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Journal of the Korean Physical Society, Vol. 67, No. 2, July 2015, pp. 389∼394

Effects of the Frequency-degree Correlation on Local Synchronization in Complex Networks Shijin Jiang∗ and Wenyi Fang School of Mathematical Sciences, Peking University, Beijing, China

Shaoting Tang,† Sen Pei, Shu Yan and Zhiming Zheng‡ Laboratory of Mathematics, Information and Behavior, School of Mathematics and Systems Science, Beihang University, Beijing, China (Received 7 May 2015, in final form 10 June 2015) We investigate the effects of the frequency-degree correlation on local synchronization in complex networks with Kuramoto oscillators. We find that a discontinuous synchronization transition occurs in the local patterns for heterogenous networks while for homogenous networks, the local synchronization transition remains continuous. Then, we extend our study to a general frequency-degree correlation case and conclude that the positive correlation does not change the local synchronization patterns while in the case of a negative correlation, the local synchronization transition degenerates to second order. Moreover, the correlation parameter α is verified to have a strong influence on the synchronization level. In particular, smaller |α| results in higher synchronization ability and faster speed to a synchronized state. Our study provides a deeper understanding of the effects of the frequency-degree correlation on network synchronization. PACS numbers: 89.75.Hc, 89.20.-a Keywords: Explosive synchronization, Kuramoto model DOI: 10.3938/jkps.67.389

I. INTRODUCTION

explosive transition to synchronization [16]. Moreover, the general case of frequency-degree correlation has been studied by Skardal [17] and Liu [18]. Their works gave more details about such kind of synchronization transition under different correlations and are good extensions of Ref. [13]. However, the mechanism of explosive synchronization transition and the effects of frequencydegree correlation are still not completely understood. In additions, most studies about synchronization dynamics focus on the global behavior of oscillator system, especially the state of complete synchronization, but how the synchronization emerges on the local scale and how the partial synchronization state evolves to the global synchronization state are not very clear yet. Only a few works deal with those issues [19–21]. In this paper, we try to understand the effects of frequency-degree correlation on the local synchronization pattern in the framework of Kuramoto model. The remainder of this paper is organized as follows. In Section II, we give a detailed introduction for Kuramoto model on complex networks. Then, in Section III, we analyze the effects of frequency-degree correlation on the local synchronization patterns in Erd¨ os-R´enyi (ER) and SF networks through numerical simulation. Section IV is devoted to a study of the general case of frequency-degree correlation and shows how the correlation parameter in-

Synchronization phenomena, which describe the emergence of collective behavior of large ensembles of coupled units, are always the focus of intense research in synthetic and natural systems [1–3]. In most cases, the interaction between the coupled units can be described as a complex network [4–7]. Previous studies have shown that the underlying network structure can strongly influence the critical point for the onset of synchronization, as well as stability of completely synchronized state [8– 10]. However, the topological features seem not to affect the order of synchronization transition, which is usually second order. Recently, a first-order synchronization transition has been reported under the framework of Kuramoto model [11,12] in random scale-free (SF) networks when a positive correlation exists between the natural frequency and degree [13]. The explosive synchronization transition soon attracted many researchers’ interest, and the same phenomenon has been observed in some other models [14, 15]. The time delay has also been found to enhance the ∗ E-mail:

[email protected] [email protected] ‡ E-mail: [email protected] † E-mail:

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Journal of the Korean Physical Society, Vol. 67, No. 2, July 2015

fluences the synchronization patterns. Finally, Section V presents conclusions.

II. THE MODEL Let us consider an ensemble of N phase oscillators interacting in a complex network via the coupling  aij sin(θj − θi ), i = 1, 2 · · · , N, (1) θ˙i = ωi + λ j

where aij is the adjacency matrix of the network, λ is the coupling strength and ωi stands for the natural frequency of node i. In the original work of Kuramoto, the phase oscillators were considered as fully interacting; that is, aij = 1, ∀i = j. The correlation between the degree and the frequency is presented as ωi ∝ kiα . Here, α can be understood as a correlation parameter. Generally, an order parameter r that measures the global synchronization level can be defined as reiΨ =

N 1  iθj e , N j=1

(2)

where Ψ represents the average phase of the system. The system is totally synchronized when r = 1 while r = 0 for the incoherence solution. The order parameter r is a good indicator that measures the global synchronization level, but it cannot give much information about the details of local synchronization. G´ omez-Garde˜ nes proposed a new parameter in Ref. [19] which is defined as rlink

 τr +Δt 1  1 = aij | lim ei(θi −θj ) dt| (3) Δt →∞ Δt τ 2Nl i,j r

and represents the average synchronization level of all pairs of collected oscillators. The parameter rlink can measure the local construction of the synchronization pattern and provides a new perspective to study the path to totally synchronization. Here, Nl is the number of links of the network. The time parameters τr and Δt should be sufficiently large to ensure that the system has achieved an equilibrium state and obtained a stable time average.

III. EFFECTS OF A LINEAR FREQUENCY-DEGREE CORRELATION In this section, we study local synchronization with a linear frequency-degree correlation on ER and SF networsk by setting ωi = ki in Eq. 1. We solve this equation by using a 4th-order Runge-Kutta method for given a λ. The initial θi are randomly drawn from a uniform

Fig. 1. (Color online) Synchronization diagram r(λ) for an ER network (left panel) and a SF network (right panel). Both forward and backward continuations in λ using δλ = 0.01. The networks size is N = 1000 and their average degree are < k >= 6.

distribution in the interval [0, 2π]. In order to capture the feature of possible first-order transitions of the parameters r and rlink , we compute two diagrams which are labeled as forward and backward for each parameter. Here “forward” means increasing λ progressively with increment δλ while “backward” means decreasing λ progressively with decrement δλ. In this paper, the network topologies we considered are limited to ER and SF network with power-law exponent γ = 3, size N = 1000 and average degree < k >= 6. Figure 1 presents the parameter r as a function of the coupling strength λ. In an ER network, the forward and the backward curves coincide and exhibit a continuous transition from an incoherent state (r ≈ 0) to a coherent state (r ≈ 1). However, in a BA network, the forward and the backward continuations do not coincide all the time. The abrupt transition of r and the strong hysteresis between the two curves show that the synchronization transition is, indeed, first order. These phase transition laws showed in our simulation agree with the results in Ref. [13]. We should mention that there are slight differences in the quantitative values of the order parameter r between our simulations and the simulations in Ref. [13]. The reasons mainly come from two aspects. One is that the rule of generating the BA networks in our manuscript is not completely the same as the one in Ref. [13]. We construct BA networks by using the classical preferential attachment rule from a small random network with 10 nodes and 25 links while the networks in the simulations of Ref. [13] are generated according to the rules in Ref. [22]. Although the two kind of networks have the same network size, mean average degree, and similar degree distribution, they are not the same in all details. The other reason is the randomness in the construction of the networks and in the numerical simulations of the Kuramoto model (different initial states, computing error, etc.), which can lead to a considerable difference in the value of r between two different simulations. As we focus on the orders of the phase transitions of global and

Effects of the Frequency-degree Correlation on Local Synchronization · · · – Shijin Jiang et al.

Fig. 2. (Color online) Local synchronization parameter rlink as a function of λ. The left panel is the diagram for an ER network, and the right panel is for an SF network. The parameters in this simulation are the same as those in Fig. 1

local synchronization behaviors of the oscillators system in this paper, a small mismatch in the values of r is not a key point. Thus, we will ignore this issue in the remainder of this paper. The curves rlink (λ) for ER and SF networks are shown in Fig. 2. In ER networks, rlink increases from a small but nonzero value to 1 with increasing coupling strength λ. There seems to be no critical transition point for rlink . However, the most interesting result is observed in SF networks. An abrupt jump appears for both the forward and the backward procedures. In the forward diagram, the order parameter rlink increases smoothly until a sudden jump appears, where rlink turns to 1. While in the case of backward continuation, the order parameter rlink remains 1 until a sudden fall. Moreover, the transitions in the two diagrams take place at different points, and rlink has a different critical value, indicating that the transition of parameter rlink is first order. This result is very different from the case without a frequency-degree correlation in Ref. [19], where rlink increases continuations without a sharp transition. Noticing that rlink and r have the same transition point λc , we can infer that the transitions of the two parameters should have the same mechanism. This sharp transition is just because of the heterogenous topology and the correlation between degree and frequency [13]. In fact, the frequencies of oscillators will be locked (θ˙i = θ˙j =const) when the coupling exceeds the critical point λc . Then,  τr +Δt i(θ −θ ) 1 limΔt →∞ Δt e i j dt = 1 for each pair of oscilτr lators i and j, and rlink jumps to 1. There are also some differences between the two synchronization parameters. Before the critical point, the parameter r remains 0, but rlink does not, implying that even in the global incoherence state, the local synchronization pattern exists. Now we investigate the local synchronization pattern further. Using the methodology in Ref. [19], we can define a “distance matrix”  τr +Δt 1 (4) Dij = aij | lim ei(θi −θj ) dt|. Δt →∞ Δt τ r The entry Dij stands for the average local synchronization level of two oscillators i and j, and rlink is the av-

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Fig. 3. (Color online) Relative size of the giant connected cluster (GCC) for ER (left panel) and SF network (right panel) as a function of λ. Forward and Backward continuations are presented for both networks. The parameters are the same as those in Fig. 1

erage of Dij for all collected pairs of oscillators. The entry Dij varies from 0 to 1. An entry of 0 indicates that the two oscillators are unconnected or totally unsynchronized while an entry of 1 indicates that the two oscillators are completely synchronized, but most entries of matrix Dij do not belong to either of these extreme situations. Therefore, we need a criterion to determine which pairs of oscillators are synchronized. As in Ref. [20], we set a threshold T to filter Dij . Those pairs of oscillators with Dij > T are regarded as synchronized while those with Dij < T are regarded as unsynchronized. Noticing that the fraction of synchronized links is rlink , we choose a proper threshold T so that the number of synchronized pairs oscillators equals 2rlink Nl . Let Dij =1 if oscillator i and j are synchronized, and Dij =0 if they are not. Then, a new network topology is created according to the matrix Dij . This network topology allows us to obtain information the about local synchronization pattern by investigating its statistic characteristics. Figure 3 presents the size of the largest synchronized component corresponding to the new network topology for ER and BA networks. The results show that the two different kinds of network topologies have much different synchronization patterns. The GCC (giant collected component) seems to have a zero critical coupling, and increase from 0 to 1 continuously in ER network. However, in BA networks, there exists a nonzero percolation threshold where the giant synchronized component occurs. We can infer that before the threshold, the synchronized pairs of oscillators are divided into some micro clusters and none of them have a visible scale in BA networks. This means that in the weak coupling case (λ is small), large synchronization clusters are more easily formed in ER networks than BA ones. Compared to ER topology, the curve for the BA network still has a jump at the critical point λc . This jump is certainly caused by the sharp transition from a global incoherent state to a completely synchronized state. Furthermore, we should notice that before the critical point λc , the giant synchronized cluster has obtained above 60% of the oscillators. It shows that the local synchronization pattern has a non-trivial structure even in the global incoherent state.

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Fig. 4. (Color online) Number of synchronized clusters for both ER network and SF network are shown in this figure. The synchronized clusters here are connected components with at least two oscillators(we ignore the isolated oscillators). Parameters are the same as in Fig. 1.

In order to obtain an insightful understanding, we plot the number of synchronized clusters (NC) as a function of the coupling strength in Fig. 4. Ignoring small fluctuations, the curve for the ER network displays a downward trend with increasing coupling strength while the curve for the BA network has a peak. This result provides detailed information about how the synchronized clusters evolve. As the coupling increases, large clusters are formed and the total number of clusters rapidly decreased for the ER network. However, this is not the case for the SF network, where there is a coupling interval in which many new clusters emerge and the number of clusters are sharply enlarged with λ. Moreover, ER networks start with more synchronized clusters and larger GCC than SF ones, indicating that homogenous networks seem to present higher synchronizability than heterogenous ones in the low coupling region. Figure 5 is presented to get an intuitive understanding of the evolution of local synchronization patterns. The difference of the behaviors between the two kind of topologies is rooted in the growth of GCCs. For ER networks, many different clusters aggregate into a giant one when coupling increases while for SF networks, oscillators join into the GCC practically one-by-one in terms of λ. Moreover, the GCC in SF networks is unique and contains much more oscillators than other clusters, but for some coupling λ, more than one GCC exist in ER networks and their sizes are equal. The reason behind this behavior lies in the character of network topology. In a homogenous network, the oscillators have similar statuses, and they nearly have equal probabilities of belonging to the GCC. When the coupling is weak, the oscillators tend to form serval clusters with similar structures. Then, these clusters will join together into a unique giant cluster as the coupling increases. This case is changed by the heterogenous topologies of SF networks, where the hubs have many more collections than other oscillators and, thus, have a higher probability of taking part in the giant synchronized clusters. The above results show the local synchronization patterns for the SF and ER networks. Compared to the

Journal of the Korean Physical Society, Vol. 67, No. 2, July 2015

Fig. 5. (Color online) Synchronized clusters for several values of λ for an ER and an SF network. In order to get a sizeable picture, the network size here is set as N = 100.

previous results without a frequency-degree correlation, the most important difference is the explosive local synchronization transition in SF networks, which is a result of a linear correlation between local dynamics (natural frequency) and local heterogenous structure (degree). We are interested in whether this phenomenon will hold when the correlation is nonlinear, especially when it is sub-linear or negative, and what synchronization patterns will occur in these cases. We will discuss this in the following.

IV. GENERAL CASE OF THE FREQUENCY-DEGREE CORRELATIONS In light of the results in the above section, we have extended our study from the case of a linear frequencydegree correlation to a more general correlation case. The model we discuss here is based on the following equation [18]:  ki α +λ aij sin(θj − θi ), i = 1, 2 · · · , N, (5) θ˙i = β j where the parameter α measures the tendency of the correlation between the frequency and degree, and β =  α  ki is a normalized coefficient to avoid the homogei ki nous of ωi when |α| is small. Here, we focus on the effects of frequency-degree correlations of synchronization patterns of ER and SF networks by decreasing α from 1 to −1. According to the results of Ref. [18], the phase transition of the order parameter r is very different in the cases α > 0 and α < 0. Therefore, we discuss the two cases separately in the following.

1. Case 0 < α < 1

In this case, the order of phase transition of the parameters r holds in both homogenous and heterogenous networks. It is worth mention that the critical points for

Effects of the Frequency-degree Correlation on Local Synchronization · · · – Shijin Jiang et al.

Fig. 6. (Color online) Diagrams of the order parameter in the sublinear correlation case: (a) r versus λ for an ER network. (b) r versus λ for a BA network. (c) rlink versus λ for an ER network. (d) rlink versus λ for a BA network. In each panel, curves for α = 0.3 (blue line), α = 0.6 (red line) and α = 0.9 (green line) are plotted.

both kind of networks get smaller as α decreases [18]. The diagram of the parameter rlink for both ER and SF networks are similar to the case α = 1, showing a continuously rising trend in ER networks and an abrupt jump in SF networks. Although the parameter α does not change the main feature of the curves for r and rlink , it affects the global and the local synchronization levels. For smaller α, r and rlink have higher values. This means that the oscillators are more synchronized because small α leads to a homogenous distribution of the natural frequency ωi , which will enhance the synchronization ability. The above results are confirmed in Fig. 6. The GCC and the NC for several α are presented in Fig. 7. Just like the parameters r and rlink , the GCC and NC show behaviors similar to those for the case α = 1 , so we will not repeat it here. An important fact is that smaller α leads to larger GCC for both kinds of networks and leads to smaller NC for ER network for each λ in the range of [0, 2]. Although for BA networks, the curve of NC for α = 0.3 is not always below the other two, it shows a faster speed to the local synchronized state (NC = 1). Larger GCC and smaller NC means oscillators tend to form larger clusters rather than many smaller clusters. This fact shows that decreasing α enhances the local synchronization level and enriches the local structure of synchronized oscillators.

2. Case −1 < α < 0

In this case, the degree and frequency are negatively correlated. Simulation results in Fig. 8 and Fig. 9 have some differences with the sub-linear correlation case. The most visible difference is the transition order of parameter r for BA networks. In this negative

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Fig. 7. (Color online) Size of the giant collected cluster (GCC) and number of the synchronized clusters (NC) as functions of coupling strength in the sublinear correlation case. Panels (a) and (c) are for ER networks. Panels (b) and (d) are for BA networks. Three curves are plotted in each panel (α = 0.3 (blue line), α = 0.6 (red line) and α = 0.9 (green line)).

Fig. 8. (Color online) r and rlink for ER (panel (a) and panel (c)) and BA networks(panel (b) and panel (d)) in the negative correlation case. The blue, red and green line correspond to α = −0.3, α = −0.6 and α = −0.9, respectively.

correlated case, the explosive transition vanishes, and a second-order transition occurs instead. Moreover, the rlink , GCC and NC both continuously depend on the coupling strength λ without a sharp jump. These results show that in the case of a negative correlation, the system does not have an abrupt transition in terms of the global and the local synchronization levels or the structure of the synchronization pattern. The influence of α can be observed in Fig. 8 and Fig. 9. Graphically, curves with larger α have higher r, rlink , GCC and lower NC for both ER and BA networks. This indicates that the systems with larger α have higher global and local synchronization levels. This is also because the distribution of ωi becomes homogenous when

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fects of frequency-degree correlations can be studied in other synchronization models, for example, R¨ ossler units [15]. More work is needed on these issues, and we will investigate them in future papers.

ACKNOWLEDGMENTS This work is supported by the Major Program of National Natural Science Foundation of China (11290141), NSFC (11201018) and the International Cooperation Project No. 2010DFR00700. Fig. 9. (Color online) GCC and NC for ER and BA networks in the negative correlation case. Panel (a) and panel (c) are for ER networks while panel (b) and panel (d) are for BA networks. The three lines in each panel strand for the case α = −0.3 (blue line), α = −0.6 (red line) and α = −0.9 (green line),respectively.

α is close to 0.

V. CONCLUSION In summary, we have investigated local synchronization of Kuramoto model with frequency-degree correlations in complex networks. Our results show that the local synchronization level presents a first-order transition in heterogenous networks, which is different from the previous results without frequency-degree correlations. Moreover, we have also generalized our study to a nonlinear frequency-degree correlation case. The synchronization pattern remains unchanged in the sublinear correlation case (0 < α < 1) for both homogenous and heterogenous networks, but in the negativecorrelation case (−1 < α < 0), the synchronization transition degrades into a second-order one for heterogenous networks. The correlation parameter α influences the local and the global synchronization levels, and decreasing |α| can enhance the synchronization ability for both kinds of network. Compare to previous works [13, 18], our study provides a further understanding of the effects of frequencydegree correlations on network synchronization. In particular, we obtain useful information about both global and local constructions of synchronization patterns and that should guide our comprehension of these phenomena. However, in this paper we have only considered ER and BA networks with fixed sizes and average degrees. How the network size and the average degree influence the synchronization pattern is not very clear (Ref. [21] has discussed this issues in the case without the frequency-degree correlation). In addition, the ef-

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