Phase Synchronization on Spacially Embeded Duplex Networks with Total Cost Constraint
PPhase Synchronization on Spacially
Embeded Duplex Networks with Total CostConstraint
Ruiwu Niu ∗ , Xiaoqun Wu ∗ , ‡ , Jun-an Lu ∗ , Jianwen Feng † Abstract
Synchronization on multiplex networks have attracted increasing attention inthe past few years. We investigate collective behaviors of Kuramoto oscillatorson single layer and duplex spacial networks with total cost restriction, which wasintroduced by Li et. al [Li G., Reis S. D., Moreira A. A., Havlin S., Stanley H. E.and Jr A. J.,
Phys. Rev. Lett.
Keywords:
Synchronization; multilayer network; spacial network.
I Introduction
Synchronization as a natural phenomenon, which can be easily observed in a wide vari-ety of biological, chemical and physical systems, has drawn extensive attention of scien- ∗ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. † College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, PR China ‡ To whom correspondence should be addressed: [email protected] § This work was supported in part by the National Natural Science Foundation of China under Grants61573262. a r X i v : . [ n li n . C D ] N ov ists in the last few decades. A large amount of research achievements have been madeabout synchronization properties of small-world, scale-free and other types of complexnetworks [1–17]. Aside from these researches, the spacial network has become more andmore popular ever since it was proposed in 2000 [18]. As we know, many natural orman-made infrastructures are embedded in space, such as power grids, oil pipelines, com-munication networks and so on. So, it is important to investigate the synchronizationproperties of spacially embedded networks.Among various studies on network synchronization, the Kuramoto model has beenextensively employed [19]. It can describe collective behaviors caused by interaction be-tween coupled oscillators. Since then, many works have been done to analyze synchroniza-tion among coupled oscillators. Watts and Strogatz found that a few shortcuts betweenKuramoto oscillators in a network can greatly improve the synchronizability of the net-work [20, 21]. Moreno and Pacheco applied Kuramoto oscillators to scale-free networksand indicated that the hubs play a vital role in determining network dynamics [22]. Afterthat, some significant analytical and numerical results demonstrated the relation betweensynchronization and network topologies [8, 9]. In 2011, G´omez et. al discovered a discon-tinuous synchronization transition by introducing a correlation between network structureand local dynamics [23]. This special phenomenon called explosive synchronization offersa new perspective for investigating synchronization of the Kuramoto model.It is noteworthy that many synchronization phenomena, as in social networks, do notinvolve a single network in isolation but rely on the behaviors of a collection of smallernetworks [24, 25]. And more generally, beyond single networks, we are now understandingthat interactions between networks are playing an increasingly important role in deter-mining the dynamical processes [26–30].In the last few years, a new type of networks called multilayer networks has been paidincreasing attention to. Numerous results indicate that interaction between networkscan greatly affect dynamics on interacting networks, such as spreading, diffusion andsynchronization [24, 31–42]. For example, in 2014, Aguirre et al. [32] investigated theinfluence of the connector node degree on the synchronizability of two star networks withone inter-layer link and showed that connecting the high-degree (low-degree) nodes ofeach network is the most (least) effective way to achieve synchronization. In 2016, Weiet al. [43]worked on cooperative epidemic spreading on two interacting networks andfound that the global epidemic threshold is smaller than the epidemic thresholds of thecorresponding isolated networks and the cooperative interaction between networks can2nhance the final infection fraction. Very recently, Tang et. al [44] studied inter-layer,intra-layer and global synchronization in two-layer networks, and showed that for anygive nodal dynamics and network structure, the occurrence of intra-layer and inter-layersynchronization depend mainly on the coupling functions of nodes within a layer andacross layers, respectively.In the real world, many complex networks can be geographically represented or spa-tially embedded. In 2000 [18], Kleinberg proposed a spatial network model by addingshortcuts with probability P ( r ij ) ∼ r − αij on a regular lattice, where r ij is the Manhattandistance between nodes i and j on the lattice, and α is the spacial exponent. He foundthat proper addition of shortcuts ( α = d , where d is the spatial dimension) can lead toformation of a complex network possessing the small-world property. After that, con-sidering the cost of connecting different nodes, Li et. al improved Kleinberg’s model byrestraining the total length of shortcuts, which is named as Li network afterwards [45, 46].They found that when α = d + 1 , the network has the small-world property. These re-sults have inspired a lot of studies on this interesting phenomenon of the emergence ofthe small-world property on lattices by shortcuts addition [47–50]. Nevertheless, few ofthem focused on synchronous behaviors in spatial networks. In this paper, we considerthe specific Li network of Kuramoto oscillators and find that proper addition of shortcutscan greatly improve the synchronizability of spacial networks.In order to investigate synchronous behaviors on mutilayer spatial networks, we pro-pose a spatially embedded duplex network model with total cost constraint. Here, thesame set of nodes interact on different layers, and links in different layers represent differ-ent link types. By employing the Kuramoto model, we study inter-layer, intra-layer andglobal synchronizability of the networks. We show how coupling strength between lay-ers influences synchronization, and point out that large inter-layer coupling strength cangreatly enhance the inter-layer, intra-layer as well as global synchronization. We furtherinvestigate the synchronization processes. Interestingly, we find that, even for a smallvalue of inter-layer coupling strength, inter-layer synchronization can always happen inthe first place. In addition, for single layer networks, the nodes with large degrees alwaysfirstly arrive at synchronization. While for duplex networks, this phenomenon becomesless obvious due to inter-layer connections. At last, we reduce the inter-links betweenlayers and find that, a small portion of inter-links can make the networks achieve globalsynchronization just as one-to-one inter-layer connections can do.3 I Results and Discussion
A Synchronization on single layer Li network
First of all, we introduce a spatial network proposed by Li et. al [45, 46]. Figure 1 shows atypical Li network, which is a regular two-dimensional square lattice with N = L × L nodes,where L is the linear size of the lattice, and long-range connections are randomly addedbetween nodes. In this model, pairs of nodes i and j are randomly chosen to generate along-range connection with probability P ( r ij ) ∼ r − αij , where r ij is the Manhattan distancebetween nodes i and j , and α is the spacial exponent. In addition, the total length ofthe long-range connections is restricted by ∧ = L × L . It is noteworthy that if exponent α = d + 1 , the network will have the small-world property [45, 46]. Besides, when α isgetting larger, the network will be closer to a regular lattice. Otherwise, the network willbecome a lattice network with few long shortcuts. In this way, we can change networktopology by altering the exponent α .Figure 1: In a two-dimensional space, each node i has four short-range connections to itsnearest neighbors ( a , b , c and d ). A long-range connection is added to a randomly chosennode j with probability proportional to r − αij .Next, consider the Kuramoto oscillators as node dynamics. Thus the evolution of the N oscillators is governed by [19]˙ θ i = ω i + N (cid:88) j =1 λ ij A ij sin( θ j − θ i ) , i = 1 , , · · · , N, (1)4here θ i denotes the phase of the i th oscillator, ω i s are the natural frequencies whichare distributed by an even and symmetric probability density function g ( ω ). A ij is theadjacency matrix of the network and λ ij is the coupling strength between nodes i and j ,which are normally considered as λ ij = λ/k i , where λ is the global coupling strength and k i is the degree of node i .To quantify the synchronous behaviors of the oscillators, Kuramoto introduced thefollowing order parameter [19] R ( t ) e iψ ( t ) = 1 N N (cid:88) j =1 e iθ j ( t ) . (2)The order parameter R ( t ) can be considered as the proportion of synchronized oscillators,and R ( t ) ∈ [0 , R = 1, while for completely incoherent situations, R = 0.Here we focus on the influence of the spacial exponent α and coupling strength λ on theemergence of global synchronization. Figure 2 shows how the dynamical parameter λ andthe topological parameter α affect the order parameter R . In the contour graph, the colourrepresents the value of order parameter R . The red region means the network reachessynchronization, and the blue one indicates that the phases of network oscillators are outof synchronization. Other colours denote that only partial nodes reach synchronization.As we can see, when the coupling strength λ is lager than some critical value λ c , phasesynchronization appears at α < α op . In the contour graph, when α > λ . As mentioned before, for Li networks, when α = α op = d + 1 (for the considered lattice, d = 2 ), the small-world property emergesin the networks. So we can say that proper addition of long-range connections on asquare lattice network, where the network topology is varying between the small-worldtype ( α = d + 1) and the random one ( α = 0), can greatly improve synchronizability ofthe network. B Synchronization on duplex Li network
In order to investigate synchronous behaviors on multilayer spacial networks, we introducea duplex Li network model. As shown in Fig. 3, the duplex network model has two layersand the nodes of each layer have the same geographic positions. The way intra-linksconnect intra-nodes follows the same rule as that for a single layer Li network. That is,the probability that two nodes locating in the same layer establish a long-range connection5igure 2: The colour variation shows the value of order parameter R as a function ofspacial exponent α and coupling strength λ , for a single layer Li networks, and the sizeof the network is N = 50 ×
50. For α < λ >
4, phase synchronization can emerge.For α > λ <
4, the network can hardly synchronize.follows P ( r ij ) ∼ r − αij , where r ij is the Manhattan distance between nodes i and j , and α is the spacial exponent. In particular, we set α = d + 1 = 3 (the network exhibitssmall-world properties) in both layers. Moreover, the inter-links are one-to-one, that is,each node in one layer is connected to a counterpart in the other layer.Figure 3: A duplex network consisting of layer K and layer L . Each layer is a two-dimensional Li network, and each node in a layer has a connection with its counterpartin the other layer.Analogously, for the duplex Li network, the oscillators evolve by the Kuramoto model,6 θ Ki = ω Ki + λ K k i (cid:88) j ∈ Λ i sin( θ Kj − θ Ki ) + λ KL sin( θ Li − θ Ki ) , i = 1 , , · · · , N, (3)˙ θ Li = ω Li + λ L k i (cid:88) j ∈ Λ i sin( θ Lj − θ Li ) + λ KL sin( θ Ki − θ Li ) , i = 1 , , · · · , N, (4)where θ Ki ( θ Li ) denotes the phase of the i th oscillator in layer K ( L ) , ω Ki ( ω Li )s are thenatural frequencies of layer K ( L )’s oscillators being distributed by an even and symmetricprobability density. λ K ( λ L ) is the coupling strength of layer K ( L ) and λ KL is the inter-layer coupling strength. Λ i represents the set of neighbors of node i within the samelayer. Using the global order parameter R global , which is considered as the proportion ofsynchronized oscillators in both layers, we can find out whether two layers evolve into thesame synchronous state. For R global = 1, oscillators in both layers are fully synchronized,while for R global = 0, most of the oscillators are not synchronized.Figure 4: The color variation shows the value of global order parameter R global as afunction of inter-layer coupling strength λ inter and intra-layer coupling strength λ intra , fora duplex Li network, where the size of each layer is N = 10 ×
10, and the spacial exponentsof layer K and L are α K = α L = 3. In the region of λ inter > .
25 and λ intra >
2, thenetwork can evolve into global synchronization.For clarity, we set λ K = λ L = λ intra , and λ KL = λ inter . Figure 4 displays the influenceof intra-layer coupling strength λ intra and inter-layer coupling strength λ inter on globalsynchronization, where α = 3. We can see that the two factors have much impact onthe order parameter R global . For λ inter > .
25 and λ intra >
2, the duplex network’ssynchronizability is greatly improved. For other values of α <
3, there exist similar7esults. In other words, large values of inter- and intra-coupling strength can lead duplexspacial networks into global synchronization.
C Synchronization process on duplex Li network
As is well known, for Kuramoto oscillators, full synchronization means the phases of allnodes modulo 2 π are the same. We can say that they all come to an identical state [19].While for multiplex networks, there are other kinds of synchronous states, as shown inFig. 5. Intra-layer synchronization means all nodes within each layer have the same valueof phase modulo 2 π , and inter-layer synchronization means each node in a layer reachesthe same state as its counterparts in the other layer. In this section, we put forward anew order parameter to describe the inter-layer synchronization as follows, R inter = (cid:80) i =1 R ( i ) N , R ( i ) > . . (5)Here, R ( i ) is the local order parameter between node i in one layer and its counterpartin the other layer, N is the total number of pairs of nodes between layers. We term R inter as the inter-layer order parameter. For R inter = 1 , the system reaches inter-layersynchronization, while for R inter = 0 , it means no inter-layer synchronization appears.Figure 5: Schematic representation of (a) intra-layer synchronization and (b) inter-layersynchronization, in a duplex spacial network.Figure 6 shows the impact of inter-layer and intra-layer coupling strength on the inter-layer order parameter. It is obvious that the inter-layer coupling strength λ inter is a veryimportant factor influencing inter-layer synchronization. When λ inter is getting larger,the inter-layer order parameter comes closer to value 1. Also, larger intra-layer couplingstrength λ intra also lead to better inter-layer synchronizability.8igure 6: Inter-layer order parameter R inter varying as a function of inter-layer couplingstrength λ inter and intra-layer coupling strength λ intra . For a duplex Li network, wherethe size of each layer is N = 10 ×
10, and the spacial exponents for both layers K and L are α K = α L = 3.Figure 7 shows the inter-layer, intra-layer and global synchronization order parameterfor different values of inter- and intra-layer coupling strength, for a duplex Li networkwith N = 10 ×
10 and α K = α L = 3. Theintra-layer synchronization order parametersare defined the same as that in Eq. (2) for nodes within each layer. For a large inter-layercoupling strength λ inter = 5 , no matter what value of intra-layer coupling strength λ intra is, inter-layer synchronization can be achieved and always be the earlier one comparedwith intra-layer or global synchronization. Since the spacial exponents α are identicalin both layers, the intra-layer and global synchronization can be reached simultaneously.While for a quite small value of inter-layer coupling strength, such as that shown inPanel (d) for λ inter = 0 .
01 , inter-layer synchronization cannot occur, neither can globalsynchronization. For a large intra-layer coupling strength, layer K and layer L can reachintra-layer synchronization. In general, for duplex spacial networks consisting of theKuramoto oscillators, inter-layer coupling strength is a crucial factor determining inter-layer as well as global synchronization. Analogously, intra-layer coupling strength is adetermining factor for intra-layer synchronization.In order to further investigate how inter-layer coupling enhances the inter-layer syn-chronization, we illustrate the evolution progress of phase oscillators in Figs. 8 - 11. In allthe figures, the horizontal axis denotes the evolution time steps and vertical axis repre-sents the serial number of nodes in each layer. While the colours mean the value of phasemodulo 2 π , and the same colour means the same synchronous state. Here, for clearer9igure 7: The inter-layer, intra-layer and global order parameters varying with time t . Ineach panel, the black squares represent the inter-layer order parameter, the blue uppertriangles and green lower triangles represent the intra-layer parameter of layer K and L ,respectively, and the red circles represent the global order parameter. Panel (a): λ inter = 5and λ intra = 10, the inter-layer order parameter reaches 1 when t ≈ t ≈ λ inter = 5 and λ intra = 1,the inter-layer order parameter reaches 1 after t ≈ λ inter = 5 and λ intra = 0, the inter-layer orderparameter reaches 1 after t ≈ λ inter = 0 .
01 and λ intra = 10, layer K and layer L can reach intra-layersynchronization, while neither inter-layer nor global synchronization happens.and more visible observations, we use the round of phases modulo 2 π instead of phasesmodulo 2 π .For λ inter = 5 and λ intra = 10 shown in Fig. 8, all nodes in both layers reach synchro-nization after a certain time steps. Moreover, for clarity, we enlarge Panel (a) in the timeslot [0, 2000], and plot the node dynamics in the way symmetric about t = 0, as shownin Panel (b). Therefore, the left half of this panel is for dynamical evolution of nodes inLayer K , and the right half is for that in Layer L . It is obvious that, due to the stronginter-layer interaction, each node in one layer is influenced by its counterpart in the other10ayer and reaches a synchronous state before its intra-layer neighbors do.Figure 8: Node dynamics in each layer evolving with time, for λ inter = 5 and λ intra = 10.(a): Each pair of nodes in different layers will arrive at the same state after t ≈ t ∈ [0 , λ inter = 5 , λ intra = 1 , Fig. 9 shows that, because of the weak intra-layer couplingstrength, only partial synchronization occurs. We can obtain from the left panel thatpart of nodes in the two layers arrive at cluster synchronization, as indicated by the samecolors emerging simultaneously. Similarly, in the zoom in Panel (b) of Fig. 9, each nodein one layer and its counterpart in the other always stay in an identical state after sometime steps.Figure 9: Node dynamics in each layer evolving with time, for λ inter = 5 and λ intra = 1.(a): Only part of nodes will arrive at the same state at the same time; (b): Enlarged plotin t ∈ [0 , λ inter = 5, and λ intra = 0, we can observe that there is no occurrence ofintra-layer or global synchronization, as shown in Fig. 10. From Panel (b), it is obvious11hat only inter-layer synchronization is achieved.Figure 10: Node dynamics in each layer evolving with time, for λ inter = 5 and λ intra = 0.(a): The nodes in the same layer or the other layer can not get into large synchronouscluster at the same time; (b): Enlarged plot in t ∈ [0 , λ intra = 10 and λ inter = 0 .
01 . It is obvious that nodes in each layer reach itsspecific synchronous state, but no inter-layer synchronization occurs. From Panel (b) ofFig. 11, we can obtain that because of weak interaction between layers, a node in onelayer can not reach the same state as its counterpart in the other layer. Figures 10 and 11imply that both inter- and intra-layer interactions are crucial factors determining globalsynchronization.Figure 11: Node dynamics in each layer evolving with time, for λ inter = 0 .
01 and λ intra =10 .(a): The nodes from the same layer are in the same synchronous cluster, and thenodes from another layer are in different cluster; (b): Enlarged plot in t ∈ [0 , k belonging to the largest synchronized connected component (GC) with respectto the coupling strength λ for single layer Li networks. The probability P GC is defined by P GC = n/m , where m is the total round of simulation and n is the number of times whena node gets into the largest synchronized connected component. The panel indicates thathighly connected nodes are more likely to reach synchronization state. While for duplexspacial networks, as shown in Panel (b), this phenomenon becomes much less obvious,which is possibly due to interaction between layers. In other words, interaction betweenlayers can greatly affect the synchronization process.Figure 12: The probability P GC for a node of degree k belonging to the largest synchro-nized connected component as a function of coupling strength λ ( λ intra ) and λ inter = 10.(a): In single layer Li networks, nodes with larger degrees will arrive at synchronizationearlier; (b): For duplex Li networks, the influence of degrees will be weakened. D The impact of inter-links on synchronization
As studied previously, the coupling strength between layers plays a dominant role in theemergence of global synchronization on multiplex networks. In other words, the interac-tion across layers can greatly change the synchronous behaviors of multiplex networks.In this section, we study how the density of links between layers (inter-links) influencesthe emergence of global synchronization. Fist of all, we use the previous duplex Li net-work model with one-to-one inter-links and randomly remove a fraction of inter-links by1 − P link , thus the density of inter-links can be denoted as P link = M/N , where M is thenumber of links between layers after the removal and N is the size of each layer. Then13e investigate the global order parameter with different inter-link density P link in the pa-rameter space of inter- and intra-layer coupling strength, with results being shown in Fig.13. As we can see, for a relatively small value of inter-link density such as P link = 0 . P link = 1).Figure 13: Global order parameter R global as a function of inter-layer coupling strength λ inter and intra-layer coupling strength λ intra for different inter-link densities P link . (a): P link = 0 .
5, the synchronization region regarding λ inter and λ intra is similar to Fig. 4; (b): P link = 0 .
1, the synchronization region begin to shrink; Panel (c). P link = 0 .
01, there isno synchronization in the parameter space for λ inter ∈ [0 ,
5] and λ intra ∈ [0 ,
10] .In Fig. 14, we further illustrate different order parameters varying with inter-linkdensity P link , where λ inter = λ intra = 5. It is obvious that, when we remove the inter-links between layers, there exists a threshold P clink ≈ .
1. For P link > P clink , the networkcan always have inter-layer, intra-layer and global synchronization, while for P link < P clink , only intra-layer synchronization appears. Therefore in duplex Li networks, a small valueof inter-link density (as small as about 0.1 for this case ) can make the whole networkreach global synchronization. 14igure 14: The inter-layer, intra-layer and global order parameter as a function of inter-link density, for λ inter = λ intra = 5. The intra-layer order parameters almost do not varywith P link . While for inter-layer and global order parameters, there exists a critical valethresholding synchronization and non-synchronization. III Conclusions
In this letter, we study synchronization on single-layer and duplex spacial networks withtotal cost constraint. Fist of all, we investigate the influence of spacial exponent α onsynchronizability of single-layer Li networks. When α > d + 1 ( d = 2) , no matterwhat values of coupling strength are assigned, the networks can barely synchronize. For α < d + 1 , the networks will change from non-synchronization to synchronization withincreasing coupling strength. Then we study the effect of inter-layer and intra-layercoupling strength on synchronous behaviors for duplex spacial networks for α = 3 withtotal cost constraint. We find that synchronization will emerge with strong enough inter-layer and intra-layer coupling. Furthermore we introduce order parameters for inter-layer, intra-layer and global synchronization to investigate synchronization processes. Wenote that when inter-layer coupling strength is larger than a certain value, no matterwhat value of intra-layer coupling strength is assigned, the inter-layer synchronizationwill always occur fist. And inter-layer interactions can improve the inter-layer and globalsynchronization. Interestingly, the synchronizability of a node is positively related with itsdegree, and for duplex networks this effect will be weakened by the inter-layer interactions.That is, inter-layer interactions play a crucial role in global synchronization for duplex Linetworks. Finally, we study the impact of the inter-link density on global synchronization,and find that the existence of sparse inter-links can already lead to global synchronization15f duplex Li networks. Those results for duplex Li networks can help us to manipulatethe synchronization for a specific purpose, such as making two different groups reach anagreement in an optimal way. References [1] Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks.
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