Evolution of microscopic and mesoscopic synchronized patterns in complex networks

Jesus Gomez-Gardenes,1, 2 Yamir Moreno,2, 3 and Alex Arenas2, 4

1Departamento de Matematica Aplicada, Rey Juan Carlos University, Mostoles (Madrid) E-28933, Spain2Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain

3Departamento de Fısica Teorica, Facultad de Ciencias, Universidad de Zaragoza, Zaragoza 50009, Spain4Departament d’Enginyeria Informatica i Matematiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain

(Dated: October 26, 2010)

Previous studies about synchronization of Kuramoto oscillators in complex networks have shown how localpatterns of synchronization emerge differently in homogeneous and heterogeneous topologies. The main differ-ence between the paths to synchronization in both topologies is rooted in the growth of the largest connectedcomponent of synchronized nodes when increasing the coupling between the oscillators. Nevertheless, a recentstudy focussing on this same phenomenon has claimed the contrary, stating that the statistical distribution ofsynchronized clusters for both types of networks is similar. Here we provide extensive numerical evidencesthat confirm the original claims, namely, that the microscopic and mesoscopic dynamics of the synchronizedpatterns indeed follow different routes.

PACS numbers: 05.45.Xt, 89.75.Fb

Synchronization is at the core of diverse collective phe-nomena spanning a variety of life scales from social toneurological contexts [1–3]. On the other hand, mostof the natural and social systems in which synchroniza-tion shows up have intricate connectivity patterns betweentheir units that are nowadays described as complex net-works [4–7]. Consequently, the question about how syn-chronization is affected by the architecture of such com-plex networks has spurred on a number of studies duringthe last decade [8]. In previous studies [9, 10], we showedthat local patterns of synchronization evolve towards thefully synchronized state differently in homogeneous andheterogeneous complex networks for the case of Kuramotooscillators. Namely, the transition to synchronization inhomogeneous networks occurs when several synchronizedclusters of similar (small) sizes collapse into one macro-scopic cluster. In contrast, in scale-free heterogeneous net-works this transition is ruled by the hubs and takes placesmoothly and centralized around the cluster that containthe hubs. Recently, the previous results have been chal-lenged by a new study that focused on the same system[11] but using coarse grained data and statistical metrics(rather than non-averaged quantities). Here we provideextensive numerical evidences that confirm our previousclaims, namely, that the microscopic dynamics of the syn-chronized patterns indeed follow different routes.

I. INTRODUCTION

Synchronization is an emergent phenomenon found in nat-ural and social networked systems, ranging from biologicalclocks to stock market crashes. Understanding synchroniza-tion processes in networks thus helps advancing on the phys-ical description of such different complex systems. Severalstudies have focused on this subject during the last few years,and as a result, we already understand some aspects aboutsynchronization in networks [8]. Features that have been an-alyzed in depth include the formation and growth of synchro-

nized clusters in networks before complete synchronization isattained, and the study of the conditions needed for the on-set of synchronization. The latter is particularly important toaddress the question of which topology is optimal for syn-chronization purposes, as far as it is intended as the ability ofthe system to form the first synchronized clusters. In fact, theubiquity of scale-free networks in natural, social and techno-logical systems has given rise to a number of studies about theonset of synchronization on this type of topology [9, 10, 12–19], which is characterized by large fluctuations in the num-ber of connections a node may have. In addition, the sameissue has also been studied on other (scale-free or not) net-work topologies with additional structural properties, such asgradient networks [20], random geometric graphs [25], clus-tered [22, 23] and modular networks [14, 24], or weightedgraphs [26]. These theoretical studies have paved the way to-wards a better understanding of complex synchronization pat-terns found in natural systems. This is the case of the onset ofsynchronization during both epileptic seizures [27] and neuraldevelopment [28], or the dynamical organization of the braincortex [29, 30].

In recent works [9, 10] we compared how local patternsof synchronization emerge differently in homogeneous andheterogeneous scale-free complex networks. To this end, weimplemented the Kuramoto model of coupled phase oscilla-tors [31–33] on top of Erdos-Renyi (ER) and Barabasi-Albert(BA) scale-free (SF) networks generated using a configura-tional model introduced in [34]. In these works, we showedthat the main difference between these topologies relies onthegrowth of the largest connected component of synchronizednodes when increasing the coupling between them. Neverthe-less, these works showed these differences in terms of averagevalues of the relevant physical quantities and thus constitutedan indirect evidence of the differences between the two micro-scopic scenarios. The above numerical evidences have beendistrusted recently by Kim et al. [11] who claim that the dif-ference in the microscopic evolution of synchronized patternsin ER and SF networks does not exist. In particular, by usinga debatable scaling ansatz borrowed from percolation theory

2

and finite size scaling to estimate the proposed scaling laws,they compute numerically the critical exponents, which turnedto be practically the same in ER and SF networks.

In this work we present the evolution of the synchronizedclusters in terms of the statistical distributions of theirde-scriptors together with the corresponding raw data as obtainedfrom extensive numerical simulations. This new descriptionallows to confirm our previous claims, namely, that the micro-scopic dynamics of the synchronized patterns follow differentroutes in homogeneous and heterogeneous networks. There-fore, will not discuss the weaknesses of the asymptotic ap-proach adopted in [11], instead we will offer clear numericalevidences that show that our previous claim was correct, andconsequently that their theory should be revised.

II. THE MODEL

In order to explore computationally the evolution of syn-chronization patterns in degree-homogeneous and degree-heterogeneous complex networks, we consider networks ofN non-identical phase oscillators evolving according to theKuramoto model [31]

dθi

dt= ωi + λ

N∑

j=1

Aij sin(θj − θi) i = 1, ..., N (1)

whereωi stands for the natural frequency of oscillatori, λ isthe coupling strength andAij is the connectivity matrix of thenetwork (defined asAij = 1 if i is linked toj andAij = 0otherwise). It is well-known that the collective dynamics ofcoupled Kuramoto oscillators [32, 33] undergoes a transitionfrom incoherent dynamics to a synchronized regime asλ in-creases. The synchronization transition can be monitored bymeans of an order parameterr defined as

reiΦ =1

N

N∑

j=1

eiθj . (2)

The value ofr changes fromr ≈ 0 to r ≈ 1 as the systemgoes from the incoherent state (at low values ofλ) to the fullysynchronized one (for large enoughλ).

When the network of interactions, encoded in the adjacencymatrix Aij , has a nontrivial underlying structure, as it is thecase for ER and SF graphs, it is interesting to analyze the syn-chronization transition by looking at the emergence of smallclusters of synchronized nodes. In particular, we focus on theevolution of these synchronized clusters, that can be seen assmall subgraphs embedded into the network, along the wholesynchronization transition. To this end, one starts at moder-ate coupling values to observe how certain parts of the sys-tem become synchronized rather fast whereas other regionsstill behave incoherently. Increasing the coupling strength,one monitors the synchronized patterns by reconstructing thesubgraphs composed of those nodes and links that share thelargest degree of synchronization. The study of the structural

properties of these subgraphs can thus be performed accu-rately as the couplingλ is increased, allowing the characteri-zation of those sets of nodes that drive the global dynamics ofthe system.

The numerical experiment designed to prove the differencebetween ER and SF networks with respect to the microscopicevolution of synchronized patterns is the following: we ana-lyze104 different realizations of the dynamics on both graphs,every realization corresponds to a network made up of103 os-cillators with average degree〈k〉 = 6. For each realization weset up a different initial condition by assigning to each oscil-lator an initial phase,θi ∈ [−π, π], and an internal frequency,ωi ∈ [−1/2, 1/2], using uniform distributions in both cases.We integrate the system of equations (1) by means of a4th

order Runge-Kutta method with a time step of∆t = 0.02, forvalues ofλ in the intervalλ ∈ [0.01, 0.15] with ∆λ = 0.01,up to achieving the stationary state. The stationary state isreached when the time evolution ofr (2) ends up in a constantvale. At this point, we measure the microscopic patterns ofsynchronization as described below.

The degree of synchrony between two connected nodesiandj is measured by quantifying the phase coherence [9, 10]as follows

Cij = limT→∞

Aij

∣

∣

∣

∣

∣

1

T

∫ τ+T

τ

ei[θi(t)−θj(t)] dt

∣

∣

∣

∣

∣

. (3)

Each of the values{Cij} are bounded in the interval[0, 1], be-ing Cij = 1 wheni andj are fully synchronized andCij = 0when these nodes are dynamically uncorrelated or physicallydisconnected. Note that for a correct computation of the ma-trix C the averaging timeT should be taken large enough (inour computationsT = 400) to obtain reliable measures of thedegree of coherence between each pair of nodes. Once the de-gree of synchronization between nodes is measured, we filterthe matrixC to construct a filtered matrixF whose elementsare eitherFij = 1 if i andj are considered as synchronizedor Fij = 0 otherwise. To unveil what links are regarded assynchronized we compute the fraction of synchronized linksas

rlink =1

2L

∑

i,j

Cij , (4)

whereL is the total number of links of the network (L =∑N

i,j=1 Aij/2). Therefore, one would expect that2L · rlink

elements of the matrixF haveFij = 1, while for the remain-ing elementsFij = 0. The former elements correspond tothe 2L · rlink links with the largest values ofCij . The ma-trix F defines a new network, composed of those synchro-nized links. This new network is, in general, composed ofseveral connected components being each of them a subgraphof the original network substrate. Thus, by means of the pro-cedure described above we can describe these synchronizedsubgraphs (or clusters) for each value ofλ. This descriptionincludes the number of synchronized clusters that coexist si-multaneously in the system as well as their sizes, as given bytheir number of nodes and links, respectively.

3

0

0.02

0.04

0.06

0.08

0.1

0.12

0 100 200 300 400 500 600 700 800 900 1000

P(N

sync

)

Nsync

λ=0.01λ=0.02λ=0.03λ=0.04λ=0.05λ=0.06

0

0.005

0.01

0.015

0.02

0.025

0.03

0 100 200 300 400 500 600 700 800 900 1000

P(N

sync

)

Nsync

λ=0.01λ=0.02λ=0.03λ=0.04λ=0.05λ=0.06

FIG. 1: Size of the giant synchronized component. We plot herethe probability of finding a giant synchronized component of a givensize, Nsync, for different values of the coupling strengthλ. Theupper and bottom figures correspond to ER and SF networks, respec-tively. N = 1000 in both cases. The results for every value ofλ

are obtained using a sample of104 different realizations. While forER networks there is a clear gap inP (Nsync), SF networks show agradual shift to the right without significant jumps inP (Nsync) asλ

is increased.

III. RESULTS

For each topology, ER and SF, we have solved numericallythe system of equations (1) for different values of the couplingconstantλ in order to follow the evolution of the synchronizedclusters. This enables to monitor the coalescence of pairs ofsynchronized oscillators as a function ofλ. In this section wewill show that the microscopic path to synchronization is in-deed different for both classes of networks. These differencesin the behavior can be traced back to the growth of the largestcluster (or subgraph) of synchronized nodes, also known asGiant Synchronized Component (GSC), as a function ofλ.It turns out that for degree-homogeneous ER networks, manydifferent clusters of synchronized pairs of oscillators mergetogether to form a macroscopicGSC when the coupling isincreased. The union of many small clusters leads to a giantcomponent of synchronized pairs that is almost of the size ofthe system (N ) once the incoherent state destabilizes. Thecase is remarkably different for SF networks. In this case, theGSC grows gradually and synchronized oscillators are incor-porated to it in an almost sequential way as theλ is increased.

0 100 200 300 400 500 600 700 800 900

1000

0 100 200 300 400 500 600 700 800 900 1000

Nsy

nc(λ

+δλ

)

0 100 200 300 400 500 600 700 800 900

1000

0 100 200 300 400 500 600 700 800 900 1000

Nsy

nc(λ

+δλ

)

Nsync(λ)

λ0.02 0.06 0.140.10

FIG. 2: Evolution of the size of theGSC when increasing thecoupling strength in ER (top) and SF (bottom) networks. To cap-ture the growth of theGSC size as the coupling strength is in-creased fromλ to λ + δλ (with δλ = 0.01) we plot the functionNsync(λ + δλ) = f [Nsync(λ)]. A total number of104 numericalcontinuations have been carried for every value ofλ ∈ [0.01, 0.14].The color of the dots denote the corresponding value ofλ as shownin the color bar.

In Figure 1 we show the probability distribution of the sizeof the GSC, P (N sync), for several values of the couplingstrengthλ for both ER and SF networks. The evolution of thedistributions withλ for ER networks reveals the transition oc-curring when different synchronized clusters merge togetherto form a singleGSC of macroscopic size. The transitionoccurs betweenλ = 0.02 and0.04 and it is mediated by thenearly flat distribution shown forλ = 0.03. On the other hand,in SF networks the distributions do not present this transitionand for all values ofλ P (N sync) is clearly peaked and hasa meaningful mean value for the average size of theGSC.Therefore, these distinct behaviors are already indicating thatthe path towards full synchronization depends on the hetero-geneity of the degree distribution.

We also performed numerical experiments of adiabatic con-tinuation inλ. For a givenλ, we drive the system to its sta-tionary state and induce a small perturbation of the couplingstrength by incrementing it toλ+ δλ (with δλ = 0.01). Then,we record the changes in the size of theGSC, i.e., the num-ber of new nodes that are incorporated into theGSC. Thiscontinuation allows us to obtain a more detailed picture of theevolution of the size of theGSC by monitoring its growthat the level of individual nodes incorporated, and only those

4

0

0.005

0.01

0.015

0.02

0.025

0.03

1000 1500 2000 2500 3000

P(L

sync

)

Lsync

λ=0.09λ=0.10λ=0.11λ=0.12λ=0.13λ=0.14

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

1000 1500 2000 2500 3000

P(L

sync

)

Lsync

λ=0.09λ=0.10λ=0.11λ=0.12λ=0.13λ=0.14

FIG. 3: Number of synchronized links in the giant synchronizedcomponent. We plot the probability of finding a giant synchronizedcomponent with a given number of links,Lsync, for different valuesof the coupling strengthλ. The upper and bottom figures correspondto ER and SF networks, respectively. The distributions are obtainedas in Figure 1.

synchronized links between them. In this way, we counthow many clusters of new nodes and synchronized links arepresent, and finally compute the size of theGSC at the newλvalue.

In Figure 2 we plot the functionN sync(λ + δλ) =f [N sync(λ)] obtained by performing the adiabatic continua-tion for several values ofλ from 0.01 to 0.15. Every singlenumerical experiment is represented by a dot in the maps offigure 2 so that, after an extensive study (104 adiabatic con-tinuations for each value ofλ) we can observe the trends ofthe functionsf(x) of the maps corresponding to SF and ERtopologies. The map obtained for ER networks confirms thatwhen making a small change of the coupling aroundλ = 0.02the GSC suddenly jumps from sizes ofN sync ∼ 102 toN sync ∼ 6 · 102 (recall that our simulations are done on topof networks of sizeN = 103). On the other hand, for SFnetworks the map does not present such dramatic changes forall the range ofλ values explored. The color of the dots inboth maps reveals that the size of theGSC of ER networks isN sync∼ N for low values ofλ (for which the global degree ofsynchronization is still low [9, 10]) while in SF networks suchsizes proportional toN are only reached for large values ofλ(in which a large global degree of synchrony is also observed[9, 10]).

0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000 2500 3000

Lsync

(λ+

δλ)

0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000 2500 3000

Lsync

(λ+

δλ)

Lsync(λ)

λ0.02 0.06 0.140.10

FIG. 4: Number of synchronized links,Lsync, after increasing thecoupling. Same philosophy as figure 2: We construct the mapLsync(λ + δλ) = g[Lsync(λ)] by following the evolution of the syn-chronizedGSC as the couplingλ is increased from0.01 to 0.15.The evolution for the ER network (top) shows two big jumps in thenumber of links incorporated into theGSC in contrast with the con-tinuous shape ofg(x) for the SF topology (bottom). Both panelsshow a number of104 numerical continuations.

We have also explored the evolution of the number of linksof theGSC. First, in Figure 3 we have plotted the probabilitydistribution of the number of synchronized links in theGSCfor several values ofλ. In the case of ER networks we observea region of the coupling strength, aroundλ = 0.12, with anearly flat distribution, thus signaling again dramatic changesin the structure of theGSC when the coupling is increasedin this parametric region. At variance with Figure 1, wherethe same phenomenology was observed, this flat distributiondoes not correspond to a growth of theGSC, but points outa fast addition of links connecting nodes that already belongto theGSC (those that joined it at lower values ofλ). There-fore, in this region ofλ, the synchronized links added do notcontribute to the growth of theGSC (it is already of orderN ), but to the decrease of the average path length between itsnodes (see below). On the other hand, the SF topology showsa smooth evolution of the probability distributions and theircorresponding mean values, pointing out a gradual additionofsynchronized links asλ increases.

By using again the adiabatic continuation inλ we can care-fully check how the addition of links to theGSC takes place.We performed5 ·103 numerical experiments of adiabatic con-tinuation fromλ = 0.01 to λ = 0.14 with ∆λ = 0.01. In

5

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

<l>

sync

(λ+

δλ)/

<l>

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

<l>

sync

(λ+

δλ)/

<l>

<l>sync(λ)/<l>

λ0.02 0.06 0.140.10

FIG. 5: Evolution of the average path length of theGSC along thesynchronization path. We show the change in theAPL of theGSC

of ER (top) and SF (bottom) networks when passing fromλ toλ+δλ

with δλ = 0.01. In both panels we plot the relation〈l〉sync(λ +δλ) = h[〈l〉sync(λ)] observed in104 numerical continuation alongthe rangeλ ∈ [0.01, 0.14]. As denoted by the dotted line in the toppanel, a sudden decrease of theAPL in theGSC of ER networksoccurs atλ ≃ 0.12 due to the fast addition of links into theGSC.Such dramatic change is not observed for SF networks and theAPL

gradually decreases as shown by the dotted line.

Figure 4 we show the scatter plots for the number of links,Lsync(λ + δλ) = g[Lsync(λ)], for both SF and ER networks.From this figure it becomes evident that the addition of linksinto theGSC in SF networks proceeds in a smooth way simi-larly to its growth in size. On the other hand, for ER networks,we find evidences of two sudden jumps during the adiabaticcontinuation. In particular, for values of the coupling in theregionλ ∈ [0.1, 0.12], for which the size of theGSC is of theorder of the size of the substrate ER network, we observed asudden addition of links that connect nodes already belongingto theGSC.

Finally, we analyzed the evolution of the average pathlength, (APL), of theGSC, 〈l〉sync, when passing from lo-cal to global synchronization. The idea here is to provide an-other indication of the structural differences in theGSC for

the different classes of networks, when changing the couplingbetween oscillators. In Figure 5 we show this variation byplotting the map〈l〉sync(λ + δλ) = h[〈l〉sync(λ)] obtainedafter104 realizations for each value ofλ for both ER and SFnetworks. In both cases theAPL is normalized to theAPLof the corresponding substrate network. From the figure, itis clear that theGSC of ER networks reachesAPL valuesremarkably larger than those obtained in SF networks alongthe synchronization path. Moreover, in ER networks and forlarge values ofλ we observe a fast collapse to the value ofthe APL of the substrate network from configurations thatare twice longer than the substrate. This sudden collapse oc-curs in the region ofλ where the massive addition of links wasobserved in Figure 4. On the other hand, in SF networks thetransition from large values of theAPL to the length of thesubstrate networks takes place smoothly since link addition isalso a sequential process bringing new nodes to theGSC.

IV. CONCLUSIONS

Summarizing, we have numerically explored the synchro-nization of Kuramoto oscillators in complex topologies as thecoupling strength is increased driving the system’s dynam-ics to the fully synchronized state. By carefully character-izing the way in which the size and composition of theGSCchanges, we have explored the paths towards synchronizationin homogeneous and heterogeneous networks. All the exten-sive simulations performed on the two topologies, ER and SF,clearly indicate that the evolution of theGSC is different inboth classes of networks.

We have presented raw data from the numerical experi-ments without any type of processing, averaging or any othercoarse-graining of the information. Therefore, our results putin evidence the statistical analysis performed by the authorsof [11] and the claimed coincidence of the scaling exponentsthere derived. The conclusion is then that the suitability ofsuch approaches have to be confirmed first and therefore can-not be used as the base of the claim raised to invalidate ourprevious findings, which are doubtlessly corroborated here.As any theory intended to explain the results of a numericalexperiment, when numerics and theory do not agree, the the-ory must be revised.

Acknowledgments

This work has been partially supported by the Spanish DG-ICYT undet projects FIS2008-01240, FIS2009-13364-C02-01, FIS2009-13730-C02-02 and MTM2009-13848, and bythe Generalitat de Catalunya under project 2009SGR0838.

[1] Winfree, A. T. The geometry of biological time(Springer-Verlag, New York, 1990).

[2] Strogatz, S. H.Sync: The Emerging Science of SpontaneousOrder (New York: Hyperion, 2003).

6

[3] Manrubia, S. C., Mikhailov, A. S. & Zanette, D. H.Emergenceof Dynamical Order. Synchronization Phenomena in ComplexSystems(World Scientific, Singapore, 2004).

[4] Strogatz, S. H. Exploring complex networks.Nature410, 268-276 (2001).

[5] Albert R. & Barabasi, A.-L. Statistical mechanics of complexnetworks.Rev. Mod. Phys.74, 47-97 (2002).

[6] Newman, M. E. J. The structure and function of complex net-works.SIAM Review45, 167-256 (2003).

[7] Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang,D.-U. Complex networks: structure and dynamics.Phys. Rep.424, 175-308 (2006)

[8] A. Arenas, A. Dıaz-Guilera, J. Kurths, Y. Moreno and C. Zhou,Physics Reports469, 93 (2008)

[9] J. Gomez-Gardenes, Y. Moreno, and A. Arenas,Phys. Rev. Lett.98, 034101 (2007)

[10] J. Gomez-Gardenes, Y. Moreno, and A. Arenas,Phys. Rev. E75, 066106 (2007).

[11] Y. Kim, Y. Ko and S-H. Yook,Phys. Rev. E81, 011139 (2010).[12] Y. Moreno and A.F. Pacheco, EPL68, 603 (2004).[13] T. Ichinomiya, Phys. Rev. E70, 026116 (2004).[14] E. Oh, K. Rho, H. Hong, and B. Kahng, Phys. Rev. E72,

047101 (2005).[15] D.-S. Lee, Phys. Rev. E72, 026208 (2005).[16] J.G. Restrepo, E. Ott, and B.R. Hunt, Chaos 16, 015107 (2006).[17] C. Zhou, and J. Kurths, Chaos 16, 015104 (2006).[18] Y.-Ch. Hung, Y.-T. Huang, M.-Ch. Ho, and Ch.-K. Hu Phys.

Rev. E77, 016202 (2008).[19] T. Pereira, Phys. Rev. E82, 036201 (2010).

[20] M. Chen, Y. Shang, Y. Zou, and J. Kurths Phys. Rev. E77,027101 (2008).

[21] X. Wang, L. Huang, Sh. Guan, Y.-Ch. Lai, and Ch. H. Lai,Chaos18, 037117 (2008).

[22] J. Gomez-Gardenes, and Y. Moreno, Int. J. Bif. Chaos17, 2501(2007);

[23] M. Brede, Eur. Phys. J. B62, 87 (2008).[24] A. Arenas, A. Diaz-Guilera, and C.J. Perez-Vicente, Phys. Rev.

Lett. 96, 114102 (2006).[25] A. Dıaz-Guilera, J. Gomez-Gardenes, Y. Moreno, and M.

Nekovee, Int. J. Bif. Chaos19, 687 (2009).[26] W.-X. Wang, L. Huang, Y.-Ch. Lai, and G. Chen, Chaos19,

013134 (2009).[27] K. A. Schindler, S. Bialonski, M.-Th. Horstmann, Ch. E. Elger,

and K. Lehnertz, Chaos18, 033119 (2008).[28] E. Fuchs, A. Ayali, E. Ben-Jacob, and S. Boccaletti, Physical

Biology 6, 036018 (2009).[29] C.S. Zhou, L. Zemanova, G. Zamora-Lopez, C.C. Hilgetag, and

J. Kurths, New J. Phys.9, 178 (2007).[30] J. Gomez-Gardenes, G.Zamora-Lopez, Y. Moreno, and A. Are-

nas, PLoS ONE5, e12313 (2010).[31] Y. Kuramoto,Prog. Theor. Phys.79, 223 (1984).[32] S.H. Strogatz,Physica D143, 1 (2000).[33] J.A. Acebron, L.L. Bonilla, C.J. Perez Vicente, F. Ritort and R.

Spigler,Rev. Mod. Phys.77, 137 (2005).[34] J. Gomez-Gardenes and Y. Moreno,Phys. Rev. E73, 056124

(2006).