COMPLEX DYNAMICAL NETWORKS: MODELLING, SYNCHRONIZATION AND CONTROL

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1 Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 11a (2004) Copyright c 2004 Watam Press COMPLEX DYNAMICAL NETWORKS: MODELLING, SYNCHRONIZATION AND CONTROL Jinhu Lü 1, Henry Leung 2 and Guanrong Chen 3 1 Institute of Systems Science, Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing , P.R. China. 2 Department of Electrical and Computer Engineering University of Calgary, Calgary, Alberta, Canada T2N 1N4 3 Department of Electronic Engineering City University of Hong Kong, Kowloon, Hong Kong, P.R. China. Abstract. Over the last few years, complex networks have been intensively studied in all fields of sciences and humanities. Today, complex networks have become a very important part of our daily lives. This paper reviews and introduces some basic mathematical models of complex dynamical networks, as well as their synchronization and control problems. Several synchronization criteria are presented. Keywords. Complex dynamical networks, mathematical modelling, synchronization, control AMS (MOS) subject classification:34c28, 34H05, 94C15, 94B99 1 Introduction Complex networks refers to large-scale networks with complicated topological structures and dynamical behaviors, including such as the Internet, the World Wide Web, local computer networks, electrical power grids, telephone call graphs, biological neural networks, food webs, cellular and metabolic networks, coauthor-ship and citation networks of scientists, and so on [1,14]. Generally speaking, a complex network is a graph with a large set of interconnected nodes, where the nodes are the basic units of the system with specific dynamical and information contents, while the links are the connections measuring the interactions among the nodes [1-14]. Today, we are building various interconnected and integrated complex networks for energy, transportation, information, and finance, among many others. These complex networks are very important and closely relative to our daily lives. The network security against failures and attacks is an extremely important issue for everyone, especially for major networks such as the Internet and various e-bank networks [6,7]. This calls for greater efforts to better understand the essential nature and fundamental behaviors of network topological structures as well as their synchronization properties, towards better design and management of real-world networks [4-7].

2 Complex Dynamical Networks 71 Complex networks with different topological structures may be classified as regular networks, random networks, small-world networks, scale-free networks, and evolving networks [1]. It has been demonstrated that many complex dynamical networks display various synchronization phenomena. In this paper, we review several fundamental mathematical models for complex networks, including the small-world network model [14], scale-free network model [1,2], local-world evolving network model [3], and general time-varying network model [4-7]. Based on the general time-varying network model, several fundamental network synchronization theorems are presented. In particular, we show that the synchronization of such a general time-varying complex dynamical network is completely determined by its inner-coupled matrix and its coupled configuration matrix specifically the eigenvalues and the corresponding eigenvectors of this coupled configuration matrix, rather than the sole eigenvalues of the coupled configuration matrix for a uniform network. Moreover, control problems [13] of complex networks will also be discussed briefly. This paper is organized as follows: In Section 2, we review several fundamental mathematical models of complex dynamical networks. The definition of network synchronization and several synchronization criteria are presented in Section 3. In Section 4, the control problem of complex dynamical networks is briefly discussed. Conclusions are given in Section 5. 2 Mathematical Models of Complex Dynamical Networks 2.1 Small-World Networks Model In 1998, Watts and Strogatz [14] proposed a single-parameter small-world network model that bridges the gap between a regular network and a random graph. The original WS model is described as follows: (i) Initialize: Start with a nearest-neighbor coupled ring lattice with N nodes, in which each node i is connected to its K neighboring nodes i±1, i±2,, i± K 2, where K is an even integer. (Assume that N K ln(n) 1, which guarantees that the network is connected but sparse at all times.) (ii) Randomize: Randomly rewire each link of the network with probability p such that self-connections and duplicated links are excluded. Rewiring in this sense means transferring one end of the connection to a randomly chosen node. (This process introduces pnk 2 long-range links, which connect some nodes that otherwise would not have direct connections. One thus can closely monitor the transition between order (p = 0) and randomness (p = 1) by adjusting p.)

3 72 J. Lü, H. Leung, G. Chen A small-world network lies along a continuum of network models between the two extreme networks: regular and random ones. Recently, Newman and Watts modified the original WS model. In the NW modelling, instead of rewiring links between nodes, extra links called shortcuts are added between pairs of nodes chosen at random, but no links are removed from the existing network. Clearly, the NW model reduces to the originally nearest-neighbor coupled network if p = 0; while it becomes a globally coupled network if p = 1. However, the NW model is equivalent to the WS model for sufficiently small p and sufficiently large N values. The WS and NW models show a transition with an increasing number of nodes, from a large-world regime in which the average distance between two nodes increases linearly with the system size, to a small-world model in which it increases only logarithmically. 2.2 Scale-Free Networks Model In 1999, Barabási and Albert [1,2] introduced a scale-free network model, which continuously grows by the addition of new nodes and these new nodes are preferentially attached to existing nodes with large numbers of connections. The BA model is described as follows: (i) Growth: Start with a small number (m 0 ) of nodes, at every time step a new node is added and is connected to m ( m 0 ) already existing nodes. (ii) Preferential attachment: When choosing the nodes to which the new node connects, assume that the probability p i that a new node will be connected to node i depends on the degree k i of node i, satisfying p i = k i. k j j (After t time steps, this network has N = t + m 0 nodes and mt links. From growth and preferential attachment, this network evolves into a scaleinvariant state: the degree distribution of network does not change over time t; that is, the probability of connecting a node with k links is proportional to the power term k 3.) 2.3 Local-World Evolving Networks Model It is noticed that the BA scale-free model only calculates the preferential attachment probability of each node, yielding an average degree value of the entire network. However, in some real-world complex networks, each node in a network only has local connections therefore only owns local information about the entire network. In 2003, Li and Chen [3] proposed a local-world evolving networks model, which is described as follows:

4 Complex Dynamical Networks 73 (i) Start with a small number m 0 of nodes and small number e 0 of links. (ii) Select M nodes randomly from the existing network, referred to as the local world of the new coming node. (iii) Add a new node with m links, connecting to m nodes in its local world determined in Step (ii), using a preferential attachment with probability P Local (k i ) defined at every time step t by P Local (k i ) = P (i Local W orld) M m 0 + t k i j Local where P (i Local W orld) = and the term local-world here refers to all nodes in interest with respect to the newly added node at time step t. At every time step t, the newly added node connects to m existing nodes, which are selected from its local world with preferential attachment, but does not connect to nodes over the global system (this is the key difference from the BA model). Note that the map from the whole system to a local-world structure can vary, depending on the actual local connectivity of the network. 2.4 A General Time-Varying Dynamical Network Model Recently, Wang and Chen [12] proposed a uniformly connected dynamical network model, which has the same coupling strength for all links and the inner coupling matrix is a constant 0 1 diagonal matrix. However, most realworld complex networks are time-varying, with different coupling strengths for different links, and their inner coupling matrix may not be a diagonal matrix. To better characterize such real-world complex networks, we introduced a general time-varying dynamical network model [4-7] as follows: k j, ẋ i (t) = f(x i (t)) + N c ij (t) A(t) x j (t), i = 1, 2,, N, (1) j=1 where x i (t) = (x i1 (t), x i2 (t),, x in (t)) T R n are the state variables of node i, A(t) = (a ij (t)) n n R n n is a time-varying inner coupling matrix between nodes at time t, C(t) = (c ij (t)) N N is the coupling configuration matrix of the network at time t, in which c ij (t) is defined as follows: if there is a connection from node i to node j (j i) at time t, then the coupling strength c ij (t) 0 at time t; otherwise, c ij (t) = 0 (j i) at time t, and the diagonal elements of C(t) are defined by c ii (t) = N c ij (t), i = 1, 2,, N. (2) j=1 j i

5 74 J. Lü, H. Leung, G. Chen Obviously, the uniformly connected network model [12] is a special case of network (1), with C being a 0 1 symmetric matrix and A a 0 1 diagonal matrix. Since real-world complex networks may be directed networks, such as the WWW, whose coupling configuration matrix C(t) is not symmetric, it is not assumed C(t) be symmetric nor its off-diagonal elements be nonnegative. Suppose that network (1) is connected in the sense that there are no isolate clusters; that is, C(t) is irreducible. When A(t), C(t) are constant matrices, network (1) becomes a time-invariant dynamical network ẋ i = f(x i ) + N c ij A x j, i = 1, 2,, N. (3) j=1 3 Synchronization of Complex Dynamical Networks DEFINITION 1: Let x i (t, X 0 ) (i = 1, 2,, N) be a solution of the dynamical network ẋ i = f(x i ) + g i (x 1, x 2,, x N ), i = 1, 2,, N, (4) where X 0 = ( (x 0 1) T,, (x 0 N )T ) T R nn, f : D R n and g i : D D R n (i = 1, 2,, N) are continuously differentiable with D R n, and g i (x, x,, x) = 0. If there is a nonempty open subset E D, with x 0 i E (i = 1, 2,, N), such that x i (t, X 0 ) D for all t 0, i = 1, 2,, N, and lim x i(t, X 0 ) s(t, x 0 ) t 2 = 0 for 1 i N, (5) where s(t, x 0 ) is a solution of system ẋ = f(x) with x 0 D, then the dynamical network (4) is said to realize synchronization and E E is called the region of synchrony for the dynamical network (4). DEFINITION 2: Let Γ = {s(t) 0 t < T } denote the set of T periodic solutions of system ẋ = f(x) in R n. A T periodic solution s(t) is said to be orbitally stable, if for each ε > 0 there exists a δ > 0 such that every solution x(t) of ẋ = f(x), whose distance from Γ is less than δ at t = 0, will remain within a distance less than ε from Γ for all t 0. Such an s(t) is said to be orbitally asymptotically stable if, in addition, the distance of x(t) from Γ tends to zero as t. Moreover, if there exist positive constants α, β and a real constant h such that x(t h) s(t) α e βt for t 0, then s(t) is said to be orbitally asymptotically stable with an asymptotic phase. THEOREM 1: Suppose that s(t) is a hyperbolic periodic solution of the individual node ẋ = f(x), and is orbitally asymptotically stable with an

6 Complex Dynamical Networks 75 asymptotic phase. Suppose also that the coupling configuration matrix C = (c ij ) N N can be diagonalized. Then, S(t) = ( s T (t), s T (t),, s T (t) ) T is a hyperbolic synchronous periodic solution of network (3), and is orbitally asymptotically stable with an asymptotic phase, if and only if the linear time-varying systems ẇ = [Df(s(t)) + λ k A] w k = 2,, N (6) are asymptotic stable about their zero solution, where Df(s(t)) is the Jacobian of f(x) at x = s(t). THEOREM 2: Let x = s(t) be an exponentially stable solution for nonlinear system ẋ = f(x), where f : Ω R n is continuously differentiable, Ω = {x R n x s(t) 2 < r}. Suppose that the Jacobian D F(t, η) is bounded and Lipschitz on Ω = { η R nn η 2 < r}, uniformly in t. Suppose also that there exists a nonsingular real matrix, Φ(t), such that Φ 1 (t) (C(t)) T Φ(t) = diag{λ 1 (t), λ 2 (t),, λ N (t)} and Φ 1 (t) Φ(t) = diag{β 1 (t), β 2 (t),, β N (t)}. Then, the synchronous solution S(t) is exponentially stable in the dynamical network (1) if and only if the linear systems ẇ = [Df(s(t)) + λ k (t) A(t) β k (t) I n ] w k = 2,, N (7) are exponentially stable about their zero solutions, where Df(s(t)) is the Jacobian of f(x) at x = s(t). THEOREM 3: Assume that F : Ω R n(n 1) is continuously differentiable on Ω = {x R n(n 1) x 2 < r}, with F (t, 0) = 0 for all t, and the Jacobian DF (t, x) is bounded and Lipschitz on Ω, uniformly in t. Assume that there exists a bounded nonsingular real matrix Φ(t), such that Φ 1 (t) (C(t)) T Φ(t) = diag{λ 1 (t), λ 2 (t),, λ N (t)} and Φ 1 (t) Φ(t) = diag{β 1 (t), β 2 (t),, β N (t)}. Then, the chaotic synchronous state x 1 (t) = x 2 (t) = = x N (t) = s(t) is exponentially stable for dynamical network (1) if and only if the linear time-varying systems ẇ = [Df(s(t)) + λ k (t) A(t) β k (t) I n ] w, k = 2,, N (8) are exponentially stable about the zero solution. REMARKS: Theorem 1 gives a network synchronization criterion for periodic orbit of time-invariant network (3). Theorem 2 presents a network synchronization criterion for any orbit (non-chaotic) of time-varying network (1). Theorem 3 provides a chaos synchronization criterion for time-varying network (1). All proofs can be found in [4-7].

7 76 J. Lü, H. Leung, G. Chen 4 Control of Complex Dynamical Networks Currently, various interconnected and integrated complex networks are building up for energy, transportation, information, and finance, etc. These complex networks are closely relative to our daily lives. The network security against failures and attacks is an extremely important issue for everyone, especially for major networks such as the Internet and e-bank networks. This calls for greater effort to better control these complex networks. Wang and Chen [13] proposed a pinning control technique for scale-free dynamical networks, where local feedback injections are applied to a small fraction of network nodes so as to control the entire networks. Feedback pinning has been a common technique for the control of spatiotemporal chaos in regular dynamical networks. There are two pinning schemes: specifically pinning scheme and randomly pinning scheme. In the specifically pinning scheme, a fraction of the most highly connected nodes (hub nodes) are pinned; while in the randomly pinning scheme, a fraction of randomly selected nodes are pinned. Due to the extremely inhomogeneous connectivity distribution of a scale-free network, it is much more effective to pin some most highly connected nodes (hub nodes) than pinning the same (even much larger) number of randomly selected nodes. 5 Conclusions This paper reviews and introduces several fundamental mathematical models for complex dynamical networks, as well as some network synchronization criteria for time-invariant and time-varying complex dynamical network models. The control problem of complex networks has also been discussed briefly. There are abundant dynamical behaviors unknown in complex networks. The discovery of small-world and scale-free properties of complex networks has led to dramatic advances in this active research direction. However, how to construct and control a robust complex network for network security, e.g., protecting against attacks to the Internet, remains an urgent and yet challenging problem for future research. 6 Acknowledgements J. Lü was supported by the National Natural Science Foundation of China no , the K. C. Wong Education Foundation, Hong Kong and the Chinese Postdoctoral Scientific Foundation. G. Chen was supported by the Hong Kong Research Grants Council under the grant CityU 1115/03E.

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