Elsevier

Automatica

Volume 45, Issue 8, August 2009, Pages 1799-1807
Automatica

Structure identification of uncertain general complex dynamical networks with time delay

https://doi.org/10.1016/j.automatica.2009.03.022Get rights and content

Abstract

It is well known that many real-world complex networks have various uncertain information, such as unknown or uncertain topological structure and node dynamics. The structure identification problem has theoretical and practical importance for uncertain complex dynamical networks. At the same time, time delay often appears in the state variables or coupling coefficients of various practical complex networks. This paper initiates a novel approach for simultaneously identifying the topological structure and unknown parameters of uncertain general complex networks with time delay. In particular, this method is also effective for uncertain delayed complex dynamical networks with different node dynamics. Moreover, the proposed method can be easily extended to monitor the on-line evolution of network topological structure. Finally, three representative examples are then given to verify the effectiveness of the proposed approach.

Introduction

Nowadays, there are numerous natural or man-made complex networks. Typical examples are the World Wide Web, Internet, communication networks, social networks, food webs, metabolic networks, and so on (Albert and Barabási, 2002, Barabási, 2002, Dorogovtsev and Mendes, 2002, Guimera et al., 2002, Jeong et al., 2000, Lü and Chen, 2005, Lü, Yu, and Chen, 2004, Lü, Yu, Chen, and Cheng, 2004, Sorrentino et al., 2007, Strogatz, 2001, Wu, 2006, Yang et al., 2006). All the above networks can be represented in terms of nodes and edges indicating connections between nodes. It is well known that complex networks pervade through almost all scientific and technological fields, including mathematics, physics, engineering, biological sciences, ecology, and social sciences.

In real-world complex networks, there exists various uncertain information, such as unknown or uncertain topological structure and node dynamics (Lu and Cao, 2005, Wu, 2008, Yu et al., 2006, Yu and Cao, 2007, Yu et al., 2007, Zhou and Lu, 2007, Zhou et al., 2006). Moreover, time delay often appears in various complex networks, such as communication networks, neural networks, and metabolic networks, in either the state variables or the coupling coefficients (Chen et al., 2004, Lu and Chen, 2004, Pyragas, 1998, Zhang et al., 2008). Time delay is often caused by finite signal transmission speeds or memory effects (Chen et al., 2004, Pyragas, 1998, Yu and Cao, 2006, Yu and Cao, 2007). Therefore, the issue of network structure and parameter identification is of theoretical and practical importance for uncertain complex dynamical networks with time delay. That is, can we estimate the uncertain topological structure and system parameters of a specific complex networks by using its dynamical behaviors? In fact, in numerous real-world complex networks, the identification of network structure becomes a key problem, such as the protein-protein or protein-DNA (Deoxyribonucleic Acid) interactions in the regulation of various cellular processes. It is well known that the protein-DNA interactions often play pivotal roles in many cell processes, such as DNA replication, modification, repair and RNA (Ribonucleic Acid) transcription (Goto et al., 1998, Horne et al., 2004, Jin et al., 1999, Zhu et al., 2005). Another typical example is biological neural networks. In addition to modeling the neurons, the exact topological structure of a biological neural network often plays an important role and is the prime research interest (Yu and Cao, 2006, Zhu et al., 2005). However, due to the nonlinear, complex, and high dimensional nature of the practical complex networks, it is very difficult to exactly identify its topological structure by using the traditional approaches.

Recently, some new advances have been reported in the structure identification of complex networks (Huang, 2006, Wu, 2008, Yu et al., 2006, Zhou and Lu, 2007). However, most of the aforementioned approaches are only valid for several kinds of typical complex networks with known system parameters (Lu and Cao, 2005, Wu, 2008, Yu et al., 2006, Zhou and Lu, 2007). In fact, in many practical settings, it is often difficult to exactly know all system parameters beforehand. Therefore, it is very necessary to develop an effective method to identify the network topological structure and system parameters for complex networks together.

In this paper, by using network synchronization theory and adaptive control techniques (Huang, 2006, Lin and Ma, 2007, Lü, Yu, and Chen, 2004, Lü, Yu, Chen, and Cheng, 2004, Zhou et al., 2008), an effective approach is then proposed to identify unknown network topological structure and system parameters for general uncertain delayed complex dynamical networks. Furthermore, this developed method is also valid for general uncertain delayed complex dynamical networks with different node dynamics. Finally, the proposed approach can be used to monitor the on-line evolution of network topological structure and system parameters for complex networks.

The paper is organized as follows. Section 2 gives some useful preliminaries. The identification of network topological structure and system parameters is then further explored for general uncertain complex dynamical networks with coupling delay and node delay in Sections 3 Structure identification of uncertain general complex dynamical networks with coupling delay, 4 Structure identification of an uncertain general complex dynamical network with node delay, respectively. Section 5 uses three representative examples to verify the effectiveness of the proposed approach. Finally, some concluding remarks are then drawn in Section 6.

Section snippets

Preliminaries

Consider uncertain dynamical systems ẋi(t)=f̄i(t,xi(t),αi),i=1,2,,N. Rewrite systems (1) in the following form ẋi(t)=fi(t,xi(t))+Fi(t,xi(t))αi,i=1,2,,N, where xi(t)Rn are state vectors, αiRmi are unknown system parameter vectors for i=1,2,,N, in which mi are nonnegative integers. For every i, fi(t,xi(t)) is an n×1 matrix, and Fi(t,xi(t)) is an n×mi matrix.

Assumption 1 A1

Suppose that there exist nonnegative constants Li (i=1,2,,N), satisfying f̄i(t,x(t),αi)f̄i(t,y(t),αi)Lix(t)y(t), where x(t),y(t

Structure identification of uncertain general complex dynamical networks with coupling delay

Consider a complex dynamical network with time-varying coupling delay consisting of N different nodes, which is described byẋi(t)=fi(t,xi(t))+Fi(t,xi(t))αi+j=1NcijAxj(tτ(t)), where xi(t)=(xi1(t),xi2(t),,xin(t))TRn for i=1,,N is the state vector of the i-th node, and delay τ(t) is time-varying. C=(cij)N×NRN×N is an unknown or uncertain coupling configuration matrix. If there exists a link from nodes i to j(ji), then cij0 and cij is the weight or coupling strength; otherwise, cij=0. A:Rn

Structure identification of an uncertain general complex dynamical network with node delay

Consider an uncertain general complex dynamical network consisting of N different nodes with time-varying delay τ(t), called the drive network, which is described by ẋi(t)=ḡi(t,xi(t),xi(tτ(t)),βi)+j=1NcijAxj(t),i=1,,N. The node dynamics can be rewritten as follows ḡi(t,xi(t),xi(tτ(t)),βi)=gi(t,xi(t),xi(tτ(t)))+Gi(t,xi(t),xi(tτ(t)))βi, where βi (i=1,2,,N) are unknown or uncertain system parameter vectors.

Construct another controlled general complex dynamical network, called response

Numerical simulation examples

In this section, several representative examples are then given to verify the effectiveness of the proposed structure identification and parameters estimation approach based on network synchronization in Sections 3 Structure identification of uncertain general complex dynamical networks with coupling delay, 4 Structure identification of an uncertain general complex dynamical network with node delay.

Conclusion

We have proposed a novel adaptive feedback control approach to simultaneously identify the unknown or uncertain network topological structure and system parameters of uncertain delayed general complex dynamical networks together. Based on Lyapunov theory and Barbaˇlat Lemma, several useful identification criteria are then attained. In particular, the proposed methods are also valid for the structure identification of complex networks with nonidentical nodes. Several representative numerical

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 70771084, 60574045, 60772158, and 60821091, the National Basic Research Program (973) of China under Grants 2007CB310805, the Important Direction Item of Knowledge Innovation Project of Chinese Academy of Sciences under Grant KJCX3-SYW-S01, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

Hui Liu received the B.Sc. Degree and M.S. Degree in mathematics, both from Wuhan University, Wuhan, China, in 2004 and 2007, respectively.

Currently, she is a Ph.D. candidate with the School of Mathematics and Statistics, Wuhan University, Wuhan, China. Her main research interests are complex networks, nonlinear systems, chaos control and synchronization, and multi-agent systems.

References (37)

  • S.N. Dorogovtsev et al.

    Evolution of networks

    Advances in Physics

    (2002)
  • S. Goto et al.

    Chemical database for enzyme reactions

    Bioinformatics

    (1998)
  • R. Guimera et al.

    Optimal network topologies for local search with congestion

    Physical Review Letters

    (2002)
  • A.B. Horne et al.

    Constructing an enzyme-centric view of metabolism

    Bioinformatics

    (2004)
  • D. Huang

    Adaptive-feedback control algorithm

    Physical Review E

    (2006)
  • H. Jeong et al.

    The large-scale organization of metabolic networks

    Nature

    (2000)
  • H.K. Khalil

    Nonlinear systems

    (1996)
  • T.C. Lee et al.

    A generalization of Krasovskii-LaSalle theorem for nonlinear time-varying systems: Converse results and applications

    IEEE Transactions on Automatic Control

    (2005)
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      The research of finite-time topological identification was considered in [11]. The network structure has been studied from different angles and a variety of network models were proposed, such as the weighted networks [12], the networks with uncertain system parameters [13], and the networks with time delay [14]. However, the above-mentioned researches mainly concentrated on constructing the auxiliary network that has the identical size of nodes with the original network.

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    Hui Liu received the B.Sc. Degree and M.S. Degree in mathematics, both from Wuhan University, Wuhan, China, in 2004 and 2007, respectively.

    Currently, she is a Ph.D. candidate with the School of Mathematics and Statistics, Wuhan University, Wuhan, China. Her main research interests are complex networks, nonlinear systems, chaos control and synchronization, and multi-agent systems.

    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 nonlinear systems, chaos control and synchronization, complex networks, and scientific and engineering computing. He has published more than 150 journal papers in the above fields. He 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 State Natural Science Award from the State Council of China in 2008.

    Jinhu Lü received the Ph.D. Degree in applied mathematics from the Chinese Academy of Sciences, Beijing, China in 2002.

    Currently, he is an Associate Professor with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. He held several visiting positions in Australia, Canada, France, Germany, Hong Kong and USA, and was a Visiting Fellow in Princeton University, USA from 2005 to 2006. He is the author of two research monographs and more than 70 international journal papers published in the fields of nonlinear circuits and systems, complex networks and complex systems. He served as a member in the Technical Committees of many international conferences and is now serving as a member of IFAC Technical Committee on Large Scale Complex Systems and a Vice-Chair of Technical Committee of Complex Systems and Complex Networks of CSIAM. He is also an Associate Editor of IEEE Transactions on Circuits and Systems II, Journal of Systems Science and Complexity, ARI — the Bulletin of the Istanbul Technical University and DCDIS-A. Dr. Lü received the prestigious Presidential Outstanding Research Award from the Chinese Academy of Sciences in 2002, the National Best Ph.D. Theses Award from the Office of Academic Degrees Committee of the State Council and the Ministry of Education of China in 2004, the First Prize of the Science and Technology Award from the Beijing City of China in 2007, the First Prize of the Natural Science Award from the Ministry of Education of China in 2007, the Lu Jiaxi Youth Talent Award from the Chinese Academy of Sciences in 2008, the Second Prize of the State Natural Science Award from the State Council of China in 2008. He is the co-author of the Most Cited SCI Paper of Chinese Scholars in the field of mathematics during the periods of 2001–2005 and 2002–2006. He is also an IEEE Senior Member.

    David John Hill received the B.E. and B.Sc. Degrees from the University of Queensland, Australia, in 1972 and 1974, respectively. He received the Ph.D. Degree in Electrical Engineering from the University of Newcastle, Australia, in 1976. He is currently a Professor and Australian Research Council Federation Fellow in the Research School of Information Sciences and Engineering at The Australian National University. He is also Deputy Director of the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems. He has held academic and substantial visiting positions at the universities of Melbourne, California (Berkeley), Newcastle (Australia), Lund (Sweden), Sydney and Hong Kong (City University). He holds honorary professorships at the University of Sydney, University of Queensland (Australia), South China University of Technology, City University of Hong Kong, Wuhan University and Northeastern University (China). His research interests are in network systems science, stability analysis, nonlinear control and applications. He is a Fellow of the Institution of Engineers, Australia, the Institute of Electrical and Electronics Engineers, USA, the Society for Industrical and Applied Mathematics, USA and the Australian Academy of Science; he is also a Foreign Member of the Royal Swedish Academy of Engineering Sciences.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Miroslav Krstic.

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