The changes on synchronizing ability of coupled networks from ring networks to chain networks

Science in China Series F: Information Sciences 007 SCIECE I CHIA PRESS Springer The changes on synchronizing ability of coupled networks from ring networks to chain networks HA XiuPing 1 & LU JunAn 1, 1 School of Mathematics and Statistics, Wuhan University, Wuhan 43007, China; State Key Laboratory of Software Engineering, Wuhan University, Wuhan 43007, China In this paper, two different ring networks with unidirectional couplings and with bidirectional couplings were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures by cutting a link are completely opposite. The synchronizing ability will decrease if the change is from bidirectional ring to bidirectional chain. Moreover, the change on synchronizing ability will be four times if the number of is large enough. However, it will increase obviously from unidirectional ring to unidirectional chain. It will be /(π ) times if the number of is large enough. The numerical simulations confirm the conclusion in quality. This paper also discusses the effects on synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. Theoretical results are accordant to numerical simulations. Synchronization is an essential physics problem. These results proposed in this paper have some important reference meanings on the real world networks, such as the bioecological system networks, the designing of the circuit, etc. complex networks, synchronizing ability, unidirectional couplings, bidirectional couplings, ring, chain, coupling matrix 1 Introduction Recently, two significant discoveries stimulate great interests in studying the complex networks. Firstly, in 1998, in order to describe the transition from a completely regular network to a completely random network and the qualities of short route between most pairs of nodes in the large network, Watts and Strogatz [1] introduced the concept of small-world network in their paper published in ature. Then, Barabasi and Albert [] published a letter in Science in 1999. They pro- Received August 4, 005; accepted February 6, 007 doi: 10.1007/s1143-007-0048-z Corresponding author (email: jalu@whu.edu.cn) Supported by the ational Basic Research 973 Program of China (Grant o. 006CB70830) and the ational atural Science Foundation of China (Grant os. 60574045 and 90604005) www.scichina.com www.springerlink.com Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64

posed that a number of real-world complex networks are scale-free network, which have a degree distribution with tails that decay as a power-law, and these power-laws are free of any characteristic scale. Then, many people are interested in studying complex networks and some important progress results have been obtained [3]. Complex dynamical networks can be looked at as large coupled nonlinear systems. Collective behaviors in coupled nonlinear systems have many expression forms. Moreover, synchronization is the basic dynamical behavior of them. The mechanisms of many collective behaviors have relations with synchronization. In 1673, Huygens firstly introduced the concept of synchronization by describing the synchronization between two coupled pendulums. Synchronization phenomena have been found in acoustics, periodic router information, laser oscillator, nano-oscillators, electric current, etc. Plus, it also exists in some biological systems. For example, many fireflies fire in unison in the tree, fish swarm, birds fly in mass, and there is synchronization between the circadian and the rhythm of the environment. The appearance and progress of the linguistics caused the synchronization in conversation. The thinking or standpoints in a group converge to consistent. The combat command is harmony with different environments, the synchronization of the different organizations, work efficiency, etc. Synchronization is important in secure communication, the appearance of the harmonic oscillator, modeling populations of interacting biological systems, etc. [4]. The synchronizing ability of the complex network stimulates more and more researchers interests. For the past years, some criterions have been given to determine the synchronizing ability of the network. Consider the symmetric and diffusion coupled continuous time-varying networks. The eigenvalues of the coupling matrix are 0 = λ 1 > λ λ. The ratio of the smallest nonzero eigenvalue and the largest one R = λ /λ can be used to judge the synchronizing ability of the network if the synchronized region is bounded. The smaller value means the better synchronization [5,6]. For the boundless case, the largest nonzero eigenvalue λ can play the same role; the smaller value λ implies stronger synchronization [7]. Wu [8] supplied another decision criteria in graph theory, and translated the synchronization of the unsymmetrical coupled network into an optimization problem. However, it is not easy to operate. It was shown in ref. [9] that it is difficult to synchronize for the unidirectional coupling ring networks with > 6. In this paper, two different network structures, the ring networks with unidirectional couplings and bidirectional couplings, were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures by cutting a link are completely opposite. The synchronizing ability will decrease if the change is from bidirectional ring to bidirectional chain. Moreover, the change on synchronizing ability will be four times if the number of is large enough. However, it will increase obviously from unidirectional ring network to unidirectional chain network. It will be /(π ) times if number of is large enough. This paper also studied the effects on synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. The effort to synchronizing is symmetric about the length of d for the unidirectional couplings. Description of problem and theoretical analysis The ring and chain are the basic structures in complex networks. The two structures exist in many fields, such as natural science, engineering, etc. In addition, the coupling mechanisms of some 616 HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64

T behaviors in social science have relations with those two structures. Then, the researches of the characters of them are very important. Many practical problems, such as the DPT double rings structure networks and the synchronization accelerator, are based on the ring structures. Foreign citizen of Chinese origin scientists discovered the new nano structure single-crystal nanorings. The nanoring appeared to be initiated by folding of a nanobelt coaxial, and uniradial loopby-loop winding of the nanobelt formed a complete ring. It has semiconductor and piezoelectricity characters. It can be applied to detect micron, nano-sensors, the biologic cells, etc. The ring structure is a stable case in circuit, and the chain is a cascade pattern. In 1997, Matias et al. [10] observed and simulated the appearance of periodic discrete rotating waves in rings of unidirectionally coupled chaotic oscillators by coupled circuits experiments. They found that the collective periodic wave is -3 scalar faster than the uncoupled oscillator in time scale. The mechanisms, allowing for dynamical collective behaviors that are much faster than those characteristic of the isolated units, might be useful in explaining why synchronization and segmentation among areas of the brain that are quite apart from the point of view of the velocity of propagation of information is possible, as suggested by physiologists. The authors reported on the experimental observation of a recently predicted behavior of coupled chaotic Lorenz oscillators. Chaotic synchronization has been observed experimentally and numerically in arrays of Chua s circuits, arranged in both linear and ring geometries [11]. For open linear chain structure, the chaotic cells are seen to synchronize consecutively as a synchronization wave spreads through the array. In the case of closed loops, the behavior is more complex due to the presence of feedback. It is found that there are a critical number of cells above which the uniform synchronized state will not be stable. Other coupled patterns for oscillators can also be considered, such as cascade, etc. These structures and methods might allow people to design in a more systematic and compact way circuits, which can be employed as chaotic filters for secure communications, or as cells potentially useful for arrays of cellular neural networks [1,13]. The studies of the kinetic and mechanistic of Carboxylic Acid-Bisoxazoline reactions are the studies of the chain pattern [14]. It can be found from the research of stability and fragmentation of complex structures in ferrofluids when quaternion molecular dynamics is used to determine the equilibrium structure, the stability, and the dynamics of fragmentation of magnetic particle with complex one-dimensional structures. That rings are more stable than chains at low fields and beyond a critical size. Rings orient perpendicularly to an applied field and can be activated to fragment into chains [15]. Consider coupled oscillators in a ring geometry, which has several interesting applications in a number of biological systems. Especially when dealing with the so-called central pattern generators (CPGs), which have been shown to play important roles in peripheral neural systems, locomotion, etc. [16,17]. Consider the linear coupled networks with nodes; each one is an n-dimensional dynamical system x i = fi( xi, t), where xi = ( xi1,, xin) ( i = 1,,, ). The coupled systems are described by x 1 = f1( x1, t) + D1, ixi, i= 1 (1) x = f( x, t) + D, ixi, i= 1 HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64 617

T where xi = ( xi1,, xin) ( i = 1,,, ) and Di, j R. T i 1, i, i Assume x = ( x,, x ), ( i = 1,,, n). Then, system (1) can be rewritten as where D i, j R. n n n x 1 = f 1( x 1, t) + D 1, ix i, i= 1 () n x n = f n( x n, t) + D n, ix i, i= 1 t1 0 0 0 t 0 Let f1 = f = = f = f, and T = is the feedback gain matrix such that 0 0 t n the origin of the system x = f( x, t) Tx is uniformly asymptotic stable, where t i 0, i = 1,, n. Then, we have the following lemma: Lemma 1 [18]. Suppose the coupling matrices D ii, are either zero or a normal matrix such T that D ii, + D ii, is an irreducible matrix with the zero sum of each row. Assume also that all the T matrices D ii, + D ii, are of the same form, i.e., there exists α i,j for each i, j such that D ii, + T T D = α ( D + D ). Then, the coupled system () is uniformly asymptotically synchronized if i, i i, j j, j j, j real parts of the eigenvalues of nonzero D ii, is less than or equal to t i. Consider a familiar coupling structure. Assume all the D, = 0, i j, D, = cd, c > 0, i = 1,,, n, where D is the coupling structure matrix of networks. In addition, D is a symmetric c1 0 0 0 c 0 matrix with negative diagonal entries and zero sum of each row. Denote C = 0 0 c n as the coupling strength matrix. Assume 0 = λ1, λ, λ3,, λ as the eigenvalues of D, and let 0= λ1 > Re( λ) Re ( λ3) Re( λ ). Denote R = Re( λ )/ Re( λ) = λ / λ when D is a symmetric diffusion coupled matrix. For a given network, the smaller R implies the better synchronizing ability and the bigger R is bad to synchronize. Moreover, it is pointed out in ref. [5] that the largest nonzero eigenvalue λ of 0 = λ1 > λ λ3 λ can be looked at as the criteria on determining the synchronizing ability of the coupled networks. Plus, the network with smaller λ is easy to synchronize. There are complex eigenvalues if the matrix is not symmetric. It can be obtained by Lemma 1 that the coupled systems will synchronize if ci Re( λ k ) ti, k = 1,,,, i = 1,,, n. That is ci ti / Re( λk), k = 1,,, i = 1,,, n. Then, each c i must be at least ti /Re( λ ). Therefore, the network with smaller Re( λ ) is easy to synchronize. i j ii i i 618 HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64

The practical meanings of many real-world networks decide their coupling directionality, such as using a neural network model simulates the dynamics of the lamprey spinal cord by computer. The directionality is very important in many fields. The real-time interaction between neuromorphic electronic circuit and the spinal cord were interfaced in unidirectional and bidirectional modes. There are unidirectional congestions of cordis fibre in bioengineering, etc. [19,0]. Activity of many physiological subsystems has well-expressed rhythmic character. Often, a dependency between physiological rhythms is established due to interaction between the corresponding subsystems. Some new methods of data analysis for quantification of coupling directionality can be applied to study the mutual influence of respiratory and cardiovascular rhythms. The directional analysis may be used to other interacting narrow band oscillatory system, e.g., in the central nervous system. It is an important step forward in revealing and understanding causal mechanisms of interactions [1]. The coupling directionality of the nonlinear elements contained in an optical ring cavity determine the cooperative dynamics and functions in a collective nonlinear optical element systems. The results can be applied to predict phenomena, such as a cooperative all-optical switching, multivibrator operations, flip-flop operations, as well as spatial chaos memory, etc. []. Therefore, it is necessary to discuss the coupling directionality. Different coupling matrix denotes different network structures. Consider the bidirectional coupled ring networks and the unidirectional coupled ring networks. The corresponding matrices are 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 Db = 0 0 1 0 0 and Du = 0 0 1 0 0. 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 Suppose is even, the eigenvalues of bidirectional coupling matrix are 0, k = 1,,, 1. The smallest and the largest nonzero eigenvalues are 4 and respectively. Then, parts of eigenvalues are 0, 4sin ( kπ / ), 4sin ( π / ), R = 1/sin ( π / ). In the case of the unidirectional coupled ring, the real sin ( kπ / ), k = 1,,, 1, and R = /( sin ( π / )) = 1/sin ( π / ). The matrix of the bidirectional chain coupled networks is 1 1 0 0 0 1 1 0 0 0 1 0 0 D = 0 0 1 0 0, 0 0 0 1 0 0 0 1 1 and its eigenvalues are 0, 4sin ( kπ / ), k = 1,,, 1. Then, the smallest and largest nonzero eigenvalues are 4sin (( 1) π / ) and 4sin ( π / ), respectively, and HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64 619

sin (( 1) π / ) R =. The coupling matrix of the unidirectional chain coupled network is sin ( π / ) 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 D =, 0 0 1 0 0 0 0 0 1 1 the eigenvalues are 0 and 1 ( 1 multiplicity). Its smallest and largest one all are 1 and R = 1. sin (( 1) π / ) It is easy to see that the ratior increase from R = 1/sin ( π / ) to R = sin ( π / ) as the bidirectional ring coupled networks changed to bidirectional chain networks by cutting a sin ( π / )sin (( 1) π / ) link. The ratio between the latter R and the front case is. Then, sin ( π / ) the synchronizing ability decrease as the number of the nodes is larger than 4. The ratio R of bidirectional chain networks is about four times that of the bidirectional ring case if is large enough [3]. For the unidirectional structure, the network is much easier to synchronize if Re( λ ) is much smaller. Re( λ ) will be changed from sin ( π / ) to 1 if the networks are changed from unidirectional ring coupled networks to unidirectional chain case. The synchronizing ability increases if > 4. It will be /( π ) times if number of is large enough. The numerical simulations confirm the conclusion in quality. The changes on the synchronizing ability are different for the two different ring structures change into chain structures. This conclusion is very interesting. Many such phenomena exist in real-worlds, such as the decision-making network with many people, people s cognitive network, etc. This paper also studied the effects on the synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. For given networks with nodes, denote the nodes from 1 to. Then, we can define the length of the path between node 1 and other nodes i( i ) for the two structures. For unidirectional ring coupled networks, define the distance of the path between node 1 and ii ( 1) as d = i 1. For the bidirectional ring coupled network, let the distance of the path between node 1 and ii ( 1), i 1, i, + 1 i 1, i, d =, i= + 1, as is even; d = as is odd. + 3 i+ 1, i, i+ 1, + i, The length of the side is the distance of the path d if we add one side to the network, which links node 1 and ii ( 1). The following are the numerical simulations of the efforts on synchro- 60 HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64

nizing ability for the two different rings. Figures 1 3 show the relations between the network synchronization criterion and the length of the adding link with = 500. Figure 1 displays the relation between d and 1/Re( λ ) with unidirectional ring adding a link. Figure shows the relation between d and R = Re( λ ) / Re( λ ). Figure 3 shows the relation between d and lgr( R = λ / λ ) with bidirectional ring coupled networks. Figures 4 6 show the same relations with = 1000, Figures 4 and 5 show the relation between d and 1/Re( λ ), d and R = Re( λ ) / Re( λ ), respectively, with unidirectional coupling. Figure 6 displays the relation between d and lgr( R = λ / λ ) with bidirectional case. It can be seen from the simulations that the effects on the synchronization ability of the unidirectional ring by adding a link is regular. The effort on synchronization of the network is symmetric about d = /. In addition, the largest effect is not on the longest length d = /. The syn- 1 Figure 1 Relation between d and (unidirectional Figure Re( λ ) ring = 500). ring = 500). Relation between d and Re( λ ) R = (unidirectional Re( λ ) Figure 3 Relation between d and lgr (bidirectional ring Figure 4 Relation between d and = 500). = 1000). 1 Re( λ ) (unidirectional ring HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64 61

Figure 5 Relation between d and ctional ring = 1000). Re( λ ) R = (unidire- Figure 6 Relation between d and lgr (bidirectional ring Re( λ) = 1000). chronizing ability increase obviously as d is away from 1 or 1. There are some wave changes as d converge to /. However, bidirectional case is different from that. That is, the synchronization ability cannot change if the length of link achieve a constant. 3 umerical simulations To demonstrate the above results, the chaotic Lorenz system is used as a dynamical node of the coupled networks. The bidirectional ring coupled systems and the unidirectional ring coupled systems are described by x 1 = f( x1) + C( x + x x1), x 1 = f( x1) + C( x x1), x = f( x) + C( x1 + x3 x), x = f( x) + C( x1 x), and x = f( x) + C( x1 + x 1 x), x = f( x) + C( x 1 x). The equations can be written as the following if a link is cut from the ring structure: x 1 = f( x1) + C( x x1), x 1 = f( x1), x = f( x) + C( x1 + x3 x), x = f( x) + C( x1 x), and x = f( x) + C( x 1 x), x = f( x) + C( x 1 x), where C has the same definition as front. Assume the dynamics of the ith node is x i = f( xi), and described by x i1 = α( xi xi1), x i = σ xi1 xi1xi3 xi, x i3 = xi1xi βxi3.( α, β, σ > 0). When α = 10, β = 8/ 3, σ = 8, Lorenz system has a chaotic attractor. Figures 7 and 8 show the synchronization of the unidirectional ring coupled networks and the 6 HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64

1 3 unidirectional chain coupled networks with = 100, respectively, where the coupling strengths c = 100, c = 0, c = 0. It can be seen that the unidirectional chain coupled networks synchronize easily after cutting a link from the unidirectional ring structure. Figure 7 Synchronization of unidirectional ring. Figure 8 Synchronization of unidirectional chain. Figures 9 and 10 show the synchronization of the bidirectional ring coupled systems and the bidirectional chain coupled systems with = 100, where the coupling strengths c 1 = 300, c = 300, c = 300. It can be learned that the synchronizing ability will decrease if the change is 3 from bidirectional ring to bidirectional chain. Figure 9 Synchronization of bidirectional ring. Figure 10 Synchronization of bidirectional chain. 4 Conclusion Many real-world networks have ring and chain basic structures. Moreover, the coupling directionality of the coupled nonlinear systems is very important. In this paper, two different network structures, the ring networks with unidirectional couplings and with bidirectional couplings, were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures are completely opposite by cutting a link. The synchronizing ability will decrease if the change is from bidirectional ring coupled networks to bidirectional chain coupled networks. In addition, the change on synchronizing ability will be four times if the number of is HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64 63

large enough. However, it will increase obviously from unidirectional ring coupled networks to unidirectional chain coupled networks. It will be /(π ) times if number of is large enough. The numerical simulations confirm the conclusion in quality with = 100 Lorenz coupled networks. This paper also discussed the effects on synchronization by adding one link with different length d to these two different structures. It can be seen from numerical simulation that the effects are different. These results proposed in this paper have some important reference meanings on the synchronization of real world networks, and study the robustness and frangibility of the networks, and decrease or increase the synchronization ability of the networks. It is also important for constructing networks, such as the bioecological system networks, the designing of the circuit, etc. 1 Watts D J, Strogatz S H. Collective dynamics of small world networks. ature, 1998, 393: 440-44[DOI] Barabasi A L, Albert R. Emergence of scaling in random networks. Science, 1999, 86: 509-51[DOI] 3 Wang X F, Li X, Chen G R. Theory and Application of Complex etwork. (in Chinese) Beijing: Tsinghua University Press, 006 4 Zheng Z G. Spatiotemporal Dynamics and Collective Behaviors in Coupled onlinear Systems (in Chinese). Beijing: Higher Education Press, 004 5 Hong H, Kim B J, Choi M Y, et al. Factors that predict better synchronizability on complex networks. Phys Rev E, 004, 69: 067105-1-067105-4[DOI] 6 Barahona M, Pecora L M. Synchronization in small-world systems. Phys Rev Lett, 00, 89(5): 054101-1 - 054101-4[DOI] 7 Wang X F, Chen G R. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans Circuits Systems-I, 00, 49(1): 54-6[DOI] 8 Wu C W, Chua L O. Application of Graph Theory to the Synchronization in an array of coupled nonlinear oscillators. IEEE Trans Circuits Systems-I, 1995, 4(8): 494-497[DOI] 9 Wang X F, Chen G R. Synchronization in small-world dynamical networks. Int J Bifur Chaos, 00, 1(1): 187-19[DOI] 10 Matías M A, Pére-Muńuzuri V, Lorenzo M, et al. Observation of a fast rotating wave in rings of coupled chaotic oscillators. Phys Rev Lett, 1997, 78(): 19-[DOI] 11 Lorenzo M, Marińo I P, Pére-Muńuzuri V, et al. Synchronization waves in arrays of driven chaotic systems. Phys Rev E, 1996, 54 (4): 3094-3097 [DOI] 1 Cuomo K M, Oppenhein A V. Circuit implementation of synchronized chaos with applications to communications. Phys Rev Lett, 1993, 71(1): 65-68[DOI] 13 Chua L O, Yang L. Cellular neural networks: Theory and applications. IEEE Trans Circuits Systems, 1988, 35: 157-17[DOI] 14 éry L, Lefebvre H, Fradet A. Kinetic and mechanistic studies of carboxylic acid bisoxazoline chain-coupling reactions. Macromol Chem Phys, 003, 04: 1755-1764 [DOI] 15 Jund P, Kim S G, Toma nek D, et al. Stability and fragmentation of complex structures in ferrofluids. Phys Rev Lett, 1995, 74(15): 3049-305[DOI] 16 Collins J J E, Stewart I. A group-theoretic approach to rings of coupled biological oscillators. Biol Cybern, 1994, 71: 95-103 17 Abarbanel H D I, Rabinovich M I, Selverston A, et al. Synchronization in neural networks. Phys Uspek, 1996, 39: 337-36[DOI] 18 Wu C W, Chua L O. Synchronization in an array of linearly coupled dynamical systems. IEEE Trans Circuits Systems-I, 1995, 4(8): 430-447[DOI] 19 Ranu J, Elizabeth J B, James J A. Real-time interactions between a neuromorphic electronic circuit and spinal cord. IEEE Trans eural Systems Rehab Eng, 001, 9(3): 319-36[DOI] 0 Sahakian A V, Myers G A, Maglaveras. Unidirectional block in cardiac fibers: Effects of discontinuties in coupling resistance and spatial changes in resting membrane potential in a computer simulation study. IEEE Trans on Biomed Eng, 199, 39( 5): 510-5[DOI] 1 Ralf M, Laura C, Andreas P, et al. Directionality of coupling of physiological subsystems: age-related changes of cardiorespiratory interaction during different sleep stages in babies. Am J Physiol Regulatory Integrative Comp Physiol, 003, 85: 1395-1401 Otsuka K, Ikeda K. Cooperative dynamics and functions in a collective nonlinear optical element system. Phys Rev A, 1989, 39(10): 509-58 [DOI] 3 Wang W, Slotine J J E. On partial contraction analysis for coupled nonlinear oscillators. Biol Cybern, 004, 9(1): 38-53[DOI] 64 HA XiuPing et al. Sci China Ser F-Inf Sci Aug. 007 vol. 50 no. 4 615-64