in the semi-log scale. In spirit of all these facts, we define extreme events inthis article as follows,1) These short lasting events are recurrent, aperiodic andof different amplitudes higher than the significant height H S = m + 8 σ . In order to further eliminate thesmall amplitude events, another restriction is maintained, H S > . is taken along with H S = m +8 σ [65], [82].2) They appear much more often than Gaussian statisticsand they are unpredictable almost surely with respect totime, where an event is defined as almost surely if the setof possible exceptions may be non-empty but has zeroLebesgue measure [83].3) The appearance of these large events would have verysmall probability (though the exact measure of smallneeds to be quantified) [48]. Clearly, the histogram in Fig. (7) is an approximate represen-tation of the distribution f ( x ) of the numerical data. In the next,we define a return interval k . Mathematically, a return interval k occurs if X i > H S and X i + k > H S , but X j ≤ H S for i < j < i + k , where X i is the value of the observable X at the i -th step. Suppose, W H S is the number of total return intervals. So,we have k i for i = 1 , , · · · , W H S such that not necessarily all the k i are distinct. Clearly, if k i is short then a cluster of accumulatedextreme events can be observed. In contrast, a large value of returninterval definitely portrays few occurrences of extreme events.Also, let k + k + · · · + k W HS = W , then the mean returninterval is R H S = W HS (cid:80) W HS i =1 k i = WW HS . (11)This relation (11) clearly indicates that the mean return interval R H S is relatively short if the occurrence of extreme events isfrequent (i.e., W H S is high). Similarly, if W H S is small then themean return interval is relatively high. But, this number of returnintervals W H S is dependent on various factors and thus it varieson realizations. Hence, relation (11) gives different average returninterval R H S based on different realizations. Using time seriesanalogous of Kacs Lemma [84], [85], [86], R H S can be describedin terms of the tail of the normalized distribution density f ( x ) as R H S = WW HS (cid:39) (cid:82) ∞ HS f ( y ) dy . (12)Thus, there exists a one-one relation between the chosen threshold H S and the mean return interval R H S , which is solely determinedby the normalized distribution f ( x ) . k A − k R couplingparameter space For a complete understanding of interplay between attractive andrepulsive interactions (i.e., for α (cid:54) = 0 ), a two dimensional k A − k R parameter space is presented in Fig. 8. In this figure, we vary k A from [0 . , . whereas k R is varied within [0 . , − . .Initially, in absence of repulsive coupling, i.e., for k R = 0 . ,complete synchronization is observed for suitable choices of k A where E < − . As per this Fig. 8, whenever k A reachesto . , then the introduction of small k R < . may destroythe limit cycle behavior of the oscillators and their synchronizedbehavior is also lost. The reflected region of inhomogeneous smalloscillation enlarges with further increment of positive coupling k A . Interestingly, for a fixed k A ≥ . , the decrement of k R helps to restore the periodic behavior of the attractors. Withsuitable choices of k R for fixed k A ≥ . , this transition frominhomogeneous small oscillation to complete synchronized stateoccurs through the path of attractor switching, as already shownin Fig. 5. The attractors are quenched and with further decrementof k R for appropriate k A , the trajectories switch between theco-existing attractors and finally regain their respective periodicsynchronized attractor.When k R is further decreased, the synchronization error E becomes intermittent and reveals occasional away journey fromthe synchronization manifold. This regime is depicted throughthe red and blue regions in the Fig. (8). In order to distinguishbetween these red and blue regions, we calculate the significantheight H S . When H S lies within (10 − , . , we label thosestates as the intermittent region (in red). For H S > . , wedefine those states as extreme events which obey additional fewclauses mentioned earlier. While in complete synchronization, all Fig. 8. Parameter Space k R − k A : The yellow and gray regions respectively correspond to the states E < − (cid:39) (i.e., complete synchrony)and the inhomogeneous small oscillation. The blue area stands for the extreme events whereas the red zone represents the state where the error E becomes intermittent but H S ≤ . . Finally, the area in cyan corresponds to the desynchronization state. Other parameters are N = 100 , g = α = 10 and v = 2 . . Here, independent realizations are used to obtain this parameter space. the attractors converge into a single attractor. But, during theappearance of extreme events, most of the time all the trajectoriesevolve over a converged single trajectory, but occasionally due tothe local repulsion (of suitable strength k R ) few trajectories leavethe converged attractor and follow their own path for a relativelyshort time. Hence, during those time, the error trajectory leavesthe equilibrium point E = 0 . (the synchronization manifold) andexhibit a large excursion exceeding H S = m + 8 σ . This choiceof H S is influenced by the Ref. [65], where the authors gaveanalytical logic behind the choice of such extreme event indicatingthreshold. The second variables y i of all N = 100 oscillatorsare plotted over a short time-interval and the corresponding errordynamics are also presented through an arrow in Fig. 8. Asexpected, if k R is further decreased while keeping k A fixed,desynchrony states are observed. YMMETRY - PRESERVING COUPLING
In this section, we deal with the scenario in which the localdynamical units, i.e., the SL-oscillators are coupled through boththe variables so that the coupling becomes (rotational) symmetry-preserving. Our main emphasis will be to examine how this changeaffects the collective dynamics of the networked system. We depictthe dynamical outcomes through Fig. 9 in which the phase diagramin the k R − k A parameter plane is portrayed. As can be seen fromthe figure, the presence of only attractive coupling readily inducescomplete synchronization in the system. Moreover, in contrast toour earlier observation for symmetry-breaking coupling (cf. Fig. 8), this synchrony sustains as k A increases. This is mainly becausenow the coupling function does not allow to break the symmetryyielding inhomogeneous small oscillation. On the other hand,switching the repulsive coupling strength k R on, one observesoccasional jumps in the error dynamics (for suitable strengthsof couplings) that indicates the appearance of intermittent states.More importantly, further increment of k R leads to the intermittentstates that satisfy the criteria of being considered as extremeevents. This result demonstrates that our observations of completesynchronization or the extreme events on the considered networkset-up is not limited to a specific choice of symmetry-breakingcoupling function. ISCUSSION AND C ONCLUSION
In this study, we have considered Stuart-Landau oscillators on topof mobile agents moving in a finite region of two-dimensionalplane. Due to the spatial movement of the agents, the relativedistance between any two agents always varies with time. Usingthis spatial distance between the agents, two competing interac-tions are introduced. Whenever the agents stay inside a closed ballof radius α , the oscillators are repulsively coupled bidirectionallythrough diffusive linear coupling. On the contrary, whenever themobile agents lie beyond the closed communication ball of radius α , they are coupled through attractive coupling. Under this set-upof the networked system, we have encountered diverse collectivebehaviors and provided rigorous investigation of each of them.Firstly, we have addressed the effect of sole attractive interac-tion among the dynamical agents. In order to accomplish this, we Fig. 9. Phase diagram in the k R − k A parameter plane: The yellow regioncorresponds to the states of complete synchronization. The blue area stands for the extreme events whereas the red zone represents the state where the error E becomes intermittent, but H S ≤ . . Finally, the areain cyan corresponds to the desynchronization state. Other parametersare N = 100 , g = α = 10 and v = 2 . . Here, symmetry-preservingvector coupling among the local dynamical units is considered in which independent realizations are used to obtain this parameter space. first set the repulsive strength to zero for which complete synchro-nization is found for α = 10 . This result is quite interesting asfor a static time-independent network, complete synchronizationis not possible. Mobility of the agents play the decisive roleto converge all the attractors to a single one. Analytically, thecritical attractive coupling strength for synchrony is calculated,which exhibit excellent agreement with the numerical results. Toinvestigate further the sole influence of attractive coupling, therepulsive matrix is set to be a null matrix by keeping α = 0 fixed. This different strategy leads to complete synchronizationfor suitable choice of k A and cluster synchronization beyond acertain value of k A .Next we dealt with our prime issue of coexisting attractionand repulsion. The coexistence of such competing interactionscan demolish the synchronized behavior of the whole system.Under suitable choices of both coupling strength, inhomogeneoussmall oscillation is perceived. In this state, each oscillator main-tains chaotic behavior though we set initially each oscillator intolimit cycle regime. Further reduction of repulsive strength helpsto regain their periodic behavior through the path of attractorswitching. Moreover, further lessening leaves the error dynam-ics intermittent. This irregular away journey of error trajectoryfrom the synchronization manifold gives rise to infrequent largedeviated events. Using few characterizations, we confirmed thesestates as extreme events. The mean return interval of these extremeevents is approximated using time series analogs of Kac’s lemmain relation (12). This relation indicates the dependency of averagereturn interval R H S on the threshold H S . The larger R H S clearlycorresponds to larger H S and vice-versa. The route for generatingsuch large amplitude events is also addressed in terms of on-offintermittency. An upper bound for the probability of occurrenceof extreme events is calculated in relation (10) depending on thechoice of H S .Further, to elucidate the mechanism behind inhomogeneous small oscillation of those oscillators, vector coupling is introduced.This coupling helps to preserve the rotational symmetry of SL-oscillators. Although, this vector coupling yields behavior likecomplete synchrony and extreme events, but it eliminates theappearance of such small oscillation, which is occurred duringsymmetry-breaking coupling.We have been able to portray the whole scenario of all theseemerging collective behaviors in a two-dimensional parameterspace of the competing coupling strengths. This parameter spacemanifests the fact that by tuning any one of the coupling strength,one can avoid such devastating extreme events. This fact can beused as one of the influential strategy for controlling extremeevents. Such controlled usage of antibiotics [87], [88] alreadyfound to be helpful in avoiding worsening medical cost andmortality especially for life-threatening bacteria infections.In this article, we analyzed the interplay of positive andnegative interactions in coupled identical limit cycle oscillators, byconsidering SL system on top of each mobile agent. An importantdirection of future generalization may include networks of chaoticoscillators, which is where our approach might make a difference.The chaotic dynamics [89], [90], [91], [92] are expected to revealan even wider spectrum of interesting dynamical states. Thisremains the core interesting avenue for future work. A CKNOWLEDGEMENTS
SNC and DG were supported by Department of Science and Tech-nology, Government of India (Project No. EMR/2016/001039).SNC would also like to thank Physics and Applied MathematicsUnit of Indian Statistical Institute, Kolkata for their support duringthe pandemic COVID-19. R EFERENCES [1] M. E. J. Newman,
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