Explosive synchronization in temporal networks: A comparative study
aa r X i v : . [ n li n . AO ] J u l Explosive synchronization in dynamic networks: A comparative study
Tanu Singla and M. Rivera Tecnol´ogico de Monterrey, Calle del Puente 222,Colonia Ejidos de Huipulco, Tlalpan, Ciudad de M´exico, M´exico Centro de Investigaci´on en Ciencias-(IICBA), UAEM,Avenida Universidad 1001, Colonia Chamilpa, Cuernavaca, Morelos, M´exico (Dated: July 21, 2020)We present a comparative study on Explosive Synchronization (ES) in dynamic networks consist-ing of phase oscillators. Dynamic nature of the networks has been modeled with two configurations:(1) oscillators are allowed to move in a closed two dimensional box such that they couple with theirneighbors, (2) oscillators are static and they randomly switch their coupling partners. (1) has beenfurther studied under two possible scenarios: in the first case oscillators couple to fixed numbersof neighbors while in other they couple to all oscillators lying in their circle of vision. Under thesecircumstances, we monitor the degrees of dynamic networks, velocities and radius of circle of visionof the oscillators, and probability of forming connections in order to study and compare the criticalvalues of the coupling required to induce ES in the population of phase oscillators.
I. INTRODUCTION
Synchronization is a phenomenon wherein simultane-ous evolution in the dynamics of oscillators is observed bythe virtue of coupling between them. After its discoveryby Christian Huygens by performing an experiment witha pair of pendulum clocks hung on a common supportingbeam, studies of synchronization remained dormant fora long period of time. The pioneering work of phase syn-chronization in chaotic oscillators in 1996 [1] motivatedscientists to explore synchronization in different systems.Since then, synchronization and its manifestations havebeen studied in diverse systems found in nature [2–5] aswell as in model experimental [6–9] and numerical sys-tems [10–12].In their pioneering work Kuramoto et al. studied syn-chronization in a network of globally coupled phase oscil-lators and a second order transition from unsynchronizedto synchronized state was observed. This was followed bya plethora of studies for synchronization in networks in-volving phase oscillators and other dynamical systems.Chimera state is a manifestation of synchronization innetworks where the coupling is employed between neigh-bors of the oscillators [13, 14]; this coupling scheme re-sults in partial synchronization of the population of os-cillators. Synchronization has also been reported in ran-dom networks wherein the oscillators randomly couple toeach other [15]. Finally, synchronization has also been re-ported in dynamic networks. The topology of a dynamicnetwork changes over time i.e. the oscillators keep form-ing new connections with other oscillators while main-taining and losing some of the old connections. Two dif-ferent variations of dynamic networks have been studiedand reported: in one case, the oscillators remain staticwhile switching couplings with other oscillators and inthe other case, the oscillators move in a given space andthey couple to their nearby oscillators [16, 17]. In nature,dynamic networks can be observed in a population of fire-flies where a firefly observes the blinking of its neighborsand adjust its own frequency of blinking. The adjustment of rhythms by the entire population of fireflies results inthe emergence of synchronized behavior. Dynamical net-works also have its importance in studying synchronizeddynamics of a population of robots in order execute acollective task.ES is a manifestation of synchronization in a popula-tion of interacting oscillators which is characterized bya first order transition from unsynchronized to synchro-nized state when the strength of interaction ( ǫ ) betweenthe oscillators is gradually increased. Presence of hys-teresis is another property which is associated with thefirst order transition in ES. This signifies that the valueof coupling strength ( ǫ = ǫ l ) at which oscillators un-synchronize while decreasing coupling between oscilla-tors is lower than the critical value of coupling strength( ǫ = ǫ u ) to achieve synchronization while increasing cou-pling. The principle mechanism behind this phenomenonis to suppress the formation of synchronization clusterswhich eventually results in the global synchronization ofthe population. In the initial works on ES, it has beenshown that this suppression can be achieved by imple-menting positive correlation between the natural frequen-cies of the oscillators and their degrees [18, 19]. Followingthis article, several other mechanisms to induce first ordertransition have been reported [20, 21]. In another reportby Zhang et al. [22], the role of local order parameterof the oscillators in achieving ES has been studied. Inthis study, the dynamics of every oscillator is influencedby its local order parameter; this reduces the effectivecoupling between oscillators as the local order parameterstarted to augment, which further causes the suppres-sion of synchronization clusters. Eventually, when thecoupling strength between oscillators is sufficiently high(higher than critical ǫ to observe second order Kuramototransition), ES can be observed in the population of os-cillators. In another work [23] this concept has been ex-tended to modify the width of hysteresis loop formed dur-ing ES. The mechanism of ES has also been intended toexplain a neurological disease called Fibromyalgia [24]. Afunctional network from the EEG signals of the patientswas constructed and analyzed to identify the imprints ofES. Finally, ES has also been reported in a model ex-perimental system consisting of Mercury Beating Heartoscillators [25].In this work, we present a comparative study of ESin dynamical networks. Three different configurationshave been considered to employ dynamic coupling in apopulation of phase oscillators. In one of the configura-tions, the oscillators remain static and randomly coupleto other oscillators such that the average degree of thenetworks forming at any time moment fluctuates arounda mean value. In other two configurations, the oscilla-tors execute random walk in a two dimensional closedspace and couple to their neighbors; degrees of the os-cillators for these cases are further decided on the basisof two different schemes (discussed later). Dependingon the configuration, parameters like coupling strength,velocity and vision size of oscillators and probability offorming a connection between two oscillators have beenvaried to study ES. II. COUPLING MECHANISMS
We consider a population of N coupled phase oscilla-tors to study ES in dynamical networks. A populationof coupled phase oscillators is generally represented withthe following equation: dφ p dt = ω p + ǫk p N X q =1 A pq sin( φ q − φ p ) . (1)Here, φ p and ω p are the phase and the frequency of the p th oscillator, ǫ is the coupling strength between the os-cillators, and A pq is an element of adjacency matrix A giving details about the coupling links between the oscil-lators; A pq = 1 if p th and q th are coupled and A pq = 0otherwise. Degree of an oscillator can be calculated from k p = P Nq =1 A pq . The extent of synchronization of thepopulation can be calculated using the following expres-sion: re i Φ = 1 N N X q =1 e iφ q . (2)Here, 0 ≤ r ≤ dφ p dt = ω p + ǫr p k p N X q =1 A pq sin( φ q − φ p ) . (3) Similar to Eq. (2), local order parameter of an oscillatoris defined as: r p e i Φ p = 1 k p N X q =1 A pq e iφ q , (4)Φ p is the average phase of the oscillators coupled to p th oscillator. Frequencies of the oscillators ( ω ) have beenchosen from a uniform distribution of numbers lying be-tween 0-2. Finally, to simulate dynamic networks the in-stantaneous adjacency matrices A pq have been obtainedusing following three configurations: • Configuration 1:
Moving oscillators with nearestneighbor coupling such that every oscillator of thepopulation couple to same number of nearest oscil-lators i.e. k of every oscillators remains equal andconstant. • Configuration 2:
Moving oscillators with nearestneighbor coupling such that every oscillator has acircular vision size and the oscillator couple to alloscillators lying in its vision. In this case k dependson the radius ( R ) of the circle of vision. • Configuration 3:
Static oscillators where everyoscillator randomly couples to other oscillators suchthat the average degrees of the dynamic networksremains same throughout time. In this case, k de-pends on the probability ( p ) with which oscillatorscouple with each other. x (a.u.)00.20.40.60.81 y ( a . u . ) r d FIG. 1: Demonstration of instantaneous locations of N = 500phase oscillators in the x − y plane along with the trajectoriesof oscillators with v = 0 .
002 (magenta), v = 0 .
05 (yellow), and v = 0 . R = 0 . r (a) u l k r (b) k r (c) k k (f)0 20 40 60 k u (d) 0 20 40 60 k l (e) FIG. 2: (a-c) Results of ES in dynamic networks emulated with configuration 1 for three different velocities of the oscillators:(a) v = 0 . v = 0 .
05, (c) v = 0 .
1. (d) Critical values of the coupling strength required to synchronize oscillators, (e)critical values of coupling strength required to unsynchronize oscillators, and (f) width of the hysteresis loops formed duringES. ( v = 0 . v = 0 . v = 0 . In the case of configurations involving moving oscilla-tors, their motion has been confined into a unit size twodimensional box with rigid boundaries. Oscillators arepermitted to move a δ step in either ± x or ± y directionat every iteration. Value of the step moved by oscillatorsalso defines their velocity ( v = δ ) in the closed box. InFig. 1 the initial positions (blue dots) of N = 500 oscil-lators have been shown for demonstration. The motionof three random oscillators with different velocities hasbeen represented with magenta line ( v = 0 . v = 0 . v = 0 . R = 0 . III. RESULTSA. Configuration 1
In this configuration the dynamical nature of the net-works of phase oscillators has been modeled such that theoscillators move in a closed unit size box with uniform ve-locities and all of the oscillators interact with fixed num-ber of nearest neighbors. This implies that the instan-taneous degree ( k ) of all oscillators remains same andfixed. Moreover, velocities ( v ) of oscillators are equal tothe step δ that the oscillators move in every iteration.Fig. 2 shows the results of ES in phase oscillators mod-eled with this configuration. In Fig. 2(a-c), we plot thevariation of order parameter of the entire population asa function ǫ for three different velocities of the oscillators((a): v = 0 . v = 0 .
05, and (c) v = 0 . k . It canbe noted that for sufficiently small degree of the oscilla-tors ( k = 15 and v = 0 .
002 in Fig. 2(a)), the populationof the oscillators undergoes classical second order Ku-ramoto transition in the order parameter. However, as k increases gradually, the population experiences explo-sive (first order) transitions between unsynchronized andsynchronized states; the transition is also accompaniedby its characteristic hysteresis loop. For the purpose ofdemonstration, the critical values of coupling strength atwhich oscillators synchronize (unsynchronize) have beenmarked with ǫ u ( ǫ l ) on Fig. 2(a). Furthermore, similardependences of ES on k have also been observed whenthe oscillators move with higher velocities: v = 0 .
05 and v = 0 . ǫ u (critical value of couplingrequired to observe ES) has been illustrated as a functionof k for three different velocities; for any v , lowest valueof k is the degree of the oscillators at which ES starts toappear. It can be noted that ǫ u increases uniformly with k for every velocity. In [27, 28] Kuramoto model has beenstudied by the perspective of complex networks. Accord-ing to this study, if the time dependent local parameterof an oscillator in the present situation is defined withthe following relation: r tp e i Φ p = N X q =1 A pq h e iφ q i t , (5)where, h· · · i t is the time average, then, the state of anoscillator can be represented as: dφ p dt = ω p + ǫr p r tp k p sin(Φ p − φ p ) − ǫh p . (6)Here, Φ p is the average phase of the oscillators coupled to p th oscillator and h p accounts for the time fluctuationsin the dynamics of this oscillator by the virtue of dy-namic adjacency matrices. The mathematical form of h p is given by h p = Im { e − iφ p P q A pq ( h e iφ q i t − e iφ q ) } , where“Im” stands for imaginary and h· · · i t is the time aver-age. During the onset of synchronization, r p = r tp ∼ k p and h p is expected to be of the order of p k p ( k p isthe degree of p th oscillator). Therefore, as the degree ofthe oscillators of the moving population increases, thesefluctuations also increase causing the suppression of syn-chronization clusters (as explained in the introductionsection) and requiring even larger ǫ for the populationto synchronize. Another interesting behavior that canbe observed in Fig. 2(d) is that the minimum value of k at which ES starts to appear reduces as the velocitiesof the oscillators increase. The instantaneous time spentby an oscillator in a local cluster decreases as its velocityincrease, causing less interaction among the members ofthe clusters. The results of Fig. 2(d) show that whenthe degree of oscillators is large then due to higher het-erogeneity in the cluster, it would require more time orhigher coupling strength to synchronize at higher veloci-ties. Conversely, it can be said that the as the velocity ofthe oscillators increase, the population of the oscillatorscan exhibit ES at lower degrees. In Fig. 2(e) variation of ǫ l (critical value of coupling at which oscillators unsyn-chronize) as a function of k for three different velocitieshas been shown. While ǫ l decreases with k for v = 0 . ǫ u and ǫ l constitutesthe hysteresis loops of ES and in agreement to results ofFig. 2(d and e), width of the hysteresis loops increaseswith k (Fig. 2(f)). r (a) v v (c)0.02 0.04 0.06 v u , l (b) FIG. 3: (a) Results of ES in dynamic networks emulated withconfiguration 1 for k = 15. (b) Critical values of the couplingstrength required to synchronize ( ǫ u : ), and critical valuesof coupling strength required to unsynchronize oscillators ( ǫ l :), (c) width of the hysteresis loops formed during ES. From Fig. 2(a) it can be observed that when k = 15,ES does not exhibit in the population of oscillators for v = 0 .
002 but it appears at higher velocities for the samevalue of k (Fig. 2(b and c)). To study this transition, wevary velocities of the oscillators keeping k = 15 and theresults of explosive transitions are shown in Fig. 3(a). Itcan be observed that the oscillators exhibit ES as theirvelocities increase from v = 0 .
02 to v = 0 .
03. Further-more, in Fig. 3(b), variation of ǫ u has been shown as afunction of v . In [17] it has been reported that similarto coupling strength ( ǫ ), velocity of the oscillators alsoacts as a parameter to observe second order Kuramototransition of synchronization. In the present case, whenvelocities of the oscillators augment, it results in the for-mation of synchronization clusters. From the analysis ofdependence of ǫ u on k (Eq. (6)), it can be said that thelocal order parameter of the oscillators also depends onthe velocities of the oscillators and it explains the increas-ing nature of ǫ u with v (Fig. 3(b)). Finally, variations of ǫ l and width of the hysteresis loops have been shown onFig. 3(b and c) and they are identical to their respectiveresults in Fig. 2(e and f). B. Configuration 2
In this section, the results of ES in dynamical networksobtained using configuration 1 have been compared withanother configuration in which coupling has been imple-mented such that every moving oscillator interacts withother oscillators lying in its vision circle (radius: R ). Asa consequence, degree of oscillators ( k ) in this case de-pends on R , does not remain fixed in time, and could bedifferent for every oscillator. Similar to previous subsec-tion, results have been obtained by varying R , keeping v constant and vice versa. r (a) R R (c)0 0.1 0.2 0.3 R u , l (b) FIG. 4: (a) Results of ES in dynamic networks emulated withconfiguration 2 for v = 0 . ǫ u : circles), and criti-cal values of coupling strength required to unsynchronize os-cillators ( ǫ l : stars), (c) width of the hysteresis loops formedduring ES. ( v = 0 . v = 0 . v = 0 . In Fig. 4(a), hysteresis loops for different values of R have been shown for v = 0 . R ES is not observed and that it starts toappear as R increases. The variation of ǫ u and ǫ l as afunction of R have been shown in Fig. 4(b) for v = 0 . v = 0 .
05, and v = 0 .
1. It must be noted that, while ǫ u for v = 0 .
002 and ǫ l for all velocities show similar trendas their counterparts in Fig. 2(d and e), the response of ǫ u for v = 0 .
05 and for v = 0 .
1, however, is different. Itshows that the critical value of ǫ to achieve synchroniza-tion of oscillators moving with larger velocities show aninitial fall before starting to increase with R . Given thatEq. (6) explains the relationship between ǫ u and k , thisresult deviates for smaller R ( k ∝ R ) from its analogousresults of Fig. 2(d) where the critical ǫ u for ES uniformlyincreased with k . The possible reason for this deviationlies in the fact that in the present case k of oscillators isnot constant. However, detailed theoretical and/or nu- merical analysis needs to be carried out to understandthis behavior. Finally, in Fig. 4(c), width of the hystere-sis loops as a function of R have been shown for differentvelocities and it can be observed that the width increaseswith R .For the purpose of completion, in Fig. 5, results of ESof moving oscillators have been shown by varying ǫ and v ;keeping R of the oscillators fixed at 0 .
1. Results obtainedin this case are identical to those presented in Fig. 3,where it was shown that the oscillators do not exhibitES when they move slowly and at higher velocities widthof the hysteresis loops of the ES increases as the velocitiesof the oscillators increase. r (a) v v (c)0.04 0.05 0.06 v u , l (b) FIG. 5: (a) Results of ES in dynamic networks emulated withconfiguration 2 for R = 0 .
1. (b) Critical values of the couplingstrength required to synchronize ( ǫ u : ), and critical valuesof coupling strength required to unsynchronize oscillators ( ǫ l :), (c) width of the hysteresis loops formed during ES. C. Configuration 3
In the final scenario, coupling has been implementedsuch that at every iteration the oscillators randomlyswitch coupling between each other. This signifies thatcoupling between two oscillators does not depend on thephysical distance between them and that the degree ofthe oscillators changes in every iteration. If the proba-bility with which an oscillator couples to another whileswitching coupling is given by p then the average degreeof the network will be k = pN . Moreover, when an oscil-lator switches its coupling partners, the algorithm doesnot preclude that this oscillator cannot couple again tosame oscillators it was coupled in the previous iteration.Fig. 6 shows the results of ES as a function of ǫ fordifferent values of p . Similar to previous results, for a r (a) p p (c)0 0.01 0.02 p u , l (b) FIG. 6: (a) Results of ES in dynamic networks emulated withconfiguration 3. (b) Critical values of the coupling strengthrequired to synchronize ( ǫ u : ), and critical values of cou-pling strength required to unsynchronize oscillators ( ǫ l : ),(c) width of the hysteresis loops formed during ES. sufficiently smaller p (or k ), ES is not observed in thepopulation of oscillators and as p increases, ES appears.In Fig. 6(b), variations of ǫ u and ǫ l have been shown. Sur-prisingly, similar to the result for v = 0 .
05 and v = 0 . ǫ u in the present case, initially decreaseswith p before starting to increase. On one hand, a pos-sible explanation of this behavior can be ascribed to thefact that the oscillators in this case switches coupling atevery iteration and the coupling also does not dependon the physical distance between the oscillators. This istantamount to the situation that the oscillators in thiscase are moving with very high velocities and frequentlychanging their coupling partners. On the other hand,the reason for the similarity of these results only withthe corresponding results of configuration 2 suggests thatthe reason behind this behavior lies in the fact that un-like configuration 1, in configuration 2 and 3, degrees ofthe oscillators does not remain constant. However, we modified configuration 1 by introducing fluctuations inthe degrees of the oscillators, but the behavior of ǫ u sim-ilar to that in Fig. 4(b) and Fig. 6(b) was not obtained.Therefore, initial fall of ǫ u with k remains unclear. Fi-nally, in Fig. 6(c), variation of width of the hysteresisloop has been shown as a function of p and it increasesmonotonically with p . IV. CONCLUSIONS
We presented our results on ES in dynamical networksof phase oscillators. The results were obtained and com-pared by implementing three different configurations inorder to achieve dynamical nature of the oscillators. Intwo of these configurations, the oscillators were chang-ing their coupling partners depending on the distancesbetween them while in the third coupling partners wereswitched randomly (irrespective of physical distances).Different control parameters were monitored in order toobserve and analyze ES in the oscillators. Using Eq. (6)the analytical understanding of the dependence of ES on k and v was established.The most striking differences in three configurationswere observed when degrees of the oscillators ( k ∝ R in configuration 2 and k ≈ pN in configuration 3) werevaried. In configuration 1 the critical value of couplingto observe ES ( ǫ u ) increased uniformly with k . However,in configuration 2 and 3, before starting to rise for largervalues of k , ǫ u decreased initially as k started to increasefrom its lower values. Moreover, this behavior was ob-served only in the situations when the oscillators weremoving with large velocities. It was also observed that ǫ l decreased and the width of the hysteresis loops increasedwith k in all the configurations.In our future work we will explore ES by implement-ing dynamical nature of coupling in experimental sys-tems. Configuration 1 and 2 although are more realisticto observe in nature, but they are difficult to realize inmodel experimental nonlinear oscillators. Configuration3, however, can be established easily and ES can be ex-plored. The fact that ES has already been reported in anexperimental system consisting of static Mercury Beat-ing Heart oscillators [25] makes this system a potentialcandidate to study ES in the present circumstances. [1] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys.Rev. Lett. , 1804 (1996).[2] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-tion: A Universal Concept in Nonlinear Sciences , Cam-bridge Nonlinear Science Series (Cambridge UniversityPress, 2001).[3] S. Strogatz,
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