Average Range and Network Synchronizability

Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 115 120 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 1, January 15, 2010 Average Range and Network Synchronizability LIU Chao ( ), 1, LI Rong (Ó ), 2 DUAN Zhi-Sheng (ãµ), 3 and CHEN Guan-Rong (í³â) 4 1 Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China 2 Department of Applied Mathematics, Shanghai Finance University, Shanghai 201209, China 3 State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China 4 Department of Electronic Engineering, City University of Hong Kong, Hong Kong, SAR, China (Received June 30, 2009; revised manuscript received October 30, 2009) Abstract The influence of structural properties of a network on the network synchronizability is studied by introducing a new concept of average range of edges. For both small-world and scale-free networks, the effect of average range on the synchronizability of networks with bounded or unbounded synchronization regions is illustrated through numerical simulations. The relations between average range, range distribution, average distance, and maximum betweenness are also explored, revealing the effects of these factors on the network synchronizability of the small-world and scale-free networks, respectively. PACS numbers: 05.45.Xt, 89.75.-k, 89.75.Fb Key words: complex network, synchronizability, average range, range distribution 1 Introduction A subject of great interest in recent years is how the collective dynamics of a complex network are influenced by the structural properties of the network. [1 3] As an important paradigm, the effects of these structural factors on the synchronizability of dynamical networks are under extensive investigation, partly due to the elegant analysis of Pecora and Barahona, [4 5] which distinguishes the influence of the network structure from other factors on the network synchronizability in terms of the eigenvalues of the coupling matrix. The structural factors considered today include the average distance, heterogeneity of degree distribution, maximum betweenness, and so on. [6 16] Studies have shown that small-world and scale-free networks are much easier to synchronize than regular lattices. [6 7] Thus, it is commonly believed that shorter average distance tends to enhance network synchronizability. [8 9] However, it is reported in Ref. [10] that the synchronization of scale-free networks can be suppressed as the degree distribution becomes more heterogeneous, even with a shorter average network distance. On the contrary, Ref. [11] shows that the network synchronizability of a small-world network can be improved as the heterogeneity of the degree distribution is increased or as the average distance is decreased. In that paper, the maximal betweenness was suggested as a suitable indicator for network synchronizability: the smaller the maximal betweenness, the better the synchronizability. In Ref. [12], however, it is demonstrated that the combination of short average distance and homogeneous degree distribution ensures strong synchronizability; the maximum betweenness alone can have different behaviors in some networks, therefore may not give a comprehensive description of network synchronizability. Today, it is still not clear how the structural ingredients affect the network synchronization, so the issue deserves further research. Moreover, a structural factor may have different effects on the synchronizability of the smallworld and scale-free networks. It is also worth noting that the structural factors considered in the literature, such as the average distance, heterogeneity of degree distribution, and maximum betweenness, mainly characterize the structural features regarding the nodes of a network. Therefore, a natural question is what kind of influences the structural features of network edges may have on the network synchronization. The present paper addresses this concerned issue. Recall the concept of range, [13 14] used to characterize different types of edges: the range of an edge l ij connecting nodes i and j, denoted as R ij, is the length of the shortest path between nodes i and j in the absence of l ij ; set R ij = 0 if nodes i and j are disconnected when edge l ij is deleted, i.e., if edge l ij is not in any loops. According to the definition, range has close connection with loop: range R ij is one less than the shortest loop containing edge l ij. In Ref. [15], by performing range-based attacks on edges, Supported by the Special Research Funds for Selection and Career Development of Outstanding Young Teachers in Higher Educational Institutions of Shanghai, China, the Leading Academic Discipline Program, the 211 Project for Shanghai University of Finance and Economics (the 3rd phase), the National Science Foundation of China under Grant No. 10832006, and the City University of Hong Kong under the SRG under Grant No. 7002134453 Corresponding author, E-mail: liu.chao@mail.shufe.edu.cn

116 LIU Chao, LI Rong, DUAN Zhi-Sheng, and CHEN Guan-Rong Vol. 53 it is found that short-range edges rather than long-range edges are the vital ones contributing to the small average distance in scale-free networks, which is contrary to the homogeneous small-world, and random-graph networks. An interesting result about range and synchronizability is given in Ref. [16], where a rewiring algorithm is applied to diverse networks, including random-graph, small-world, and scale-free networks, to get optimal networks with synchronizability as high as possible. It is shown that, despite the different initial networks, the optimal networks, called entangled networks, have one common topological feature: the absence of short loops, i.e., they share a narrow shortest-loop distribution with a large mean value. The above results lead to some natural questions about network range. How does the synchronizability depend on range? Will network range have different influences on the synchronizability of different kinds of networks? What are the relations between range and other topological characteristics? As an attempt to providing some answers to these questions, the relations between range, average distance, heterogeneity of degree distribution, maximal betweenness, and network synchronizability are investigated via numerical simulations in this paper, on the WS smallworld network [17] and BA scale-free network. [18] The rest of this paper is organized as follows. In Sec. 2, the description of network synchronizability in terms of eigenvalues of the coupling matrix is introduced. In addition, the definition of average range is given. The program flow is introduced in Sec. 3, including the so-called random interchanging algorithm which allows one to manipulate the average range while keeping the network s degree distribution unchanged. The main simulation results are shown and discussed in Sec. 4. The paper is concluded by the last section. 2 Network Synchronizability and Range Characteristics Consider a dynamical network consisting of N diffusively coupled identical nodes, with each node being an n-dimensional dynamical system: N ẋ i = f(x i ) c a ij H(x j ), i = 1, 2,...,N, (1) j=1 where x i = (x i1, x i2,..., x in ) R n is the state vector of node i, ẋ = f(x) governs the dynamics of each individual node, the constant c > 0 is the coupling strength, H(x) is the output function, and A = (a ij ) is the Laplacian matrix, defined to be a ij = 1 if nodes i and j are connected, a ij = 0 otherwise, and a ii = k i with k i being the degree of node i. The above conditions of diffusive connection imply that zero is an eigenvalue of A with multiplicity 1 and all the other eigenvalues of A are strictly positive, denoted by 0 = λ 1 < λ 2 λ N. (2) The dynamical network (1) is said to achieve synchronization if x 1 (t) x 2 (t) x N (t), as t, (3) in norm. Due to the master stability function method introduced in Ref. [4], the influence of network structural properties on the network synchronization can be in a sense separated from that of the node dynamics: the influence of the node dynamics is embodied on the synchronization region S, while the influence of the structural properties is characterized by the eigenvalues of the coupling matrix. More specifically, the linear stability condition on the synchronization manifold is given by cλ i S, i = 2, 3,...,N. (4) Furthermore, the network synchronizability represented by the eigenvalues of the coupling matrix are given according to different types of synchronization regions. [19 20] For networks with unbounded synchronization regions, the synchronizability is measured by the parameter β 1 = λ 2 : the larger the β 1, the better the synchronizability. For networks with bounded synchronization regions, the synchronizability is measured by the parameter β 2 = λ N /λ 2 : the smaller the β 2, the better the synchronizability. An interesting phenomenon of networks with disconnected synchronization regions is recently investigated in Refs. [21 23]. In this paper, the influence of network range on the synchronizability of both WS and BA networks are explored. Both networks with bounded or unbounded synchronization regions are considered. First, the concept of average range is formally defined. Definition Average range: the average range of a network, denoted as R, is the average value of all ranges over the total number of edges: R = 2 i k i R ij, (5) i,j i j where k i is the degree of node i, i = 1,...,N. 3 Program Flows To investigate the effect of range on the network synchronizability, the random interchanging algorithm [12,24] is used to adjust the average range while keeping the degree distribution unchanged. The procedure is as follows: (i) Randomly pick two existing edges, l 1 = x 1 x 2 and l 2 = x 3 x 4, such that x 1 x 2 x 3 x 4 and there is no edge between x 1 and x 4 nor between x 2 and x 3. (ii) Interchange these two edges, that is, connect x 1 and x 4 and connect x 2 and x 3, but remove the original edges l 1 and l 2. (iii) Ensure the interchange increases the average range. If it does, accept the new configuration, else return to the original setting.

No. 1 Average Range and Network Synchronizability 117 (iv) Repeat steps (i) (iii) until a desired value of the average range is reached. Figure 1 illustrates the random interchanging algorithm. Since this algorithm only rewires connections but does not change the degree of any node, the degree distribution as well as the degree sequence are unchanged. shows that λ 2 is more sensitive to R than to σ. With the increase of the average range R, parameter β 1 = λ 2 grows as well, which predicts better synchronizability for networks with unbounded synchronization regions. The dependance of the eigenratio λ N /λ 2 on the average range R on the WS network is exhibited in Fig. 4. As the average range R increases, parameter β 2 = λ N /λ 2 decreases, indicating better synchronizability for networks with bounded synchronization regions. Thus, for the WS network with either an unbounded or a bounded synchronization region, enlarging the average range makes the network more synchronizable. Illustration of the random interchanging algo- Fig. 1 rithm. 4 Simulation Results In this section, simulation results are presented, to show how the synchronizability depends on the average range on both WS and BA networks. The relations between average range, range distribution, average distance, and maximum betweenness are also illustrated. Figures 2 7 show the simulation results on a WS network with size N = 500, average degree k = 4, and standard deviations of degree distributions σ = 0.35, 0.7, and 1, respectively. Triangles, squares, and circles in the six figures represent the cases of σ = 0.35, 0.7, and 1, respectively. All the data are the averages over 20 different simulation runs. Fig. 3 Parameter β 1 = λ 2 vs. average range R on a WS network. Fig. 2 Maximal eigenvalue λ N vs. average range R on a WS network. It can be seen from Fig. 2 that λ N, the maximal eigenvalue of the coupling matrix A, varies only slightly with the changing average range R. It seems that σ, the standard deviation of degree distribution, has larger effect on λ N : the smaller the σ, the smaller the λ N. However, for λ 2, the second smallest eigenvalue of the coupling matrix A, the simulation results are quite different. Figure 3 Fig. 4 Parameter β 2 = λ N/λ 2 vs. average range R on a WS network. From Fig. 5, it can be seen that the standard deviation of the range distribution, denoted as D r, mainly decreases as the average range R increases on the WS network, which indicates that the network becomes more homogeneous in range distribution. Figures 3 5 show that, for the WS network, a narrow shortest-loop distribution with large mean value predicts a better synchronizability,

118 LIU Chao, LI Rong, DUAN Zhi-Sheng, and CHEN Guan-Rong Vol. 53 which is consistent with the result reported in Ref. [16]. smaller maximum betweenness indicates better synchronizability of the WS network, which is consistent with the report. [11] Figures 8 13 exhibit the simulation results of a BA network with size N = 500 and standard deviations of degree distributions σ = 17, 23, and 30, respectively. Triangles, squares, and circles in the six figures represent the cases of σ = 17, 23, and 30, respectively. All the data are the average over 20 different simulation runs. Fig. 5 Standard deviation of range distribution D r vs. average range R on a WS network. Fig. 8 Maximal eigenvalue λ N vs. average range R on a BA network. Fig. 6 Average distance L vs. average range R on a WS network. Figure 8 shows a similar result to Fig. 2: the standard deviation of the degree distribution σ has more influence than the average range R on λ N, and the smaller the σ, the smaller the λ N. However, Fig. 9 exhibits a contrary result to that of Fig. 3. For the BA network, parameter β 1 = λ 2 decreases as the average range R increases, indicating a worse synchronizability for networks with unbounded synchronization regions. As a consequence of Figs. 8 and 9, parameter β 2 = λ N /λ 2 increases with the increasing average range R on the BA network, as shown in Fig. 10, indicating a worse synchronizability for networks with bounded synchronization regions. Thus, contrary to the WS network, for a BA network with either an unbounded or a bounded synchronization region, shortening the average range makes the network more synchronizable. Fig. 7 Maximum betweenness B max vs. average range R on a WS network. Figure 6 shows that, as the average range R increases, the average distance L decreases on the WS network. From Fig. 7, it can be seen that the maximum betweenness B max decreases as the average range R increases on the WS network. Considering the varying trend of the network synchronizability, shortening the average distance enhances the synchronization of the WS network, which is consistent with the common sense. Figure 7 shows that Fig. 9 Parameter β 1 = λ 2 vs. average range R on a BA network.

No. 1 Average Range and Network Synchronizability 119 Figure 11 shows that the standard deviation of the range distribution D r increases with the increasing of the average range R on the BA network, which is contrary to the result of the WS network shown in Fig. 5. This implies that a BA network will become more heterogenous in terms of range distribution as the average range R increases. Figures 9 11 show that, for the BA network, a narrow shortest-loop distribution with a small mean value predicts a better synchronizability. have different behaviors on WS and BA networks. Fig. 12 Average distance L vs. average range R on a BA network. Fig. 10 Parameter β 2 = λ N/λ 2 vs. average range R on a BA network. Fig. 13 Maximum betweenness B max vs. average range R on a BA network. Fig. 11 Standard deviation of range distribution D r vs. average range R on a BA network. Figures 12 and 13 show the increase of the average distance L and the decrease of the maximum betweenness B max with the increasing of the average range R on the BA network, respectively. The results of Figs. 6 and 12 confirm the common belief that a shorter average distance tends to enhance the network synchronization for both WS and BA networks. The result of Fig. 13 confirms the assertion of Ref. [12]; that is, the maximum betweenness can The influences of the standard deviation of the degree distribution σ on the synchronizability of WS and BA networks are also shown in Figs. 3 4 and Figs. 9 10, respectively. For the WS network, because of the small varying range of σ, the heterogeneity of degree distribution does not have much effect on the network synchronization, as displayed in Figs. 3 4. For the BA network, it can be seen from Figs. 9 10 that with a fixed average range, homogeneous degree distribution generally indicates better synchronizability. By definition, range is a structural factor indicating the importance of edges from the aspect of node-node distance. Indeed, the removal of an edge l ij of range R ij increases the length of the shortest path between nodes i and j by R ij 1. Thus, a long-range edge can be regarded as a shortcut that connects otherwise distant nodes. The different effects of the average range on the synchronizability of WS and BA networks, as exhibited in Figs. 3 4 and Figs. 9 10, are in fact due to the exchange of the roles of short and long-range edges for these networks. With exponential degree distributions, WS networks are homogeneous networks. All the nodes in a WS network have

120 LIU Chao, LI Rong, DUAN Zhi-Sheng, and CHEN Guan-Rong Vol. 53 approximately the same connectivity. For WS networks, the small average distance, and thus the efficient communication, is mainly due to long-range connections, while short-range edges are more responsible for high clustering. Whereas, BA networks are heterogeneous as their connectivity can vary significantly from node to node. Edges connecting highly connected nodes, which tend to be shortrange edges as reported in Ref. [15], are the vital ones for the communication in these networks. In simulations, by performing the random interchanging algorithm, we adjusted the average range while keeping the degree distribution unchanged, that is, keeping the homogeneity or heterogeneity of the network unchanged. As the average range R increases, the proportion of long-range edges is enhanced, which leads to a shorter average distance and better synchronization in WS networks. However, in BA networks, raising the average range tends to cause the removal of edges connecting highly connected nodes, and thus leads to worse synchronizability. 5 Conclusion In this paper, the influence of structural properties of network edges on network synchronizability has been investigated in terms of average range. It is found that the average range has different effects on the synchronizability of WS and BA networks. For WS networks with either unbounded or bounded synchronized regions, enlarging the average range tends to enhance the network synchronizability. For BA networks with both kinds of synchronized regions, however, the smaller the average range, the better the synchronizability. Moreover, for WS networks, as the average range increases, the range distribution becomes narrower, and both the average distance and the maximum betweenness decrease, which are all contrary to the results on BA networks. Effects of other structural properties including the heterogeneity of degree distribution, maximum betweenness, and average distance on the synchronizability of WS and BA networks can also be deduced from the simulations. It is found that the homogeneous degree distribution tends to enhance the synchronizability of BA networks, while this effect is not as obvious on WS networks because of the small varying range of degree distribution. The maximum betweenness, on the other hand, can have different behaviors on WS and BA networks. Last but not least, for both WS and BA networks, shortening the average distance tends to enhance the network synchronizability in general. References [1] B. Tadić, S. Thurner, and G.J. Rodgers, Phys. Rev. E 69 (2004) 036102. [2] X. Li, G.R. Chen, and C.G. Li, Int. J. Syst. Sci. 35 (2004) 527. [3] K.I. Goh, D.S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91 (2003) 148701. [4] L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 80 (1998) 2109. [5] M. Barahona and L.M. Pecora, Phys. Rev. Lett. 89 (2002) 054101. [6] L.F. Lago-Fernández, R. Huerta, F. Corbacho, and J.A. Sigüenza, Phys. Rev. Lett. 84 (2000) 2758. [7] P.G. Lind, J.A.C. Gallas, and H. J. Herrmann, Phys. Rev. E 70 (2004) 056207. [8] X.F. Wang and G.R. Chen, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12 (2002) 187. [9] T. Zhou, M. Zhao, and B.H. Wang, Phys. Rev. E 73 (2006) 037101. [10] T. Nishikawa, A.E. Motter, Y.C. Lai, and F.C. Hoppensteadt, Phys. Rev. Lett. 91 (2003) 014101. [11] H. Hong, B.J. Kim, M.Y. Choi, and H. Park, Phys. Rev. E 69 (2004) 067105. [12] M. Zhao, T. Zhou, B.H. Wang, G. Yan, H.J. Yang, and W.J. Bai, Phys. A 371 (2006) 773. [13] D.J. Watts, Small Worlds, Princeton University Press, Princeton (1999). [14] S.A. Pandit and R.E. Amritkar, Phys. Rev. E 60 (1999) R1119. [15] A.E. Motter, T. Nishikawa, and Y.C. Lai, Phys. Rev. E 66 (2002) 065103(R). [16] L. Donetti, P.I. Hurtado, and M.A. Munõz, Phys. Rev. Lett. 95 (2005) 188701. [17] D.J. Watts and S.H. Strogatz, Nature (London) 393 (1998) 440. [18] A.L. Barabási and R. Albert, Science 286 (1999) 509. [19] L. Kocarev and P. Amato, Chaos 15 (2005) 024101. [20] J.H. Lü, X.H. Yu, G.R. Chen, and D.Z. Cheng, IEEE Trans. Circuits Syst.-I 51 (2004) 787. [21] C. Liu, Z.S. Duan, G.R. Chen, and L. Huang, Physica A 386 (2007) 531. [22] Z.S. Duan, G.R. Chen, and L. Huang, Phys. Lett. A 372 (2008) 3741. [23] A. Stefański, P. Perlikowski, and T. Kapitaniak, Phys. Rev. E 75 (2007) 016210. [24] B.J. Kim, Phys. Rev. E 69 (2004) 045101(R).