Role of time scales and topology on the dynamics of complex networks
aa r X i v : . [ n li n . AO ] A p r Role of time scales and topology on the dynamics of complex networks
Kajari Gupta and G. Ambika
1, 2, a) Indian Institute of Science Education and Research (IISER) Pune, Pune-411008,India Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati-517507,India (Dated: 23 April 2019)
The interplay between time scales and structural properties of complex networks of nonlinear oscillators cangenerate many interesting phenomena, like amplitude death, cluster synchronization, frequency synchroniza-tion etc. We study the emergence of such phenomena and their transitions by considering a complex networkof dynamical systems in which a fraction of systems evolves on a slower time scale on the network. Wereport the transition to amplitude death for the whole network and the scaling near the transitions as theconnectivity pattern changes. We also discuss the suppression and recovery of oscillations and the cross overbehavior as the number of slow systems increases. By considering a scale free network of systems with multipletime scales, we study the role of heterogeneity in link structure on dynamical properties and the consequentcritical behaviors. In this case with hubs made slow, our main results are the escape time statistics for lossof complete synchrony as the slowness spreads on the network and the self-organization of the whole networkto a new frequency synchronized state. Our results have potential applications in biological, physical, andengineering networks consisting of heterogeneous oscillators.PACS numbers: 43.25.-x, 43.25.+y, 89.75.-k, 64.60.aq, 89.75.Fb, 05.45.DfKeywords: Complex networks, Nonlinear systems, Multiple time scales, Frequency synchronization, Self-organization, Amplitude death
Complexity of real world systems are studiedmainly in terms of the nonlinearity in the intrin-sic dynamics of their sub systems and the com-plex interaction patterns among them. In suchsystems, the variability and heterogeneity of theinteracting sub systems can add a further level ofcomplexity. In this context heterogeneity arisingfrom differing dynamical time scales offers sev-eral challenges and has applications in diversefields, ranging from biology, economy, sociologyto physics and engineering. In our work, we studythe interesting cooperative dynamics in interact-ing nonlinear systems of differing time scales us-ing the frame work of complex networks.
I. INTRODUCTION
Multiple-timescale phenomena are ubiquitous in Na-ture and their in-depth understanding brings in sev-eral novel challenges. Some of the examples ofsuch phenomena in real world systems are neuronalelectrical activity , hormonal regulation , chemi-cal reactions , turbulent flows and populationdynamics etc. Although there have been iso-lated studies addressing its various aspects, there arestill many interesting questions that demand multidisci-plinary approaches. Several modelling frameworks have a) Electronic mail: [email protected] proposed methods to understand dual time scale phe-nomena in single systems, like dynamical models for neu-ronal dynamics . However, studies on collective behav-ior of connected systems that differ in their intrinsic timescales, are very minimal with many open questions. Inthis context, the framework of complex networks providesa promising tool to study nonlinear multiple time scaledynamics.The emergence of synchronization in interacting dy-namical units is important for the functionality ofmany systems and coupled oscillator networks are of-ten studied to understand their dynamics . Severaltypes of synchronization phenomena like complete, phaseand generalized synchronization , as well as clustersynchronization have been studied in various con-texts. However, frequency synchronization is of recentinterest and has relevance in many realistic situationsranging from neuronal systems to power grids , wherethe individual oscillators can have non-identical naturalfrequencies.The suppression of oscillations or amplitude death is also an emergent phenomenon that has interesting im-plications. In an assembly of coupled systems, amplitudedeath emerges mainly due to specific nature of couplinglike transmission delay, processing delay, dynamic cou-pling, nonlinear coupling, environmental coupling, etc.or due to parameter mismatch or time scalediversity . Most often, amplitude death is requiredto suppress unwanted oscillations in connected systems.In this context we note that the revival of oscillationsis an equally important and related emergent phenom-ena in coupled oscillators. Often in many systems re-covering oscillations from suppressed state is necessaryfor their proper functioning, to maintain the output in-tensity in arrays of power generators, to get maximumoutput from coupled laser systems even with transmis-sion delay, to sustain oscillations in interacting cardiaccells etc. . Recently a few mechanisms to reviveoscillations from quenched state in coupled systems havebeen reported .Interestingly, heterogeneity of interacting systemsplays an important role in the diversity and organizationin many complex systems . The dynamical processesunderlying their complexity often display phase transi-tions and analysis and prediction of such transitions arecrucial for their proper functioning. Heterogeneity canarise from the pattern of interactions or nature of con-nectivity among the systems. Thus, in scale-free net-works, the heterogeneity is mostly from the broad dis-tribution of node degrees such that there exists a fewnodes with very high degrees. In addition, heterogene-ity due to different dynamical time scales is prevalent inmany complex systems. It is shown that functional hier-archy emerges through a form of self-organization of twodistinct types of neurons, with multiple timescales andhence not only the spatial connections between neuronsbut also the timescales of neural activity are important inthe mechanisms leading to functional hierarchy in neu-ral systems . So also, widely different time scales arecommon in systems of chemical reactions. In a recentstudy in biochemical networks, the fastest time scales isshown to correspond to the chemical equilibrium betweenmetabolites while the slower ones relate to more physio-logically relevant transformations . This brings out theneed to analyse correlations between metabolites consid-ering the characteristic time scales of the network. Theheterogeneity in network connectivity is shown to resultin frequency synchronized clusters in non-identical phaseoscillators .The main motivation of the present study is to an-alyze the onset and characterization of interesting col-lective dynamics or emergent behaviour in a network ofconnected systems with differing dynamical time scales.Thus the heterogeneity in dynamical time scales thatwe introduce is different from earlier studies where con-nected systems with nonidentical intrinsic frequenciesare considered. In two coupled slow and fast systemsand in minimal networks of three or four systems, onsetamplitude death and occurrence of frequency synchro-nized states with tunable emergent frequencies are stud-ied recently . By considering two types of complexnetworks, we study the interplay between heterogeneityin link structure or connectivity among the systems andtheir differing time scales that can lead to interestingcollective behaviour. Our study primarily uses standardperiodic oscillators of R¨ossler and Landau Stuart typeso that it brings out the amplitude variations and theircross over behaviours as the slowness factor increases.We find the difference in time scales and the hetero-geneity in connectivity together can drive the whole net-work to frequency synchronized clusters. Increasing the heterogeneity in time scales by increasing the number ofslow systems or the mismatch in time scales, the wholenetwork settles to a state of no oscillations. The transi-tions to that state as well as recovery to slower oscilla-tions with cross over in amplitudes are some of the inter-esting results of the study.We also address the important question of what hap-pens if part or even one node of a scale free network ofsystems suddenly slows down and then how does it affectthe performance of the whole network. In this case, therobustness of the network to such changes is studied interms of the time taken for each node to escape from thesynchronized state leading the whole network to desyn-chronized dynamics. We find this phenomenon of loss ofsynchrony settles in a time that decreases with the degreeof the node that becomes slow first. The desynchroniza-tion transition that happens over a characteristic time isfollowed by the reorganisation of the whole network toa frequency synchronized state and this self-organizationtime depends on the difference in time scales. In addi-tion, we study the transition to amplitude death on ascale free network, with multiple time scales, drawn froma normal distribution and with time scales decided bythe connectivity of the systems. II. RANDOM NETWORKS OF SLOW AND FASTPERIODIC SYSTEMS
We construct a random network of N nodes where eachnode represents an n dimensional dynamical system. Thedynamical equation of each node is taken as˙ X i = τ i F ( X i ) + Gǫτ i N X j =1 A ij ( X j − X i ) (1)where i=1, 2, ..... N. F ( X i ) represents the intrinsic dy-namics of each node. A ij represents the adjacency matrixof connections in the network with its elements havingvalues 1, if the nodes i and j are connected and zerootherwise. The topological connectivity of the randomnetwork is defined by a parameter p , where p is the prob-ability with which any two nodes of the network are con-nected. G is diag(1,0,....) since we consider diffusivecoupling between the first variables of the systems. Thesystems on the network are identical in their intrinsic dy-namics except that they evolve at different time scales, t i ,which are scaled as t i = τ i t to get the equation1. Then τ i will function as a parameter defining the dynamicaltime scale of evolution of the i th system. We vary τ i inthe range (0.1,1) to introduce time scale diversity in thenetwork of systems.Choosing periodic R¨ossler system as nodal dynamics,the equations that govern the dynamics are˙ x i = τ i ( − y i − z i ) + τ i ǫ N X j =1 A ij ( x j − x i )˙ y i = τ i ( x i + ay i )˙ z i = τ i ( b + z i ( x i − c )) (2)With the parameters chosen as a=0.1, b=0.1 and c=4,the intrinsic nodal dynamics is periodic.First, we consider the case of dual time scales, suchthat in the network, out of N identical systems, m evolveon a slower time scale. The subset of oscillators withslower time scale is taken as S. Thus τ i = τ if i be-longs to the set S and τ i = 1 for other nodes. Then thevalue of τ indicates the extend of mismatch in time scalesbetween the two sets of oscillators, smaller values of τ corresponding to larger mismatch. We analyse how theslowness of m of the systems can affect the dynamics ofthe whole network. For this the system of equations ineqn.(2) are integrated using Adams-Moulton-Bashforthmethod, with time step 0.01 for 100,000 times and thelast 10,000 values of the x-variables are used for calcula-tions in the study. A. Suppression and recovery of oscillations
In this section, we report the general results on the cou-pled dynamics of the systems by varying the parametersinvolved, time scale of slow systems ( τ ), number of slowsystems ( m ) and coupling strength of connections( ǫ ),keeping the probability of connection of the network p = 0 .
5. We find that for sufficient time scale mismatchbetween slow and fast subsets of systems, for strong cou-pling, for a range m , all the systems go to a synchronizedfixed point. This state is generally known as amplitudedeath(AD) in the context of coupled dynamics. TheFig. 1 shows the state of amplitude death in the randomnetwork of slow and fast systems. -6-4-2 0 2 4 6 0 10 20 30 40 50 60 70 80 x t FIG. 1. (colour online) Time series of x variables of periodicR¨ossler systems in a random network of slow and fast systemsshowing amplitude death state for m =50, p = 0 . τ =0.35, ǫ = 0 . We calculate the difference between global maximumto global minimum, A diff , from the time series of eachoscillator. This averaged over all the N systems in thenetwork serves as an index to identify onset of AD in the network, since < A diff > = 0 would correspond to ADstate in the whole network. Using this we identify the re-gion for occurrence of AD for different m , the number ofslow systems present in the network, with suitable valueschosen for the other parameters, p , τ , ǫ . We plot this re-gion for two sets of values of τ and ǫ in Fig. 2. This shows m FIG. 2. (colour online)Variation of average A diff with m .Here τ =0.35 and ǫ =0.01 for red curve and τ =0.35 and ǫ =0.05for green curve. N=100, p =0.5. that a minimum number of slow systems is required forAD to occur, denoted as m . As m increases the het-erogeneity decreases and the network recovers from ADstate beyond a certain value, m . Thus, suppression ofdynamics happens as m reaches a critical minimum value m and recovery to oscillatory state happens beyond thesecond critical value, m ; both these values depend onother parameters like τ , p and ǫ of the system.With m chosen from the region of AD in Fig. 2, weisolate the region of AD in ( τ, ǫ ) plane, for a chosen p =0 .
5, and this is shown as region 1 in Fig. 3. ε τ FIG. 3. (colour online)Regions of different dynamical statesin the parameter plane ( τ, ǫ ). Region 1 corresponds to AD,2 corresponds to frequency synchronization, 3 leads to insta-bility and 4 is a transient state where systems start divergingfrom fixed point state before reaching unstable state. Here m =50, p =0.5,N=100. B. Frequency synchronized dynamics
In this section we present the possible dynamical statesoutside the region of AD in the parameter plane ( τ, ǫ ).When coupling is strong and time scale mismatch issmall, all the systems in the network settle to an orga-nized state with oscillations of differing amplitudes butsame frequency. This state of frequency synchronizationis seen in region 2 in the Fig. 3. This is identified by cal-culating the frequency of each oscillator from its x-timeseries using equation, ω = 1 K K X k =1 π ( t k +1 − t k ) (3)where t k is the time of the k th zero crossing point in thetime series of the oscillator and K is the total numberof intervals for which the zero crossings are counted. Inthis state, the oscillations of slow systems are relativelycloser in phase and so are fast oscillators among them-selves but the phase difference between slow and fast setsis relatively large. Below region 2, with low couplingstrength the oscillators show a two-frequency state andas time scale mismatch increases they become periodicwith two separate time scales. For very high couplingstrength in the region marked as 3, network becomes un-stable but before this there is region for low τ (region 4),where the systems are in a transient state, diverging fromAD (Fig. 3). C. Crossover phenomena in the emergent dynamics forlarge m When the systems are in the state of frequency syn-chronized oscillations, corresponding to region 2 in Fig. 3,the amplitudes of slow and fast sets of systems vary fromeach other. In general, for low m , we observe that am-plitudes of slow systems are smaller than those of fastsystems, while for higher m this behavior gets reversedwith the slow set having larger amplitudes than the fastone. Thus, we observe a novel phenomenon of crossoverbehavior in the amplitudes as m is varied. To show thisexplicitly, we study the average amplitude of all the slowsystems and that of all the fast systems as m is variedkeeping the values of τ and ǫ in frequency synchroniza-tion state. We find that at a critical value of m , theamplitude of slow and fast systems undergoes a reversalas shown in Fig. 4.When m=0 or N, the systems are all completely syn-chronized. For other values of m, there is approximatesynchronization among systems in each set, slow and fast,but are phase shifted between them. Then the numberof systems included in the coupling that come with mis-match or the number of systems from the other set, is ef-fectively deciding the amplitude of oscillations. Thus forsmall m, the slow systems have smaller amplitude due to m FIG. 4. (colour online)Crossover in magnitudes of amplitudesof slow and fast sets of oscillators, in the random network of100 systems. Average amplitude of slow set is shown in (red)and that of fast in green. Here p =0.5, τ = 0 . ǫ = 0 . the larger number of phase shifted and mismatched sys-tems from the fast set. This gets reversed as m becomeslarge and close to N. At a certain value of m, the crossover point, the effect of mismatched and phase shiftedterms in the coupling have equal effects for both sets.In the state of frequency synchronization, we also ob-serve a similar cross over in the synchronized frequency,which is high for low m and very low for high m . The syn-chronized frequency calculated using eqn.(3) for all theoscillators is plotted as function of m in Fig. 5. For eachparticular τ the mean frequency of the intrinsic slow andfast frequencies is also shown (black lines). Thus, whenthe synchronized frequency for any given τ crosses themean and decreases below that with larger m , we sayfrequency suppression occurs. The value of m for whichthis happens is noted as the crossover point for the emer-gent frequency of the oscillators. ω m FIG. 5. (colour online) Frequency vs m for ǫ =0.05 τ =0.6(red),0.7(green) and 0.8(blue) showing crossover to fre-quency suppressed state as the emergent frequency crosses themean value of fast and slow frequencies shown by correspond-ing black lines(increasing order of τ from below to above) . p =0.5. D. Transition to amplitude death and connectivity of thenetwork
The topology or structure of the random network used,depends on its connectivity which is decided by the prob-ability of connections p . As p is increased from 0 to 1, thetopology goes from sparsely connected to fully connectednetwork. In order to understand the role of topology orthe connectivity of the network in the transition to AD,we study the collective behavior of all the systems byvarying p , for an m value that lies in the AD region ofFig. 2 and values of τ and ǫ from the AD region in theparameter plane (Fig.3). For each value of p , we take100 realizations of the network and check what fractionof them goes to AD state for the whole network. Thisfraction of the realizations f gives the probability of tran-sitions. We plot f for different values of m , to get thecorresponding transition curves. We observe that as m increases, the transition to AD occurs at lower values of p , till it reaches a minimum and with further increase of m , the transitions move to higher values of p . This isclear from Fig.6. The threshold value for the transition,where half of the realizations go to amplitude death, istaken as p t . f p (a) 0 0.2 0.4 0.6 0.8 1-10 -5 0 5 10 f (p-p t )/ δ (b) p t m (c) FIG. 6. a)(colour online)a)Fraction of realizations f for thetransition to AD, plotted with the probability p of connectionsin the network, b) normalized transition curves for m =30(red,plus), 40(green, cross), 50(blue, star), 60(magenta, square),70(cyan, solid square), 80(black, circle) and c) Variation ofcritical p value, p t with m . Here τ = 0 .
35 and ǫ = 0 . For each value of m used, we get the width of the tran-sition curve as δ and normalize the transition curves byreplacing p with ( p − p t ) /δ . Then we find all the transi-tion curves fall on top of each other revealing a univer-sal behavior. This data crunched curve is shown in theFig. 6(b). Moreover, the threshold value p t varies with m as shown in Fig. 6(c), which indicates a minimum ata particular value of m . At this value of m , the hetero-geneity in the network is optimized or maximized andhence transition to AD can happen even with sparselyconnected network corresponding to minimum value of p, the probability of connections in the network. For therandom network of 100 periodic R¨ossler systems, p t is ob-served to be minimum when the number of slow systemsis around 40.
1. Scaling with size of network
We repeat the above analysis and obtain the transitioncurves for different network sizes, with N=100, 150, 200,300, 500, 600, keeping m /N ratio fixed at 0.5. We noticethe larger the size of the network lower the value of p at which transition takes place. To study the scalingproperties of these transitions, with system size, we fiteach transition curve with the functional form f = ( p − p c ) α . (4)Here the value of p c is chosen as the one where the func-tion gives best fit, and then the corresponding value ofthe scaling index α is calculated for each transition curve.Our results indicate that the index α varies with the net-work size N. To get the value of α in the large size limit,i.e. as N approaches infinity, we plot the calculated α vs1/N and take the asymptote as 1/N goes to zero. Thiscomes out to be 0.68, which within numerical errors, canbe taken as 2/3. (Fig. 7) f p (a) 0 0.2 0.4 0.6 0.8 1-20 -10 0 10 20 f (p-p t )/ δ (b) 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0 0.005 0.01 α FIG. 7. (colour online)Fraction of realizations f forthe transition to AD plotted with the probability p forN=100(red, plus), 150(green, cross), 200(blue, star), 300(ma-genta, square), 500(cyan, solid square), 600(black, circle), τ = 0 . ǫ = 0 .
01. b) Normalized transition curves p − p t /δ for different N. c) Variation of the scaling index with 1/N witherror bar shown in red. E. Transitions to amplitude death in random networkswith non-uniform probabilities of connections
In the study presented in sections IIA-IID, the proba-bility of connections for generating the random network p is kept the same for slow and fast nodes. We nowconsider a much more heterogeneous case of random net-works, generated with three different probabilities andstudy the effect of slow and fast dynamics on it. This isdone by taking the probability with which a slow systemconnects to another slow system as p , while a fast sys-tem connects with another fast system with p and a slowsystem connects with a fast system with p . We computethe fraction of realizations resulting in amplitude deathin this random network of slow and fast systems by vary-ing p for different sets of values of p and p .It is interesting to note that amplitude death happenseven in a bipartite network, with p = 0 and p = 0 butnon-zero p . However, having non-zero values for p and p helps the network to reach amplitude death state atlower values of probability p and the minimum p forthis transition becomes smaller with increasing p and p (Fig. 8). We also study the special cases when with f p (a) f p (b) FIG. 8. (colour online)Fraction of realizations for transitionto AD for random network of heterogeneous probabilities forvarying p a)p1=0,p2=0, b)p1=0.3,p2=0.3. Here τ = 0 . ǫ = 0 . m =30(red, plus), 40(green, cross), 50(blue, star),60(magenta, square), 70(cyan, solid square), 80(black, circle). f p (a) 0 0.2 0.4 0.6 0.8 1 0.3 0.6 0.9 f p (b) FIG. 9. (colour online)Fraction of realizations for transitionto AD for a random network of heterogeneous probabilities forvarying p a)p1=0,p2=0.8, b)p1=0.8,p2=0, keeping τ = 0 . ǫ = 0 . m =30(red, plus), 40(green, cross), 50(blue, star),60(magenta, square), 70(cyan, solid square), 80(black, circle). p = 0, p is varied keeping p fixed as well as p = 0and p is varied with p fixed. The results shown are for p = 0 . p = 0 . p isvaried (Fig. 9a,b). ε τ (a) f p (b) FIG. 10. (colour online)a)Region showing AD in τ, ǫ planefor Landau-Stuart systems on a random network, p = 0 . m = 50, N = 100, b) variation of fraction of realizations with p for τ = 0 . ǫ = 0 . m =30(red), 40(green), 50(blue),60(magenta), 70(cyan). F. Random network of Landau Stuart systems
We do a similar study for slow and fast Landau Stuartoscillators on random networks given by the equations˙ x i = τ i (( a − x i − y i ) x − ωy i ) + τ i ǫ N X j =1 A ij ( x j − x i )˙ y i = τ i (( a − x i − y i ) y + ωx i ) (5)The results are qualitatively similar for amplitude deathand oscillatory behaviors. The region of AD is numeri-cally calculated and shown for p =0.5, m = 50, N=100 in τ, ǫ (Fig.10a). Choosing the τ and ǫ from the region ofAD one can show the variation for fraction of realizationsof AD while p varies. In this case also an optimum num-ber of slow systems exists for which the transition occursfor minimum value of probability p as shown in Fig.10b.
1. Stability analysis for amplitude death state
We find the onset of amplitude death is due to the sta-bilisation of the unstable fixed point at (0,0). Hence thetransition to AD can be obtained by a detailed stabilityanalysis. We note that stability of AD state is analysedin the case of two coupled systems of differing time scales,globally coupled networks and minimal networks or mo-tifs of 3 or 4 systems . In the present study, withheterogeneity in the pattern of connections also, AD oc-curs due to interplay of time scales and topology.We can write Jacobian of n-dimensional slow and fastsystems on a network of size N as J = ( τ · I ) ⊗ F ′ + ( τ · L ) ⊗ H (6)where τ is an NxN matrix in which τ ij corresponds to τ i of eqn(1) for all j. I is NxN identity matrix. Dotproduct ( · ) is defined here by the element wise productof two matrices, and cross product ( ⊗ ) is defined as eachelement of the former matrix being multiplied by the latermatrix as a block . L is the laplacian matrix, L = D − A (7)where D is the diagonal degree matrix of the network,and A is adjacency matrix. F ′ is Jacobian of the intrinsicsystem around fixed point and is an nxn matrix, wheren is the dimension of a single system on each node. H isnxn coupling matrix.In the case of Landau-Stuart oscillators systems withcoupling function as given in eqn(5) and AD at the fixedpoint (0,0), we have F ′ = (cid:18) − a − ωω a (cid:19) , H = (cid:18) ǫ
00 0 (cid:19)
We construct the matrix J following the equation (6)and calculate its eigenvalues. Scanning the whole plane of τ ǫ , we estimate the values of τ and ǫ , where the real partof the largest eigenvalue crosses zero. The plot of thesevalues in ( τ, ǫ ) plane gives the transition curve to AD orthe boundary for the stable region for AD. This is shownin black dots in Fig.10a. We note that the boundariescalculated using the above stability analysis agrees wellthose obtained by direct numericalGraph partitions andcluster synchronization in networks of oscillators simula-tions. III. SCALE FREE NETWORKS OF SLOW AND FASTSYSTEMS
As is well known, scale-free networks are inherentlymore heterogeneous than random networks, with broaddistribution of nodal degrees and a few nodes with veryhigh degrees, called hubs. Hence the emergent dynamicsdue to interactions among slow and fast dynamical sys-tems on such a scale free network will be interesting. Forthis we generate several realizations of scale free networksusing Barab´asi-Albert algorithm and consider the dy-namics on each node of the network, as that of periodicR¨ossler systems. The equations for the dynamics on sucha network will be the same as eqn( 2) with all parame-ters taken the same way. But the adjacency matrix A ij is taken as per the scale free network topology obtainedfrom the Barab´asi-Albert algorithm.In a typical calculation, we take a network of size 100,with a set of m nodes evolving at the slower time scale.Since in a scale free network, hubs play the role of controlnodes, we mostly concentrate on cases where hubs followslower dynamics. Hence in this case the number of slowsystems required for AD to occur, is much smaller. Thuson a scale free network of 100 systems even with eightof the higher degree nodes or hubs having a time scalemismatch of τ , we find the dynamics of all the systemscan be suppressed to AD state (Fig.11). We isolate theregion of AD in ( τ, ǫ ) plane as the region where the dif-ference between the global maxima and global minima ofall the oscillators goes to zero. This is shown in Fig. 12.We also estimate the minimum number of slow systemsrequired to induce AD starting with the highest degreeas slow and increasing the number one by one. For each -8-6-4-2 0 2 4 6 8 0 50 100 150 200 250 300 350 x t FIG. 11. (colour online) Time series of few typical x-variablesare plotted, showing amplitude death in a scale free networkof periodic R¨ossler systems for τ = 0 . ǫ = 0 . m =8. ε τ FIG. 12. (colour online) AD region in ( τ, ǫ ) plane for 100periodic R¨ossler systems coupled on a typical realization ofscale free network with 8 hubs taken from the high degreeend slow in dynamics. case, the average amplitude differences of all the oscilla-tors is calculated. The plot of this averaged amplitude( < A diff > ) with the number of slow hubs m gives thisas the value of m at which ( < A diff > ) becomes zero.This is repeated for different realizations and shown inFig.13. Here the values of τ and ǫ are chosen from theAD region in Fig. 12.We repeat the above study using Landau-Stuart oscil- m FIG. 13. (colour online) < A diff > vs m showing the mini-mum value for number of slow hubs required for AD to occurfor periodic R¨ossler systems on a scale free network. Here τ = 0 .
12 and ǫ = 0 .
25 Different colors correspond to differentrealizations. -1-0.5 0 0.5 1 480 490 500 510 520 530 540 550 var t FIG. 14. (colour online)Variance of few typical periodicR¨ossler oscillators (var) with time to show that each oscilla-tor takes a different time to move away from the synchronizedstate. τ = 0 . ǫ = 0 . lator as nodal dynamics. We find qualitatively similarresults for this case also with AD state and frequencysynchronization. A. Spreading of slowness on scale free networks
When there is no time scale mismatch in the dynamicsof systems, all the systems on a scale free network, canbe completely synchronized with a sufficiently strong cou-pling strength. Starting with such a state, after givingsufficient time so that all the oscillators settle to completesynchronization, we make one of the nodes, called sourcenode, slower in its dynamics. Clearly this can disrupt thedynamics of all other nodes as the slowness spreads overthe network. Consequently, all oscillators will then moveaway from the state of complete synchronization.Due to the heterogeneity of connections in the scalefree network, the time taken by each oscillator to moveaway from synchronization will not be the same. We an-alyze this scenario in terms of the degree of the nodeand shortest path from the source node, in the followingtwo ways. We calculate the change in the variance of alloscillators in time. When they are completely synchro-nized, the variance would be zero as shown in Fig. 14.When one node is made slow, the variation of each os-cillator from the mean of anticipated synchronized os-cillations(the synchronized oscillation they would havefollowed if this node was not made slow), is nonzero in-dicating onset of desynchronization. From the Fig. 14 itis evident that for each oscillator the time taken for thevariance to go to a non zero value ± ν is different, withthe source node taking the least time obviously. Thistime, t ν , for each oscillator to reach a specific value ± ν (typically -0.01 or 0.01) for its variation is plotted as afunction of the degree of the nodes. It is easy to see that t ν increases as the shortest path of that node from thesource node increases. We repeat this for several nodesas sources, including hubs and low degree nodes. Fig. 15shows the plot of t ν against degree of nodes for the twocases with a hub as the source node and a low degreenode as the source node for a typical realization. In thecase where a hub is the source of slowness, we see mostof the nodes move away in much shorter times since theshortest path from the source node is small. time k i (a) -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 time k i (b) FIG. 15. (colour online)Time taken for each oscillator to moveaway from synchrony is plotted with its degree ( k i ) when onesource node becomes slow. This is shown for a particular re-alisation of the network of 100 systems where a) the highestdegree hub is made slow with degree 47 and in b) the low-est degree node is made slow with degree 2. Different colorsrepresent different shortest path lengths from the source nodewith shortest path 1(red, plus), 2(green, cross), 3(blue, star).Here τ is 0.3 for the source node and ǫ is taken as 0.03. N s time (a) 0 5 10 15 20 25 30 35 0 20 40 60 80 100 120 N s time (b) FIG. 16. (colour online) Number of systems that move awayfrom synchrony( N s ) in a range of time is plotted with time.The results shown are averaged over six realisations. Here a)corresponds to the highest hub as source of slowness for eachrealisation and b) corresponds to the lowest degree node beingmade slow. In this case also, τ is 0.3 for the source node and ǫ = 0.03. We repeat the study for different realizations of thenetwork and calculate the number of systems that getdesynchronized in a given time and this number averagedover the realizations varies as shown in Fig. 16, for thetwo cases, when a hub and a low degree node are thesources of slowness. It is interesting to note that thetotal time taken for all the systems to move away fromeach other decreases with the degree of the source nodeas a power law. This is clear from Fig. 17 shown fora typical realization of the network. We find that thiscurve can be fitted with a function as T ( k ) = ( a/k ) + b (8)with a = 180 and b = 18. The desynchronization transi-tion is followed by a reorganization of the whole network. B. Self organization of the network to frequencysynchronized state
Once synchrony is disturbed as discussed above dueto a single node going slow, de-synchronization sets in
20 30 40 50 60 70 80 90 100 110 0 5 10 15 20 25 30 35 40 45 50 total time degree
FIG. 17. (colour online)Total time taken for all oscillators inthe network to move away from synchrony is plotted againstthe degree of source node for periodic R¨ossler systems on scalefree network. The source node has time scale as τ = 0 . ǫ = 0 . -6-4-2 0 2 4 6 490 500 510 520 530 540 550 560 x t FIG. 18. (colour online)Time series of few typical x-variablesof periodic R¨ossler systems are plotted to show the frequencysynchronized state reached from the state of complete syn-chronization after one node is made slow. Here τ = 0 . ǫ = 0 . characteristic times depending on the degrees of nodesin the network. Subsequent to this, given sufficient time,all the oscillators are found to reorganize themselves intoa frequency synchronized state. (Fig. 18). This is an in-teresting and novel phenomenon of self-organization ,where the network goes from a collective behavior ofcomplete synchronization to another less ordered but co-herent emergent state of frequency synchronization byre-adjusting the dynamics of all the nodes, after the net-work is perturbed by making one node slow.To characterize this process, we calculate the fre-quency of each oscillator using eqn.3 and plot them withtime(Fig.19a). The figure shows the synchronized fre-quencies in the beginning, the de-synchronized frequen-cies just after one node is made slow at t=500, and finallythe re-adjusted lower frequency after self-organization tofrequency synchronized state. We also study the timetaken for self-organization, called self organization time,( t so ), averaged over several realizations for τ varying inthe range 0 . − . ǫ =0.1, (Fig.19b). Weobserve that t so decreases with the increase in the timescale mismatch introduced on the source node, along astraight line. frequency time (a)
10 20 30 40 50 60 70 80 90 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t so τ (b) FIG. 19. (colour online)a) Self-organization of oscillators intofrequency synchronized state for the whole network for a typi-cal realization of scale free network after one hub is made slow.On the x axis the time and on the y axis the correspondingfrequency are shown. At time=500 one hub is made slow, b)Time taken to organize into the new synchronized state, t so ,averaged over six realizations is plotted for different τ valueswith ǫ kept at 0 . t so falls off with increasing mismatch τ asa straight line. C. Scale free network with multiple time scales
In this section, we study the collective dynamics ofnonlinear systems on a scale free network where the timescale of each node varies with its degree following therelation τ i = 2 /k i (9)This is chosen such that the node with highest degreewill have the slowest time scale and the time scale in-creases as degree decreases. Since the network is scalefree, the number of different degrees, k i and hence thenumber of different time scales, τ i , will be less than Nbut still will have a multiplicity of time scales. Thus, forone typical realisation of 1000 nodes, we get 30 differenttime scales in the network. We find the presence of mul-tiple time scales, forces the whole network to collapse toa state of AD. The onset of AD, after a threshold cou-pling strength, is evident from the plot of < A diff > , fordifferent values of ǫ , in Fig. 20 with three different sizesN=100,500 and 1000.For lower values of ǫ , prior to onset of AD, we see os-cillations with differing amplitudes. But even with mul-tiple time scales, the connectivity through the networkmakes the systems organize into three groups, high de-gree nodes with lower time scales having smaller ampli-tudes, low degree nodes with faster time scales havinglarger amplitudes and an intermediate group with ampli-tudes in between. This is clear from the distribution ofamplitudes of all the oscillators in the network from sixrealizations of the network of size N=1000 (Fig.21).Similarly we do the analysis for another distribution oftime scales on nodal dynamics of scale free network. Inthis case we choose the i th node to follow a time scale τ i drawn from a normal distribution with mean 0 . .
15. Since in this case there is nocorrelation between the degree of the node and its timescales and the network is hierarchical, we take multiple0 -2 0 2 4 6 8 10 12 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ε FIG. 20. (colour online) < A diff > vs ǫ in a typical realizationof scale free network showing onset of AD due to multiplicityof time scales for periodic R¨ossler systems with τ i = 2 /k i forN=100(red), 500(green), 1000(blue). N a FIG. 21. (colour online)Distribution of amplitudes of 1000periodic R¨ossler systems with multiple time scales on a scalefree network for ǫ = 0 .
03, with the time scales distributed aseqn. 9. The different colours indicate different realizations ofthe network. realizations of the distribution and analyse the dynamics.The onset of AD state on a scale free network of periodicR¨ossler systems and the threshold value of ǫ is shown in(Fig. 22a). We also estimate the transition probabilityas the fraction of realizations of time scale distributionsfor which AD occurs, for different values of ǫ and plotthis fraction (f) with ǫ to see the transition curve to AD.(Fig. 22b) IV. CONCLUSION