Recovering Structures of Complex Dynamical Networks ... - IEEE Xplore

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Recovering Structures of Complex Dynamical Networks Based on Generalized Outer Synchronization Shuna Zhang, Xiaoqun Wu, Jun-An Lu, Hui Feng, and Jinhu Lü, Fellow, IEEE

Abstract—The topological structures of complex networks play a crucial role in determining their evolutionary mechanisms and functional behaviors, and may have significant consequences for many real-world applications. Many researchers focused on the geometric features, collective behaviors and control of complex networks provided with precisely known structures. However, the exact topology of a network is usually unknown or uncertain in practical situations. Therefore, accurate and timely topology identification is of great necessity and importance. This paper presents a novel scheme for topology identification. Specifically, an auxiliary complex network with a very general form is constructed and some adaptive controllers are designed to recover the topology of the considered network upon generalized outer synchronization. Different from previous schemes, the network constructed can be composed of any kind of nodes. If the network with an unknown topology has very complicated node dynamics or a high node dimension, one can construct a response network consisting of nodes with much simpler dynamics, which is very practical for circuit design. The effectiveness of the theoretical findings has been illustrated by three numerical examples. This work provides a convenient approach to recovering network topology, which can facilitate the selection of practical circuits and reduce application costs. Index Terms—Adaptive control, complex network, generalized outer synchronization, topology identification.

I. INTRODUCTION

S

INCE the 20th century, the exploration of complex networks has been substantially extended from pure mathematical analysis to various scientific fields, such as physical sciences, engineering sciences, life sciences, biology sciences and so on [1]–[9]. People began to investigate the basic characteristics of complex networks which have a huge number of nodes and links, and practically complicated structures. In every scientific discipline, complex networks have attracted wide research interest. The nature of complex networks is complexity, including topological structures, dynamical evolution, node diversities, and so on. At the initial stage, research focus is mainly placed Manuscript received January 23, 2014; revised April 10, 2014; accepted May 18, 2014. Date of publication July 25, 2014; date of current version October 24, 2014. This work was supported by the National Science and Technology Major Project of China under Grant 2014ZX10004001-014, the 973 Project under Grant 2014CB845302, the National Natural Science Foundation of China under Grants 61025017, 61174028, 11172215 and 91130022. This paper was recommended by Associate Editor M. Porfiri. S. Zhang, X. Wu, J.-A. Lu and H. Feng are with the School of Mathematics and Statistics, Wuhan University, Hubei, 430072, China (e-mail: xqwu@whu. edu.cn). J. Lü is with the Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSI.2014.2334951

on the geometric features, collective behaviors and control of complex dynamical networks provided with precisely known topologies [10]–[17], which has provided an in-depth insight into the interplay between the topologies and functions. It has been found that the topological structures of complex networks play a crucial role in determining their evolutionary mechanisms and functional behaviors, and may have significant consequences for many real-world applications. For example, the underlying network structure will affect traffic on the Internet, the performance of a Web search engine, and the dynamics of social or biological systems. However, inevitably, the topological structure of a complex network is usually unknown or uncertain. Therefore, to understand and explain the connections of interacting systems built upon a network, it is of great necessity to gain the knowledge of the underlying topology of the network. In the past decades, due to the awareness of necessity and importance in inferring network topologies, there has been a steady growth of approaches regarding this topic and far a few methods have been developed, such as the method based on adaptive control and outer synchronization [18]–[24], compressive sensing [25], dynamical correlation [26], Bayesian networks [27], [28], recurrence [29], [30], Granger causality test [31], [32], transfer entropy [33], partial mutual information [34], and so on. Compressive-sensing based methods identify node dynamics and connections from a set of basis functions, thus the knowledge of the type of basis functions of node dynamics and coupling patterns is needed [25]. The methods based on dynamical correlation are incapable of distinguishing between direct and indirect interactions, which in many situations do not provide very satisfactory results [26]. The methods based on recurrence properties can only deal with very small-scale networks [29], [30]. The approaches based on Granger causality test [31], [32], transfer entropy [33], and partial mutual information [34] fall within the general problem of causality detection and are usually computationally expensive and require a huge amount of memory and time for computation. Moreover, most of the above-mentioned approaches can only tell whether a link between two nodes exists but not the coupling strength [26]–[34]. In this paper, we will investigate the existence as well as the coupling strengths of links in a complex network based on the synchronization method. In previous synchronization-based identification methods, identical synchronization between complex networks is exploited to estimate the unknown topological parameters. Specifically, a response network is constructed and some adaptive controllers are designed so that the auxiliary response network can synchronize with the uncertain network and topological parameters can be simultaneously estimated. This kind of synchronization between two networks is called complete

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ZHANG et al.: RECOVERING STRUCTURES OF COMPLEX DYNAMICAL NETWORKS BASED ON SYNCHRONIZATION

outer synchronization, which was first investigated by Li et al. [35]. Later, complete outer synchronization was investigated by many researchers using different control techniques [36]–[39]. In these papers, it is assumed that the corresponding nodes in the two networks have the same dynamics and manifest completely identical behaviors upon synchronization. However, all nodes in different networks cannot have identical dynamics due to parameter mismatch or structural discrepancy, while the two networks may still behave in a synchronous way. This kind of synchronization is called generalized synchronization, which represents another degree of coherence and arouses a certain degree of interest [40]–[44]. Due to the simplicity of controllers and effectiveness, the identification method based on synchronization has been paid wide attention to. A problem lies in that, when the nodes in the network with an unknown topology have a high dimension or complicated dynamics, it is practically costly to construct a response network with identical node dynamics. Therefore, it is more realistic and practical to consider topology identification of a complex network by constructing a response network consisting of much simpler node dynamics [45]. Motivated by the above discussions, topology identification of complex dynamical networks is systematically investigated via generalized synchronization. Based on adaptive control and the Barbalat’s lemma, a response network with a general form is constructed to recover the unknown topology of the considered network based on generalized outer synchronization. The rest of the paper is organized as follows. The network model and some preliminaries are given in Section II. Topology identification based on generalized outer synchronization is presented in Section III. In Section IV, three numerical examples are provided to illustrate the effectiveness of proposed control technique. Finally, some conclusions are given in Section V. II. NETWORK MODEL AND PRELIMINARIES Some necessary notations that will be used throughout the paper are first introduced. denotes the transpose of a matrix or a vector. is the 2-norm of the vector , represents the Kronecker product. is the maximum eigenvalue of the matrix . represents the identity matrix with order .

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It is well-known that network topology plays a pivotal role in determining the emergence of collective behaviors and governing the main features of relevant processes that take place in complex networks. Therefore, to understand a complex network, it is necessary to first gain knowledge of the intrinsic topology, which is usually unknown or uncertain in many practical situations. In network (1), the node dynamics and the coupling matrix are supposed to be known, and our purpose is to identify the unknown configuration matrix . To recover the underlying topology of network (1), one can construct an auxiliary complex dynamical network containing dynamical nodes as follows:

(2) where is the state is a smooth nonvector of the th node, and linear vector-valued function governing the evaluation of the th isolated . is the inner coupling matrix, and is the estimation of the unknown coupling matrix . is the th controller to be designed. In the following, the network given by (1) is regarded as the drive network and that by (2) as the response network. B. Preliminaries For the purpose of recovering network topology based on generalized outer synchronization, two definitions, two assumptions and a lemma are introduced as follows. Definition 2.1: Let be a continuously differentiable vector map which maps a vector into another vector . Then the Jacobian Matrix of the map is defined as .. .

..

.

.. .

Definition 2.2: Let be continuously differentiable vector maps. Network (1) is said to achieve generalized outer synchronization with network (2), if

A. Network Models Consider a general weighted complex dynamical network dynamical nodes with linear couplings, which consisting of is described by (1) is the state where vector of the th node, is a smooth nonlinear vector-valued function governing the dynamics of the th node in the absence of interactions with other nodes, is the inner coupling matrix linking coupled variables, and is the coupling configuration matrix representing the topological information of the network, in which is defined as follows: if there is a directed connection from , then ; otherwise, . The node to node diagonal elements of matrix are defined as:

Assumption 2.1: For functions exist positive constants

such that

, there

holds for any and . , , Assumption 2.2: For each , , are linearly independent on the orbit of the outer synchronization manifold . III. TOPOLOGY IDENTIFICATION VIA THE GENERALIZED OUTER SYNCHRONIZATION In this section, the main results of topology identification based on generalized outer synchronization between the drive and response networks are presented. Theorem 3.1: Suppose that the assumptions 2.1 and 2.2 hold. Then the uncertain configuration matrix of network (1) can

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be identified by the estimated values trollers and updating laws

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 61, NO. 11, NOVEMBER 2014

via the following con-

Taking

, then one gets

Therefore, one has bounded. Furthermore, one has (3) (4) , is the Jacobian matrix of the where , and . Furthermore, map network (2) reaches generalized outer synchronization with network (1), that is, Proof: Since , together with the controllers (3), one can obtain the error dynamical network from networks (1) and (2) as follows

(5) . Next, one will show that the topology where identification is achieved based on the generalized outer synchronization between networks (1) and (2), that is, the stability of the origin of the error dynamical network (5). . Consider the following Let Lyapunov candidate function (6) Obviously,

. Since Assumption (2.1) holds, one has . Taking the time derivative of along the trajectories of the error dynamical network (5), together with the updating laws (4), one obtains

, thus

is

. Thus, . Obviously, is bounded, i.e., . According to exists and is bounded the error dynamical system (5), . Therefore, according to Barbalat’s lemma for , which implies [46], one obtains for . exists, then one obtains Suppose that since converges to a constant as . Together with system (5) and Assumption 2.2, one obtains that for Therefore, the network (1) and (2) asymptotically achieve generalized outer synchronization. Simultaneously, the unknown coupling configuration matrix can be successfully identified by matrix . The proof is thus completed. Remark 3.1: It should be noted that the unknown coupling configuration matrix need not be symmetric or irreducible, and are not necesand the known inner coupling matrices sarily symmetric and identical. In addition, nodes inside one network can have diverse dynamics. Thus the proposed approach is widely applicable to various kinds of complex dynamical networks. Remark 3.2: The feedback gain can be chosen appropriately beforehand to adjust the rate of topology identification and generalized outer synchronization. Theoretically, a larger will lead to faster identification and synchronization. However, the is only a sufficient condition, which inequality condition is not necessary. can be chosen Remark 3.3: The gains properly to adjust the updating speed of . Some sufficiently would lead to faster synchronization and large values of quicker topology identification, while for small values, the time to achieve successful identification may be quite long. Remark 3.4: New controllers and updating laws are designed so that the unknown network topology can be recovered via generalized outer synchronization between the drive and response networks. Thus, the control technique can be regarded as an improvement on many results about outer synchronization between networks, such as those in [35], [44]. Remark 3.5: The linear independence condition as specified in Assumption (2.2) is the key to guaranteeing successful topology identification. However, the verification of the linear independence condition is very difficult. In [23], the authors clarified that synchronization within the drive network is an obstacle to topology identification. Actually, identification fails even if some nodes of the unknown network evolves into generalized synchronization. Furthermore, as illustrated in [22], it is often difficult to recover the topology for an unknown network with identical nodes, since the network easily reaches some kind of inner synchronization such that Assumption (2.2) cannot be satisfied. From this view point, diverse dynamics and chaotic behaviors of the nodes within the unknown network can facilitate topology identification. Corollary 3.1: Let assumptions (2.1) and (2.2) hold. If nodes in the response network have identical dimension with that in , and the inner coupling matrices the drive network, i.e.,

ZHANG et al.: RECOVERING STRUCTURES OF COMPLEX DYNAMICAL NETWORKS BASED ON SYNCHRONIZATION

, then the unknown network configuration matrix can be estimated by upon complete outer synchronization with the following controllers (7) and updating laws (8) where . and Proof: Let the error dynamics are

,

, and

, , then

where . Consider the Lyapunov function as

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proper dimensions. The outer synchronization error between the two networks, , is defined as , and denotes the topology identification error. The ordinary differential equations are numerically solved using the Runge-Kutta 4th order method. Example 1: Firstly, consider topology identification via comis taken as an plete outer synchronization. That is, the map . Suppose that identity map representing the unknown drive network is a directed ring network consisting . Therefore, for of 30 nodes with the coupling strength being the configuration matrix representing the network topology, , , for one has and , otherwise. The node dynamics in the drive network is supposed to be the well-known Lü system [47], [48], and that in the constructed response network be the well-known Lorenz system [49]. Then, the original and constructed response networks are respectively described by

and According to the Assumption (2.2), along with the controllers (7) and updating laws (8), one has where

,

,

and

It is easy to verify that for [44], that is to say, Assumption 2.1 is satisfied. any For the identity map, one has where , . The rest of the proof is the same as that of Theorem 3.1. Furthermore, if the constructed response network has the same node dynamics as that of the drive network, Corollary 3.1 is reduced to the following corollary, which is also the result presented in some earlier papers [19], [20]. Corollary 3.2: Suppose that assumptions (2.1) and (2.2) hold. , and the inner coupling matrices If , then the unknown network configuration matrix can be estimated by upon complete outer synchronization with the following controllers and updating laws:

with a sufficiently large , and

.

IV. NUMERICAL SIMULATIONS In this section, three numerical examples are presented to illustrate the effectiveness of the proposed identification methods. and are supposed to be identity matrices with For brevity,

The controllers and updating laws are designed according to (3) and (4). The initial values for numerical simulations are ran. domly set, and the feedback gain is taken as Fig. 1 displays the topology identification performance, where panel (a) is the generalized outer synchronization error between the two networks and panel (b) is the identification error. It is clearly observed that the unknown topology of the drive network is successfully recovered by the proposed control technique. Fig. 2 shows the phase diagrams of node 5 in the drive and response networks, where the transients are discarded. One can see that the two nodes exhibit completely identical dynamics, which verifies that the two networks reach outer synchronizaalong with tion. Panel (a) of Fig. 3 displays the evolution of time. To take a clearer view, one presents in panel (b) the colormap of the recovered coupling matrix, where the blue blocks represents the diagonal elements, the red blocks are the existent

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Fig. 1. Identification of the directed ring network with . (a): the outer synchronization error between the two networks; (b): the topology identification error.

Fig. 3. (a): Time evolution of matrix.

. (b): The colormap of the recovered coupling

Example 2: Next, consider topology identification via generalized outer synchronization, with being defined as . Then

(9) The proposed method is tested on a small-world network consisting of 10 nodes. The algorithm proposed by Newman and Watts [50] is employed here to generate the network. Specifically, start from a ring-shaped network with 10 nodes, in which each node is connected to its 4 nearest neighbors. For every pair of originally unconnected nodes, with probability , add an edge to connect them. A small-world network generated with , as shown in Fig. 4, is used here for illustration. The coupling strength is set to be 0.5, then the unweighted configu, is ration matrix, that is,

Fig. 2. Phase diagrams for node 5, with the outer synchronization manifold . (a): projection in the plane of node 5 being plane of node 5 in the in the drive network; (b): projection in the response network.

edges, and green the zero elements. The figure further demonstrates the validity of the proposed method.

The Lü system and the Lorenz system are still supposed to be the node dynamics in the drive and response networks, respecare then designed tively. The controllers according to (3) and (4). The feedback gain is taken as .

ZHANG et al.: RECOVERING STRUCTURES OF COMPLEX DYNAMICAL NETWORKS BASED ON SYNCHRONIZATION

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Fig. 6. Colormap of the recovered unweighted coupling matrix. Fig. 4. A 10-node network generated with the small-world algorithm proposed by Newman and Watts.

Fig. 7. Phase diagrams for node 5, with the outer synchronization manifold . (a): projection in the being plane of node 5 in the drive network; (b): projection in the plane of node 5 in the response network. Fig. 5. Identification of the small-world network with . (a): the outer synchronization error between the two networks; (b): the topology identification error.

Fig. 5 displays the perfect identification performance, where the generalized outer synchronization error and the identification error quickly go to zero after a short transient period. Fig. 6 depicts the colormap of the recovered unweighted coupling ma, which further illustrates the effectiveness of the trix proposed method. Fig. 7 shows the phase diagrams of node 5 in the drive and response networks. It is obvious that the two attractors are similar in a certain mode. To take a clearer view at the relationship between dynamics of nodes in the two networks, we also display the subvariplane, as shown ables of corresponding nodes in the in Fig. 8, where the transients are discarded. It is obviously seen that corresponding nodes in the two networks reach generalized synchronization as defined in the vector map .

Example 3: Finally, take the hyperchaotic Lü system [51] as the node dynamics of the drive network, and the Lorenz system as the dynamics of the constructed response network. Let the outer synchronization manifold be . A star network consisting of 10 nodes is used for illustration, as shown in Fig. 9. Thus, for the configuration and for matrix , one has and all the other elements are zeros. It is easy to obtain the following Jacobian matrix

Fig. 10 shows the identification performance, where panel (a) is the generalized synchronization error, and panel (b) is the is shown identification error. The detailed time evolution of in panel (a) of Fig. 11. The colormap of the recovered coupling

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Fig. 8. Relationships between the subvariables of node 5 in the drive and re; (b): . sponse networks. (a):

Fig. 10. Identification of the star network with . (a): the outer synchronization error between the two networks; (b): the topology identification error.

Fig. 9. A 10-node star network.

matrix is displayed in panel (b) of Fig. 11(b). This figure further validates the effectiveness of the proposed method. Remark 4.1: As can be seen from this example, when the nodes of the unknown network have a high dimension, one can construct a response network consisting of nodes with comparatively simpler dynamics to recover the unknown topology. In this regard, the proposed control technique can greatly simplify practical design in recovering network topologies compared to traditional control schemes. V. CONCLUSION Recovering the unknown topological structures of networks is of practical importance. In this paper, a control scheme is proposed to recover the underlying topology of a network by constructing a corresponding response network and adaptive controllers to achieve topology identification upon generalized outer synchronization between the two networks. If the network with an unknown topology has very complicated node dynamics

Fig. 11. (Color online) (a): The time evolution of recovered unweighted coupling matrix.

. (b): The colormap of the

or a high node dimension, one can construct a response network composed of much simpler node dynamics for identification, which is very practical for application. The proposed control scheme for recovering topologies reduces to a simpler

ZHANG et al.: RECOVERING STRUCTURES OF COMPLEX DYNAMICAL NETWORKS BASED ON SYNCHRONIZATION

form if the constructed network is designed to exhibit identical behaviors with that of the drive network. Furthermore, if the constructed network has the same node dynamics as that of the unknown network, the proposed control scheme is even simpler and contains some recent results as a special case. Therefore, the control scheme in this paper is practical to a great variety of real complex dynamical networks. Additionally, there is no constraint imposed on the inner coupling matrix and the unknown coupling matrix. The applicability of the theoretical findings has been illustrated by three numerical examples. For large-scale networks, the proposed control scheme still provides a theoretical framework for recovering the topological structures. However, for an unknown network composed of nodes, one has to construct an auxiliary network with identical observers to estimate the unknown number of nodes and coupling matrix. Therefore, with the increase of the size of the unknown network, the number of constructed circuits sharply . Thus, inferring the unknown topology of increases to large-scale network employing this method will result in high computational complexity and design cost. An interesting topic for future work would be to probe into structure identification of large-scale dynamical networks with reduced control cost, which is known to be a very challenging task. ACKNOWLEDGMENT The authors wish to thank the Editor-in-Chief, the Associate Editor and the anonymous reviewers for their helpful comments and suggestions. REFERENCES [1] D. Watts and S. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature, vol. 393, no. 6684, pp. 440–442, 1998. [2] A. L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999. [3] H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A. L. Barabási, “The large-scale organization of metabolic networks,” Nature, vol. 407, no. 6804, pp. 651–654, 2000. [4] C. Jin, I. Marsden, X. Chen, and X. Liao, “Dynamic DNA contacts observed in the NMR structure of winged helix protein-DNA complex,” J. Mol. Biol., vol. 289, pp. 683–690, 1999. [5] M. E. J. Newman, “The structure and function of complex networks,” SIAM Rev., vol. 45, no. 2, pp. 167–256, 2003. [6] I. Belykh, E. D. Lange, and M. Hasler, “Synchronization of bursting neurons: What matters in the network topology,” Phys. Rev. Lett., vol. 94, p. 188101, 2005. [7] I. Belykh, M. D. Bernardo, J. Kurths, and M. Porfiri, “Evolving dynamical networks,” Phys. D, vol. 267, pp. 1–6, 2014. [8] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, “Complex networks: Structure and dynamics,” Phys. Rep., vol. 424, no. 4, pp. 175–308, 2006. [9] P. D. Lelllis, G. Polverino, G. Ustuner, N. Abaid, S. Macri, E. M. Bollt, and M. Porfiri, “Collective behaviour across animal species,” Sci. Rep., vol. 4, p. 3723, 2014. [10] X. Wang and G. Chen, “Synchronization in scale-free dynamical networks: Robustness and fragility,” IEEE Trans. Circuits Syst. I:–Reg. Papers, vol. 49, no. 1, pp. 54–62, 2002. [11] Y. Chen, J. Lü, X. Yu, and Z. Lin, “Consensus of discrete-time second order multi-agent systems based on infinite products of general stochastic matrices,” SIAM J. Control Optim., vol. 51, no. 4, pp. 3274–3301, 2013. [12] W. K. Wong, W. Zhang, Y. Tang, and X. Wu, “Stochastic synchronization of complex networks with mixed impulses,” IEEE Trans. Circuits Syst.–I:Reg. Papers, vol. 60, no. 10, pp. 2657–2667, 2013. [13] M. Porfiri, D. J. Stilwell, and E. M. Bollt, “Synchronization in random weighted directed networks,” IEEE Trans. Circuits Syst.–I:Reg. Papers, vol. 55, no. 10, pp. 3170–3177, 2008. [14] P. DeLellis, M. D. Bernardo, and F. Garofalo, “Adaptive pinning control of networks of circuits and systems in Lur’e Form,” IEEE Trans. Circuits Syst.–I:Reg. Papers, vol. 60, no. 11, pp. 3033–3042, 2013.

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Shuna Zhang received the B.Sc. degree in information and computing science from Hubei Normal University, China, in 2012. She is working toward the M.S. degree in computational mathematics at Wuhan University, Wuhan, China. Her current research interest is complex networks.

Xiaoqun Wu received the B.Sc. degree in applied mathematics and the Ph.D. in computational mathematics both from Wuhan University, Wuhan, China, in 2000 and 2005, respectively. She is currently a Professor with the School of Mathematics and Statistics, Wuhan University, Wuhan, China. She has held several visiting positions in Hong Kong and Australia over the last few years. Her research interests include complex networks, nonlinear dynamics, and chaos control. She has published more than 40 SCI journal papers in the above fields. Dr. Wu received the Second Prize of the Natural Science Award from the Hubei Province, China in 2006, the First Prize of the Natural Science Award from the Ministry of Education of China in 2007, and the First Prize of the Natural Science Award from the Hubei Province, China in 2013.

Jun-An Lu received the B.Sc. degree in geophysics from Peking University, Beijing, China, and the M.Sc. degree in applied mathematics from Wuhan University, Wuhan, China, in 1968 and 1982, respectively. He is currently a Professor with the School of Mathematics and Statistics, Wuhan University, Wuhan, China. His research interests include complex networks, nonlinear systems, chaos control, and scientific and engineering computing. He has published more than 200 journal papers in the above fields. Prof. Lu received the Second Prize of the Natural Science Award from the Hubei Province, China in 2006, the First Prize of the Natural Science Award from the Ministry of Education of China in 2007, the Second Prize of the National Natural Science Award of China in 2008, and the First Prize of the Natural Science Award from the Hubei Province, China in 2013.

Hui Feng received the B.Sc. and M.Sc. degrees in mathematics from Wuhan University, Wuhan, China, in 1987 and 1990, respectively, and the Ph.D. degree in computational mathematics from Peking University, Beijing, China, in 1994. He is currently a Professor with the School of Mathematics and Statistics, Wuhan University, Wuhan, China. His research interests include numerical methods for partial differential equations and inverse problems. He has authored 22 journal papers in the above fields. Dr. Feng received the New Century Excellent Talents in University Award from the Chinese Ministry of Education in 2006.

Jinhu Lü (F’13) received the Ph.D. degree in applied mathematics from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in 2002. Currently, he is a Professor of the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He was a Visiting Fellow in Princeton University, Princeton, NJ, USA from 2005 to 2006. He is the author of three research monographs and more than 110 international journal papers published in the fields of complex networks and systems, nonlinear circuits and systems, with more than 7000 SCI citations and h-index 41. He also has two authorized patents. He is the Past Chair of Technical Committees of Neural Systems and Application and Chair of the Technical Committees of Nonlinear Circuits and Systems in the IEEE Circuits and Systems Society. He served and is serving as Editors in various ranks for 11 SCI journals including the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS–I: REGULAR PAPERS, the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS–II: BRIEF PAPERS, the IEEE TRANSACTIONS ON NEURAL NETWORKS, the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, the International Journal of Bifurcation and Chaos, and the Asian Journal of Control. Dr. Lü received the prestigious National Science Award twice from the Chinese government, the First Prize of Science and Technology Award from the Beijing City of China, the First Prize of Natural Science Award from the Ministry of Education of China, the 11th Science and Technology Award for Youth of China, and the Australian Research Council Future Fellowships Award. Moreover, he attained the National Science Fund for Distinguished Young Scholars and 100 Talents Program from the Chinese Academy of Sciences.