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Nonstationary pattern in unsynchronizable complex networks

Xingang Wang,1, 2 Meng Zhan,3 Shuguang Guan,1, 2 and Choy Heng Lai4, 2

1Temasek Laboratories, National University of Singapore, 117508 Singapore2Beijing-Hong Kong-Singapore Joint Centre for Nonlinear & Complex Systems (Singapore),

National University of Singapore, Kent Ridge, 119260 Singapore3Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

4Department of Physics, National University of Singapore, 117542 Singapore(Dated: February 1, 2008)

Pattern formation and evolution in unsynchronizable complex networks are investigated. Due to the asym-metric topology, the synchronous patterns formed in complex networks are irregular and nonstationary. Forcoupling strength immediately out of the synchronizable region, the typical phenomenon is the on-off inter-mittency of the system dynamics. The patterns appeared in this process are signatured by the coexistence of agiant cluster, which comprises most of the nodes, and a few number of small clusters. The pattern evolutionis characterized by the giant cluster irregularly absorbs or emits the small clusters. As the coupling strengthleaves away from the synchronization bifurcation point, the giant cluster is gradually dissolved into a number ofsmall clusters, and the system dynamics is characterized bythe integration and separation of the small clusters.Dynamical mechanisms and statistical properties of the nonstationary pattern evolution are analyzed and con-ducted, and some scalings are newly revealed. Remarkably, it is found that the few active nodes, which escapefrom the giant cluster with a high frequency, are independent of the coupling strength while are sensitive to thebifurcation types. We hope our findings about nonstationarypattern could give additional understandings tothe dynamics of complex systems and have implications to some real problems where systems maintain theirnormal functions only in the unsynchronizable state.

PACS numbers: 89.75.-k, 05.45.Xt

I. INTRODUCTION

Synchronization of complex networks has aroused many in-terest in nonlinear science since the discoveries of the small-world and scale-free properties in many real and man-madesystems [1, 2]. In this study, one important issue is to ex-plore the inter-dependent relationship between the collectivebehaviors of the complex systems and their underlying topolo-gies. In particular, many efforts have been paid to the con-struction of optimal networks, and a number of factors whichhave important affections to the synchronizability of complexnetworks have been gradually disclosed. Now it is known thatrandom networks, due to their small average distances, aregenerally more synchronizable than regular networks [3, 4];and scale-free networks, with weighted and asymmetric cou-plings, can be more synchronizable than homogeneous net-works [5, 6]. In these studies, the standard method employedfor synchronization analysis is the master stability function(MSF), where the network synchronizability is estimated byan eigenratio calculated from the coupling matrix, and systemwhich has a smaller eigenratio is believed to be more synchro-nizable than that of larger eigenratio [7]. Inspired by this, toimprove the network synchronizability, the only task seemstobe upgrading the coupling matrix so as to decrease the eigen-ratio, either by changing the network topology [4] or by ad-justing the coupling scheme [5, 6].

The MSF method, while bringing great convenience to theanalysis, overlooks the temporal, local property of the sys-tem and reflects only partial information about the systemdynamics. Specifically, from MSF we only know ultimatelywhether the network is globally synchronizable or unsynchro-nizable, but do not know how the global synchronization is

reached if the network is synchronizable, or what’s the pat-tern and how it evolves if the network is unsynchronizable.These evolution details, or the transient behavior in systemdevelopment, contain rich information about the system dy-namics and may give additional insights to the organizationof complex systems. For instance, the recent studies aboutsynchronization transition have shown that, in the unsynchro-nizable states, heterogeneous networks are more synchroniz-able (have a higher degree of coherence) than homogeneousnetworks at small couplings, while at larger couplings the op-posite happens [8]. This crossover phenomenon of networksynchronizability are difficult to understood if we only lookat the final state of the system, but are straightforward if welook at the transient behaviors of their evolutions [8]. Besidesrevealing the synchronization mechanisms, the transient be-havior of network synchronization can also be used to detectthe topological scales and hierarchical structures in the realsystems, e.g., the detection of cluster structures in social andbiological networks [9, 10]. However, despite of its theoreti-cal and practical significance, the study of transient dynamicsof complex networks is still at its infancy and many questionsremain open, say for example, the pattern evolution of unsyn-chronizable complex networks.

Pattern formation in unsynchronizable but near-synchronization networks has been an important issuein studying the collective behavior of regular networks[11, 12]. By setting the coupling strength nearby thesynchronization bifurcation point, the system state sharesboth the dynamical properties of the synchronizable andunsynchronizable states: a state of high coherence but is notsynchronized. The bifacial dynamical property makes thisstate a natural choice in investigating the transition process ofnetworks synchronization. Previously studies about regular

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networks, say for example the lattices [11], have shown that,when the coupling strength is slightly out of the synchroniz-able region, although global synchronization is unreachable,nodes are still synchronized in a partial sense. That is, nodesare self-organized into a number of synchronous clusters.The distribution of these clusters, also called the synchronouspattern, is determined by a set of factors such as the couplingstrength, the system size and the coupling scheme. As thecoupling strength leaves away from the bifurcation point, thepattern structure becomes more and more complicated and thesystem coherence will be decreased, and finally reaches theturbulence state. It is worthy of note that the patterns arisenin regular networks have two common properties: spatiallysymmetric and temporally stationary. More specifically, thecontents of each cluster are fixed and the clusters are oftranslation symmetry in space. For this reason, we say thatthe patterns formed in regular networks are symmetric andstationary. These two properties, as have been discussed inthe previous studies [11, 12], are rooted in the symmetrictopology of the regular networks. This makes it interestingto ask the following question: how about the patterns inunsynchronizable complex networks?

Different to the regular networks, in complex networks weare not able to find any symmetry from their topologies. Theasymmetric topology, according to the pattern analysis devel-oped in studying regular networks [11], will induce two sig-nificant changes to the patterns: 1) the synchronous clusters, ifthey exist, will be asymmetric; and 2) all the possible patterns,including the one of global synchronization, are linearly un-stable under small perturbations. In other words,the patternsformed in complex networks are expected to be asymmetricand nonstationary. Our mission of this paper is just to under-stand and characterize the nonstationary patterns arisen in thedevelopment of complex networks. Specifically, we are tryingto investigate the following questions: 1) is there any patternarises during the system evolution? 2) the pattern is stationaryor nonstationary? if nonstationary, how is it evolving and howis it reflected from the system dynamics? 3) What happensto the pattern properties during the transition of network syn-chronization? and 4) How the coupling strength and bifurca-tion type affect the pattern properties? By investigating thesedynamical and statistical properties, we wish to have a globalunderstanding to the dynamics of unsynchronizable complexnetworks.

Our main findings are: 1) for coupling strength immedi-ately outside of the synchronizable region, the system dynam-ics undergoes the process of on-off intermittency. That is,most of the time the system stays on the global synchroniza-tion state (the ”off” state) but, once in a while, it developsinto a breaking state (the ”on” state) which is composed bya giant cluster and a few number of small clusters (hereafterwe call it the giant-cluster state). As the system develops,the giant cluster changes its shape by absorbing or emittingthe small clusters, leading to the ”off” or ”on” states, respec-tively; 2) the few active nodes which escape from the giantcluster with the high frequencies are coupling-strength inde-pendent but are bifurcation-type dependent. That is, in theneighboring region of a fixed bifurcation point, the locations

of these active nodes do not change with the coupling strength;if we change the coupling strength from nearby another bi-furcation point (the two bifurcation points will be explainedlater), their locations will be totally changed; 3) as couplingstrength leaves away from the bifurcation point, the giant clus-ter is gradually dissolved and more small clusters are gener-ated from it. Eventually, the giant cluster disappears and thepattern is composed by only the small clusters (hereafter wecall it the scattering–cluster state). During the course ofsys-tem evolution, each small cluster may either increase its sizeby integrating with other small clusters or decrease its sizeby breaking to even small clusters, but it can never reach tothe global synchronization state; 4) besides the giant cluster,the giant- and scattering-cluster states are also distinctin theirsmall clusters. For giant-cluster state the size of the smallclusters follows a power-law distribution, while for scattering-cluster state it follows a Gaussian distribution.

The rest of the paper is going to be arranged as follows. InSec. II we will give our model of coupled map network and,based on the method of MSF, point out the two bifurcationpoints and the transition areas that we are going to study with.In Sec. III we will employ the method of finite-time Lyapunovexponent to predict and describe the intermittent system dy-namics in the bifurcation regions. Direct simulations abouton-off intermittency will be presented in Sec. IV. By intro-ducing the method of temporal phase synchronization, in Sec.V we will investigate in detail the dynamical and statisticalproperties of the nonstationary pattern. Meanwhile, proper-ties of the giant- and scattering states will be compared andthe transition between the two states will be conducted. InSec. VI we will discuss the phenomenon of active nodes andinvestigate their dependence to the network properties. Dis-cussions and conclusions about pattern evolution in complexnetworks will be presented in Sec. VII.

II. COUPLED MAP NETWORKS AND THE TWOBIFURCATION POINTS

Our model of coupled map network is of the following form

xi(t + 1) = F(xi(t)) − ε∑

j

Gi,jH [f(xj(t))] . (1)

wherexi(t + 1) = F(xi(t)) is a d-dimensional map repre-senting the local dynamics on nodei, ε is a global couplingparameter,G is Laplacian matrix representing the couplings,andH is a coupling function. To facilitate our analysis, weadopt the following coupling scheme [13]:

Gi,j = −Ai,jk

βj

∑N

j=1 Ai,jkβj

, (2)

for j 6= i andGi,i = 1, with ki the degree of nodei andA the adjacent matrix of the network:Ai,j = 1 if nodei andj are connected andAi,j = 0 otherwise. In comparison withthe traditional coupling schemes, one important advantagewebenefit from this coupling scheme is that the synchronizabil-ity of the network, i.e. the eigenratio of the coupling matrix

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described in Eq. [2], can be easily adjusted by the parameterβ, while the network topology is kept unchanged. This advan-tage brings many convenience in network selection since fora given network topology, even though it is unsynchronizableunder the traditional schemes, can now be synchronizable byadjustingβ in Eq. [2]. This convenience is of particular im-portance when our studies of network dynamics are focusedon the bifurcation regions, where the network synchronizabil-ity should be deliberately arranged in order to demonstrateboth the two types of bifurcations. We note that the adop-tion of Eq. [2] is only for the purpose of convenient analy-sis, the findings we are going to report are general and canalso be observed by other coupling schemes given the net-work is properly prepared. In practice, we use logistic mapF(x) = 4x(1−x) as the local dynamics and adoptH(x) = xas the coupling function.

We first locate the two bifurcation points of global syn-chronization. The linear stability of the global synchroniza-tion state{xi(t) = s(t), ∀i} is determined by the correspond-ing variational equations, which can be diagonalized intoNblocks of form

y(t + 1) = [DF(s) + σDH(s)] y(t), (3)

with DF(s) andDH(s) the Jacobian matrices of the corre-sponding vector functions evaluated ats(t), andy representsthe different modes that are transverse to the synchronousmanifold s(t). We haveσ(i) = ελi for the ith block, i =1, 2, ..., N , andλ1 = 0 ≤ λ2 ≤ ... ≤ λN are the eigenvaluesof matrixG. The largest Lyapunov exponentΛ(σ) of Eq. [3],known as the master stability function (MSF) [7], determinesthe linear stability of the synchronous manifolds(t). In par-ticular, the synchronous manifold is stable if onlyΛ(ελi) < 0for eachi = 2, ..., N . The set of Lyapunov exponentsΛ(ελi)govern the stability of the synchronous manifold in the trans-verse spaces, and a positive value ofΛ(ελi) represents theloss of the stability in the transverse space of modei. It wasfound that for a large class of chaotic systems,Λ(σ) < 0is only fulfilled within a limit range in the parameter spaceσ ∈ (σ1, σ2). This indicates that, to make the global synchro-nization state linearly stable, all the eigenvaluesλi should becontained within range(σ1, σ2), i.e., λN/λ2 < σ2/σ1. Forthe logistic map employed here, it is not difficult to prove thatσ1 = 0.5 andσ2 = 1.5. Therefore, to achieve global syn-chronization, the coupling matrixG should be designed witheigenratioR ≡ λN/λ2 < σ2/σ1 = 3 = Rc.

Besides the condition ofR < Rc, to guarantee the synchro-nization, we also need to set the coupling strength in a properway: either small or large couplings may deteriorate the syn-chronization. Ifε < ε1 = σ1/λ2, the couplings are too weakto restrict the node trajectories to the synchronous manifold;while if ε > ε2 = σ2/λN , the couplings will be too strongand actually act as large perturbations to the synchronizationmanifold. Therefore, to achieve the global synchronization,we also requireε1 < ε < ε2. The two critical couplingsε1 andε2, which are named as the long-wave (LW) [14] andshort-wave (SW) bifurcations [15] respectively in the studiesof regular networks, thus stand as the boundaries of the syn-chronizable region. Our studies about network synchroniza-

0.80 0.85 0.90 0.95 1.000.4

0.5

1.2

1.4

1.6

2 = 0.95127

1 = 0.83488

2,

N

FIG. 1: For scale-free network ofN = 1000 nodes and of averagedegree< k >= 8, a schematic plot on the generation of the twobifurcation points as a function of the coupling strength. The long-wave bifurcation occurs at aboutε1 ≈ 0.83488 which is determinedby the conditionελ2 = σ1 = 0.5 (the lower line). The short-wavebifurcation occurs at aboutε2 ≈ 0.95127 which is determined bythe conditionελN = σ2 = 1.5 (the upper line).

tion will be focused on the neighboring regions of the twobifurcation points, i.e., the region ofε . ε1 or ε & ε2.

By the standard BA growth model [1], we construct a scale-free network of103 nodes and of average degree〈k〉 = 8. Bysettingβ = 2.5 in Eq. [2], we haveλ2 ≈ 0.6 andλN ≈ 1.58.Because ofR = λN/λ2 ≈ 2.6 < Rc, the network is glob-ally synchronizable. Also, because ofλ2 > σ1 andλN > σ2,both the two bifurcations can be realized by adjusting the cou-pling strength within rangeε ∈ (0, 1). In specific, whenε < ε1 ≈ 0.835, we haveελ2 < σ1 and ελN < σ2, thesynchronous manifold loses its stability at the lower boundaryof the synchronizable region and LW bifurcation occurs; andwhenε > ε2 ≈ 0.95, we haveελ2 > σ1 andελN > σ2, thesynchronous manifold loses its stability at the upper bound-ary of the synchronizable region and SW bifurcation occurs[Fig. 1]. In the following we will fix the network topologyand the parameterβ, while generating the various patterns bychanging the coupling strengthε nearby the two bifurcationpoints.

III. FINITE-TIME LYAPUNOV EXPONENT

Before direct simulations, we first give a qualitative de-scription (prediction) on the possible system dynamics in bi-furcation regions. To concrete our analysis, in the followingwe will only discuss the situation of SW bifurcation (ε . ε1),while noting that the same phenomena can be found at theLW bifurcation as well (ε & ε2) . In preparing the unsynchro-nizable states, we only letΛ(λ2) be slightly puncturing intothe unstable region, while keeping all the other exponents stillstaying in the stable region, i.e.,Λ(λ2) & 0 andΛ(λi) < 0for i = 3, ...N . With this setting, the synchronous manifoldis only desynchronized in the transverse space of mode2. As

4

such, the system possesses only two positive Lyapunov expo-nents, one isΛ(λ0) which is associated to the synchronousmanifold itself and another one isΛ(λ2). Noticing thatΛ(λ)are asymptotic averages, and, as so, they account only for theglobal stability properties, but do not warrant the possible co-herent sets arising in the system evolutions. These coherentsets, for regular networks, refer to the stationary, symmetricpatterns to which the system finally develops. While for com-plex networks, these sets can be the temporal, irregular clus-ters emerged in the process of system evolution.

In the region ofε . ε1, although global synchronization isunreachable, the system may still keep with the high coher-ence due to the existence of the synchronous clusters. Espe-cially, there could be some moments at which all the trajecto-ries are restrained to a small region in the phase space, veryclose to the situation of global synchronization. This vary-ing system coherence, however, can not be reflected from theasymptotic valueΛ(λ). To characterize the variation, we needto employ some new quantities which are able to capture thetemporal behavior of system. One of such quantities is thefinite-time Lyapunov exponent (FLE), a technique developedin studying chaos transition in nonlinear science [16]. In steadof asymptotic average, FLE measures the diverging rate ofnearby trajectories only in a finite time intervalT .

Λi =1

T

iT∑

t=(i−1)T

lnDH(s(t)). (4)

As our studies are focused on the situation of one-modedesynchronization, the stability of the synchronous manifoldand the temporal behavior that it displays are therefore ex-pected to be more reflected from the variation ofΛ2,i, theFLE that associates with mode2. With ε = 0.83 andT = 100, we plot in Fig. 2 the time evolution ofΛ2,i. Itis found that, although with a positive asymptotic value about〈Λ2,i〉 ≈ 6 × 10−3, the instant value ofΛ2,i penetrates to thenegative region at a high frequency. According to the differentsigns ofΛ2,i, the system evolution is divided into two types ofintervals: the divergent interval and the contractive interval.In the divergent intervals we haveΛ2,i > 0 and the systemdynamics is temporarily dominated by the divergence of thenode trajectories from the synchronous manifold; while in thecontractive intervals we haveΛ2,i < 0 and the system dynam-ics is temporarily dominated by the convergence of the nodetrajectories to the synchronous manifold.

The variation ofΛ2,i, reflected on the process of patternevolution, characterizes the travelling property of the sys-tem dynamics among the neighboring regions of two differentkinds of states: the desynchronization state and the synchro-nization state. In Fig. 2, the minimum value ofΛ2,i is about−0.07, during this contractive interval the node trajectorieswill converge to the synchronous manifold by an amount ofeminΛ2,iT ≈ e−7 ≈ 10−3 on average. Assuming that beforeentering this interval the average distance between the nodetrajectories is∆ (for logistic map we always have∆ < 1),then at the end of this interval the average distance is de-creased to∆ × 10−3, a small value which is usually over-shadowed by noise in practice. Due to this small distance,

0 400 800 1200 1600 2000-0.08

-0.04

0.00

0.04

0.08

2, i

i

FIG. 2: Forε = 0.83 in Fig. 1, the time evolution of the finite-time Lyapunov exponentΛ2,i calculated on intervals of lengthT =

100. It is observed that, while having the positive asymptotic value〈Λ2,i〉 > 0, the temporal value ofΛ2,i is penetrating into the negativeregion frequently.

the system can be practically regarded as already reached thesynchronization state. On the other hand, if the system entersa divergent interval, the node trajectories will diverge fromeach other and, at the end of this interval, their average dis-tance will be increased by an order of103. This large distancewill deteriorate the ordered trajectories (or the high coherenceof the system dynamics) that achieved during the contractiveintervals, and leading to the incoherent, breaking state. Thepattern of the breaking state, however, is not unique. Depend-ing on the initial conditions and the divergence intervals,thepattern may assume the different configurations. Therefore,based on the observations ofΛ2,i [Fig. 2] the dynamics ofunsynchronizable networks can be intuitively understood asan intermittent travelling among the synchronization state andthe different kinds of desynchronization states.

IV. ON-OFF INTERMITTENCY DESCRIBED BYCOMPLETE SYNCHRONIZATION

We now investigate the system dynamics by direct simula-tions. To implement, we first prepare the system to be stayingon the synchronization state. This can be achieved by adopt-ing a large coupling strength from the synchronizable region,i.e. ε1 < ε < ε2. After synchronization is achieved, we thendecreaseε to a value slightly below the bifurcation pointε1

and, in the meantime, an instant small perturbation is addedon each node. In practice, we take i.i.d (independent iden-tically distributed) noise of strength1 × 10−5 as the pertur-bations. After this, we release the system and let it developaccording to Eq. (1). Sinceε < ε1, the synchronization stateis unstable and, triggered by the noises, the node trajectoriesbegin to diverge from each other. The divergent trajectories,however, will frequently visit the neighborhood of the syn-chronous manifold, especially during those contractive inter-vals of smallΛ2,i [Fig. 2]. The intermittent system dynamics

5

is plotted in Fig. 3(a), where the average trajectory distance∆X = 1

N

∑N

i=1 xi − ~x is plotted as a function of time. Aswe have predicted from LLE, the system dynamics indeed un-dergoes an intermittent process. To characterize the intermit-tency, we plot in Fig. 3(b) the laminar-phase distribution ofthe∆X sequence plotted in Fig. 3(a). It is found that the lam-inar lengthτ (the time interval between two adjacent burstsof amplitude∆X(t) > 10−3) and the probabilityp(τ) for itto appear follow a power-law scalingp(τ) ∼ τ−γ . The fittedexponent is aboutγ ≈ −1.5 ± 0.05, with a fat tail at largeτdue to the finite simulating time.

In chaos theory, intermittent process of laminar-phase expo-nent−3/2 is classified as the ”on-off” intermittency, a typicalphenomenon observed in dynamical systems with a symmet-ric invariant set [17]. On-off intermittency is also reportedin chaos synchronization of regular networks, where the in-variant set refers to the synchronous manifold, and the ”off”state refers to the long stretches that the system dynamics isstaying nearby the synchronous manifold and the ”on” staterefers to the short bursts that the system dynamics is stayingaway from the synchronous manifold. Therefore, in terms oflaminar-phase distribution, the intermittency we have foundin complex networks [Fig. 3] has no difference to the that ofthe regular networks, despite of the drastic difference betweentheir topologies. We have also investigated the transitionbe-havior of the averaged distance〈∆X(t)〉 nearby the bifurca-tion points. As shown in Fig. 3(c), a linear relation between〈∆X(t)〉 andε is found in the region ofε . ε1. This lineartransition of the system performance, again, is consistentwiththe transition of regular networks [18]. Therefore, in terms ofcomplete synchronization, the on-off intermittency we havefound in complex networks has no difference to that of theregular networks.

V. PATTERN EVOLUTION IN COMPLEX NETWORKS

To reveal the unique properties of the system dynamics thatinduced by the complex topology, we go on to investigatethe pattern formation of unsynchronizable networks by themethod of temporal phase synchronization (TPS).

A. Temporal phase synchronization

TPS is defined as follows. Letxi(t) be the time sequencerecorded on nodei, we first transform it into a symbolic se-quenceθi(t) according to the following equations

θi(t) =

{

0, if xi(t) < 0.5,

1, if xi(t) ≥ 0.5.(5)

Then we divideθi(t) into short segments of the equal lengthn. Regarding each segment as an new element, we there-fore have transformed the long, variable sequencexi(t) intoa short, symbolic sequenceΘi(t

′). If at momentt′ we haveΘi(t

′) = Θj(t′), then we say that TPS is achieved between

0.820 0.825 0.830 0.835

0

3

6

9

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0 2000 4000 6000 8000 10000

0.00

0.05

0.10

10 100 1000

1

10

100

(c)

< X

>

(a)

x

t(b)

p

FIG. 3: (Color online) The on-off intermittency of the system dy-namics nearby the LW bifurcation atε = 0.83. (a) The time evolu-tion of the average trajectory distance∆X. (b) The laminar-phasedistribution of∆X, which follows a power-law scaling with the fit-ted exponent around3/2. (c) The transition behavior of the averagedistance〈∆X〉 nearby the LW bifurcation pointε1, where a linearrelation is found between the two quantities.

the nodesi and j. The collection of nodes which have thesame value ofΘ at momentt′ are defined as a temporarilysynchronous cluster, and all the synchronous clusters consti-tute the temporarily pattern of the system. During the courseof system evolution, the clusters will change their shapes andcontents and the pattern will change its configuration.

In comparison with the method of complete synchroniza-tion, the advantage we benefit from TPS is obvious: it makesthe synchronous pattern detectable. With complete synchro-nization, it is almost impossible for two nodes to have ex-

6

actly the same variable at the same time. Despite the fact thatat some moments the system has already reached the high-coherence states (formed during those contractive intervals inFig. 3(a)), with complete synchronization we are not able todistinguish these states from those low-coherence ones quanti-tatively (formed during those divergent intervals in Fig. 3(a)).(A remedy to this difficulty seems to define the clusters bythe method of threshold truncation, i.e., nodes are regardedas synchronized if the distance between their trajectoriesissmaller than some small value. However, this definition ofsynchronization will induce the problem of cluster idenfica-tion, as the same state may generate different patterns if wechoose the different reference nodes.) On the contrary, TPSfocuses on the loose match (phase synchronization) betweenthe node variables over a period of time. By requiring an ex-act match of the discrete variableΘ, the synchronous patternis uniquely defined; while by requiring the match of the longsequences ofθ, the ”synchronous” nodes are guaranteed witha strong coherence.

B. Pattern evolution of the giant-cluster state

With the same set of parameters as in Fig. 3(a), by themethod of TPS we plot in Fig. 4 the time evolutions of two ba-sic quantities of pattern evolution: the number of synchronousclustersnc and the size of the largest clusterLmax. It isfound that, similar to the phenomenon in complete synchro-nization [Fig. 3(a)], on-off intermittency is also found intheTPS quantitiesnc andLmax. In Fig. 4(a) it is shown that mostof the time the system is broken into only a few number ofclusters,nc = 2 or 3, while occasionally it is broken into aquite large number of clusters,10 < nc < 50, or united to thesynchronization state,nc = 1. The intermittent pattern evo-lution is also reflected on the sequence ofLmax [Fig. 4(b)],where most of the time the size of the giant cluster is aboutLmax ≈ N , while occasionally it decreases to some smallvalues ofLmax < N/2. As we have discussed previously,the main advantage we benefit from TPS is in identifying theclusters. This advantage is clearly shown in Figs. 4(a) and(b), where for any time instant the two quantitiesnc andLi

are uniquely defined. Besides cluster identification, we alsobenefit from TPS in quantifying the synchronization degrees.In specific, the different coherence states shown in Fig. 3(a)now can be clearly quantified: high coherence states are thoseof smallernc or largerLmax. Specially, the synchronizationstate now is unambiguously defined as the moments ofnc = 1in Fig. 4(a) or, equally, the moments ofLmax = N in Fig.4(b).

We go on to investigate the pattern evolution by statisticalanalysis. The first statistic we are interested is the laminar-phase distribution of the synchronization state, i.e. the timeintervals thatnc = 1 in Fig. 4(a) orLmax = N in Fig. 4(b).In its original definition, laminar phase refers to the time inter-vals τ that all node trajectories stays within a small distancefrom the synchronous manifold, therefore the actual value ofτ is varying with the predefined threshold distance. This un-certainty is overcome in TPS. As shown in Fig. 4(a), in TPS

0 1x105 2x105 3x105

0

25

50

0 1x105 2x105 3x105

200

400

600

800

1000

(a)

n c

t'

(b)

L max

t'

FIG. 4: For the same set of parameters as in Fig. 3(a). The timeevolutions of the TPS quantities. (a) The number of the synchronousclustersnc and (b) the size of the giant clusterLmax. The synchro-nization state is defined as the momentsnc = 1 in (a) orLmax = Nin (b).

the ”off” state refers to the moments ofnc = 1 specifically.The laminar-phase distribution ofnc is plotted in Fig. 5(a). Inconsistency with the distribution of complete synchronization[Fig. 3], the laminar-phase distribution ofnc also follows apower-law scaling and has the same exponentγ ≈ −1.5±0.1.Therefore the use of TPS, while bringing convenience to thepattern analysis, still capture the basic properties of theon-offintermittency. The second statistic we are interested is the sizedistribution of the largest cluster, an important indicator forthe coherence degree of the system. For theLmax sequenceplotted in Fig. 4(b), in Fig. 5(b) we plot the size distributionof Lmax. It is seen that the probability of finding large clus-ter Lmax ≈ N is much higher than that of small cluster ofLmax < 500. In particular, the probability for finding clustersof Lmax > 990 is about20 percent and forLmax > 990 itis about70 percent. Therefore, in the region ofε . ε1, thedistinct feature of the system patterns is the existence of agi-ant cluster. Due to this special feature, we call these states thegiant-cluster state.

Besides the giant cluster, we are also interested in the prop-erties of the small clusters. We plot in Fig. 5(c) the distribu-tion of nc and in Fig. 5(d) the size distribution of the smallclustersLi that surround the giant cluster in the pattern. Asshown in Fig. 5(c), the distribution ofnc follows a power-law

7

100 101 10210-1

101

103

105

0 500 100010-1

100

101

102

103

1 10 10010-1

101

103

105

1 10 100 1000

102

103

104

105

p' (

)(a)

L max

(b)

p L

nc

p nc

(c)

p Li

L i

(d)

FIG. 5: (Color online) Statistical properties of the on-offintermit-tency plotted in Fig. 4. (a) The power-law scaling of the laminar-phase distribution ofnc. The fitted slope is about−2.3 ± 0.05. (b)The size distribution of the size of the giant cluster. (c) The power-law distribution of the number of small clustersnc. The fitted slopeis about−3±0.1. (d) The power-law scaling on the size distributionof the small clusters. The fitted slope is about−1.2 ± 0.01.

scaling with the fixed exponent is aboutγ ≈ −3 ± 0.05. Theheterogeneous distribution ofnc indicates that, in the giant-cluster state, the system is usually broken into only a few num-ber of clusters. An interesting finding exists in the size distri-bution of the small clusters. As shown in Fig. 5(d), in rangeLi ∈ [1, N/2] a power-law scaling is found betweenPLi andLi, with the fitted exponent is aboutγ ≈ −1.1 ± 0.05. Thedistribution ofLi confirms the finding of Fig. 5(c) that thesmall clusters which join or separate from the giant clusterareusually of small size.

Combining the findings of Fig. 4 and Fig. 5, the pictureof pattern evolution in the bifurcation regionε . ε1 now be-comes clear. Generally speaking, the evolution can be dividedinto two opposite dynamical processes happening around thegiant cluster: the separation and integration of the small clus-ters. During the separation process, the small clusters arees-caped from the giant cluster, which weakens the dominant roleof the giant cluster and makes the pattern complicated. How-ever, the separated clusters occupy only a small proportionofthe nodes [Fig. 5(c)], the majority nodes are still attachedtothe giant cluster, which sustains the synchronization skeletonand keeps the system on the high coherence states. At somerare moments the giant cluster may disappears, and the pat-tern is composed by only small clusters ofLi < N/2. Atthese moments, the synchronization skeleton is broken, thepattern becomes even complicated and the system coherencereaches its minimum. In contrast, during the process of clusterintegration, the giant cluster will increase it size by attractingthe small clusters , and gradually towards the state of globalsynchronization. It should be noticed that the separation andintegration processes are uneven and are typically occurringat the same time. For instance, during the separation process,while the system evolution is dominated by the separation of

new small clusters from the giant cluster, there could be somesmall clusters rejoin to the giant cluster.

C. Pattern evolution of the scattering-cluster state

As we further decrease the coupling strength fromε1, thepicture of pattern evolution will be totally changed. Withε = 0.79, we plot in Fig. 6 the same statistics as in Fig. 5.The first observation is the loss of the global synchronizationstate, as can be found from the time variation ofnc plotted inFig. 6(a). The loss of global synchronization becomes evenclear if we compare Fig. 6(a) with Fig. 4(a): in Fig. 6(a),except the moment att = 0, the system can never reach thesynchronization state atnc = 1 and very often it is broken intoa large number of small clusters at aboutnc ∼ 102. The factthat the pattern is decomposed into a large number of smallclusters is also manifested by the distribution ofnc, as plottedin Fig. 6(b). Instead of the power-law distribution found inthegiant-cluster state, in the scattering-cluster statenc follows aGaussian distribution [Fig. 6(b)]. Asε further decreases fromε1, the mean value ofnc will shift to the larger values, asindicated by theε = 0.78 curve plotted in Fig. 6(b). The sec-ond observation is the disappearance of the giant cluster. Asshown in Fig. 6(c), the size distribution of the largest clusteralso follows a Gaussian distribution, with its mean value lo-cates at〈Lmax〉 < N/2. The distribution of Fig. 6(c) is verydifferent to that of Fig. 5(b), where in Fig. 5(b) the largest(gi-ant) cluster has sizeLmax ≈ N in most of the time. Asε de-creases, the mean value of the largest cluster〈Lmax〉 will shiftto small values and the variance ofLmax will be decreased, asindicated by theε = 0.78 curve plotted in Fig. 6(c). Similar toplot of Fig. 5(d), we have also investigated the distribution ofLi, the sizes for all the small clusters appeared in the systemevolution [Fig. 6(d)]. It is found that the distribution ofLi fol-lows a power-law distribution forLi < N/2, while having anexponential tail forLi > N/2. Numerically we find that theexponent of the power-law section, i.e. in rangeLi ∈ [1, 200),is about−2± 0.05, while the fitted exponent for the exponen-tial section is about−4.5 × 10−3 ± 2 × 10−5. These twoexponents, however, are changing withε. As ε decreases, thetwo exponents will shift to some small values.

Combining Fig. 5 and Fig. 6, we are able to outline thetransition process of network synchronization nearby the bi-furcation points, i.e., the transition from the giant-cluster stateto the scattering-cluster state asε leaves away fromε1. Inthe region ofε . ε1, the pattern is composed by a giantcluster and a few number of small clusters, i.e. the giant-cluster state. Asε decreases fromε1 gradually, more andmore small clusters will be emitted out from the giant clus-ter and, as a consequence, both the size of the giant clusterand the fraction of synchronization time will be decreased.Then, at aboutεc ≈ 0.832, the giant cluster disappears andthe pattern of the system is composed by several larger clus-ters, of sizeLmax . N/2, together with many small clustersof heterogenous size distribution, i.e. the scattering-clusterstate. After that, asε decreases fromεc, the clusters shrinktheir size by breaking into even small clusters, and the pat-

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nc

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T

FIG. 6: (Color online) The dynamical and statistical properties ofpattern evolution forε = 0.79. (a) The time evolution ofnc. (b) TheGaussian distribution of the number of the small clustersnc. (b) TheGaussian distribution of the size of the largest clusterLmax. (d) Thetwo-segment distribution on the size of the small clustersLi. In theregion ofLi < 200, Li follows a power-law distribution with fittedexponent is about−2 ± 0.05; while for Li > 200, the distributionis exponential with the fitted exponent is about−4.5 × 10

−3 ± 2 ×10

−5. As ε further decreases fromε1, the largest cluster becomeseven smaller and more small clusters are emitted out from it.Asillustrated by theε = 0.78 curves plotted in (b), (c) and (d).

tern becomes even complicated. The detail transition from thegiant-cluster state to the scattering-cluster state is presentedin Fig. 7, where the average number of clusters that the sys-tem is broken into〈nc〉, Fig. 7(a), and the average size ofthe largest cluster〈Lmax〉, Fig. 7(b), are plotted as a func-tion of the coupling strength in the LW bifurcation region.The transition is found to be smooth and steady, just as wehave expected. Besides the giant cluster, another differencebetween the giant-cluster and scattering-cluster states existsin their pattern evolutions. In the giant-cluster state, whilethe configuration of the giant cluster is continuously updatedby emitting or absorbing the small clusters, its main contentsare stable and do not change with time. In contrast, in thescattering-cluster state the small clusters integrate with or sep-arate from each other in a random fashion. Although occa-sionally there could be some large clusters show up in the pat-tern of the scattering-cluster state [Fig. 6(d)], these ”large”clusters, however, are very fragile and will break into smallclusters again in a short time. This quick-dissolving prop-erty stops the scattering-cluster state from having a high co-herence.

VI. CHARACTERIZING THE ACTIVE NODES

In the giant-cluster state, most of the nodes are organizedinto the giant cluster while few nodes, either in forms of smallgroup or isolated node, are separating from or joining to thegiant cluster with a high frequency. These active nodes, al-

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FIG. 7: The transition process of the network synchronization nearbythe LW bifurcation pointε1. (a) The average number of clusters thatthe system is broken into as a function of coupling strength.(b) Theaverage size of the largest cluster as a function of couplingstrength.Each date is an averaged result over10

8 time steps.

though are few in amount, play an important role in networksynchronization. Clearly, a proper characterization of thesenodes will deepen our understandings on the system dynam-ics and give indications to the improvement of network perfor-mance. For instance, to improve the synchronizability of thesystem, we may either remove the few most active nodes fromthe network, or update their coupling strengths specifically.

In characterizing the active nodes, the following propertiesare of general interest: 1) what’s the dependence of the nodeactivity on the network topology? can we characterize thesenodes by the known network properties such as node degreeor betweenness? 2) are their locations sensitive to the couplingstrength? and 3) what’s the effect of bifurcation type on theirlocations? In the following we will explore these questionsbynumerical simulations.

We first try to characterize the active nodes by their topo-logical properties. For the giant-cluster state describedin Fig.4, we plot in Fig. 8(a) the probabilitypu1 that each node staysin the giant cluster. While the majority nodes stay in the gi-ant cluster with a high probabilitypu1 ≈ 1, few nodes areof unusually small probabilities:1 percent of the nodes havepu1 < 0.8. One important observation of Fig. 8(a) is thatthe locations of the active nodes are entangled with those of

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i

FIG. 8: (Color online) The properties of the active nodes. (a) For thegiant-cluster state shown in Fig. 4, the probability that node stays inthe giant cluster versus the node index. (b) A segment of (a) but withdifferent coupling strengths nearby the LW bifurcation point ε1. (c)For the giant-cluster state (ε = 0.952) nearby the SW bifurcationpoint, the probability that node stays in the giant cluster versus thenode index. (d) A segment of (c) under different coupling strengthsnearby the SW bifurcation pointε2.

the stable nodes. Noticing that in the BA growth model nodeof higher index in general assume the smaller degree, the ob-servation of Fig. 8(a) therefore indicates the independence ofthe node degree to the node stability, or the inaccuracy of us-ing degree to characterize the node activity. Specifically,inFig. 8(a) the5 most unstable nodes, by a descending order ofpu1, are those of degreesk = 47, 36, 26, 10, 4, respectively.Except the one ofk = 4, all the other nodes have higher de-grees. Another well-known topological property of complexnetwork is the node betweenness, which counts the numberof shortest pathes that pass through each node and actuallyevaluates the node importance from the global-network pointof view. This global-network property, however, is also inca-pable to characterize the active nodes. In Tab. 1 we list thedetail information about the5 most active nodes in Fig. 8(a),where the inaccuracy of node degree or node betweenness incharacterizing the active nodes are summarized.

TABLE I: For the attaching probabilitypi plotted in Fig. 8(a), welist the5 most unstable nodes and try to characterize them by a set oftopological quantities including the node indexi, the attaching prob-ability pi, the stability rankpi rank, the node degreeki, the degreerank ki rank, the node betweennessBi, and the betweenness rankBi rank.

Node indexi pi pi rank ki ki rank Bi Bi rank

615 0.72797 1 5 39→537 1301 280

762 0.74424 2 5 39→537 1375 356

680 0.75416 3 4 1→338 1254 680

372 0.7591 4 6 538→645 1440 406

938 0.75972 5 4 1→338 1215 159

We go on to investigate the affection of the couplingstrength on the locations of the active nodes. In Fig. 8(b) wefix the network topology and compare the node activities un-der different coupling strengths nearby the bifurcation pointε1. It is found that, despite of the changes inpu1, the lo-cations of the active nodes are kept unchanged. That is, theactive nodes are always the first ones to escape from the gi-ant cluster whenever the network is unsynchronizable. Wehave also investigated the affection of the bifurcation type onthe locations of the active nodes. By choosing the couplingstrength nearby the SW bifurcationε = 0.952 & ε2, we plotin Fig. 8(c) the node attaching probabilitypu2 as a functionof the node indexi. An interesting finding is that, comparingto the situation of LW bifurcation [Fig. 8(a)], the locationsof the active nodes have been totally changed in Fig. (c). InTab. 2 we list the detail information about the5 most activenodes in Fig. 8(c), again their locations can not be predictedby the node degree or betweenness. Similar to the LW bifur-cation, the locations of the active nodes are also independentto the coupling strength at the SW bifurcation, as shown inFig. 8(d).

TABLE II: Similar to Tab. I but for the attaching probabilitypi plot-ted in Fig. 8(c). Comparing to Tab. I, one important observation isthe changed locations of the active nodes due to the changed bifurca-tion type.

Node indexi pi pi rank ki ki rank Bi Bi rank

43 0.78196 1 9 779→813 2847 813

35 0.78969 2 18 936→940 8513 953

714 0.795 3 4 1→338 1215 158

130 0.79652 4 13 886→901 4200 886

154 0.19944 5 10 814→846 2898 815

Previous studies about network synchronization haveshown that, while individually it is difficult to predict thedy-namical behavior of each node, the average performance ofan ensemble of nodes of the same network properties do havesome reliable characters. For instance, it has been shown thatin complex networks the high-degree nodes are on averagemore synchronizable than the low-degree ones [19]. Regard-ing to the problem of node activities, it is natural to ask thesimilar question: are the high-degree nodes more synchro-nized than the low-degree nodes? In Fig. 9 we plot the av-erage attaching probability〈pu1〉k as a function of degreek.Still, we can not find a clear dependence of〈pu1〉k onk.

VII. DISCUSSIONS AND CONCLUSION

It is worthy of note that our studies of active nodes are onlyfocused on the giant-cluster state, and the purpose is to under-stand their dynamics and reveal their properties. By ensemble

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FIG. 9: (Color online) The average attaching probability〈pu1〉k asa function of node degreek. On average, the5 most unstable nodesare those of degreesk = 47, 36, 26, 10, 4. Still, we can not find aclear dependence between〈pu1〉k andk.

average, we may able to improve our prediction of the activenodes, say for example the dependence of〈pu1〉k onk in Fig.9 may be smoothed if we average the results over a large num-ber of network realizations. Such an improvement, however,comes at the cost of the decreased prediction accuracy due tothe increased candidates. Taking Fig. 9 as an example, al-though it is noticed that nodes ofk = 4 in general are moreactive than those of other degrees, only one of them is listedas the5 most unstable nodes [Tab. 1]. In specific, among thetotal number of338 nodes which have degreek = 4, mostof them are tightly attracted to the giant cluster (90 percent ofthem have attaching probabilitiespu1 > 0.95). Therefore, interms of precise predication, the average method is infeasiblein practice.

Beside node degree and betweenness, we have also checkedthe dependence of the property of node activity to some otherwell-known network properties such as the clustering coeffi-cient, the modularity, and the assortativity. However, none ofthem is suitable to characterize the active nodes, their perfor-mance is very similar to that of the node degree described inTab. 1 and Tab. 2. Our study thus suggests that, to give a pre-cise prediction to the active nodes, we may need to developsome new quantities.

Despite of the amount of studies carried on network syn-chronization, to the best of our knowledge, we are the first tostudy the nonstationary pattern in unsynchronizable complex

networks. In Ref. [9] the authors have discussed the transientprocess of global synchronization in complex networks, buttheir study are concentrating on the synchronizable state inwhich, during the course of system evolution, small clustersintegrate into larger clusters monotonically and finally reachthe synchronization state. After that, the system will alwaysstay on the synchronization state. Our works are also differentto the studies of Refs. [10, 20]. Similar to our works, in thesestudies the authors also consider the problem of pattern for-mation in unsynchronizable networks, but their interests arefocused on thestationary pattern of the system. That is, thesize and contents of the clusters do not change with time. Incontrast, in our studies both the size and contents of the clus-ters are variable.

In summary, we have reported and investigated a kind ofnew phenomena in network synchronization: the nonstation-ary pattern. That is, the final state of the network settles nei-ther to the synchronization state nor to any stationary stateof fixed pattern, the system is travelling among all the possi-ble patterns in an intermittent fashion (the pattern can be ofany configuration, but its probability of showing up is pattern-dependent). We attribute this nonstationarity to the asymmet-ric topology of the complex networks, and its dynamical ori-gin can be understood from the property of the finite-timeLyapunov exponent associated to the desynchronized mode.Two types of synchronization formats, the complete synchro-nization and the temporal phase synchronization, have beenemployed to detect the nonstationary dynamics. For couplingstrength immediately out of the stable region, the pattern evo-lution is characterized by the process of on-off intermittencyand the existence of the giant-cluster; while if the couplingstrength is far away from the bifurcation points, the patternevolution is signatured by the random interactivities amongthe number of small clusters. A remarkable finding is that,in the giant-cluster state the locations of the active nodesareindependent of the coupling strength but are sensitive to thebifurcation types. The active nodes, however, can not be char-acterized by the currently known network properties, furtherinvestigations about their identification are necessary. Whilewe are hoping our studies about nonstationary pattern couldgive some new understandings to the dynamics of coupledcomplex systems, we also hope that our findings about unsyn-chronizable networks could be used to some practical prob-lems where system maintains their normal functions only un-der the unsynchronizable states, for example the problem ofepileptic seizers [21].

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