Finite-time topology identification and stochastic synchronization of complex network with multiple time delays☆
Introduction
Complex networks with multiple time delays are universal in real world such as communication network, transportation network and relationship network, etc. [1], [2], it means that the complex network can be divided into some sub-networks based on different multiple time delays and each of them has its own property, we can also call it multi-links network. It is worth mentioning that this multi-links network is not equivalent to a weighted network, multi-links with different link properties cannot be simply merged into a link. Therefore, a weighted network cannot reflect the performance of the multi-links network. Essentially, time-delays are introduced to split the multi-links complex network into sub-networks in order to describe the time-delay property of the networks with multi-links. Taking communication network (transportation network) for example, communication network (transportation network) can be divided into telephone network (airline network), internet network (railway network) and mail network (expressway network) by different time-delays, which is shown in Fig. 1. In addition, single link complex network is a special case of multi-links complex network, different from the single-link network in Ref. [3] and the mixed time-delays in Ref. [4], the research on complex network with multiple time delays is more realistic and representative.
In the previous researches, complex network with single delay is investigated to achieve complete synchronization [5], Ref. [6] considers distributed synchronization via randomly occurring control. Besides, the different synchronization behaviors including anti-synchronization [7], cluster synchronization [8], projective synchronization [9], robust synchronization [10] and etc. are also studied in the field of systems dynamics behavior. Thus it can be seen that synchronization is a typical collective behavior in the complex network which has very important application merits. With the development of the field of systems and control engineering, different kinds of control techniques are widely used in research for system dynamical behavior, including continue control [11], [13], [14], [15], [16], [17] and discontinue control [12], [13] etc..
Particularly, since the finite time was introduced, the studies about the finite-time stability and synchronization are of great significance in practical applications. Compared with asymptotic synchronization, finite-time synchronization has lower time complexity and synchronization can be realized in a setting time. Therefore, the finite-time stabilities [18], the finite-time complete synchronization [19], [20], [46], lag synchronization [21], synchronization [22] and cluster synchronization [23], etc., have been studied based on different kinds of control techniques in complex networks. Furthermore, the corresponding finite-time synchronization is also investigated in memristor-based neural networks [24], [25]. However, these researches are given based on the known network models. It is well known that there will be many uncertain factors in the real systems, including stochastic perturbations, uncertain parameters, unknown topology, and etc. In Refs. [26], [27], [28], [29], [30], researchers consider the influence of stochastic perturbation for the asymptotic synchronization of network, and finite-time stochastic synchronization are investigated in Refs. [31], [32]. Meanwhile, the uncertain parameters or unknown topology as the uncertain factors are investigated in these papers [1], [2], [33], [47], [48], [49]. Refs. [37], [38] are respectively studied issues of topology identification and others [39], [40]. Besides, for the research of uncertain parameters in a finite time, a class of Markovian jump complex networks with partially unknown transition rates is also considered to achieve the finite-time stability or synchronization characteristics of systems [34], [35], [36]. Mei et al. [41], [42] study the finite-time topological identification and synchronization of drive-response (master-slave) system based on an effective control input and a feedback control with an updated law respectively. Wang et al. [43] investigates finite-time synchronization and parameter identification problem for uncertain Lurie systems based on the finite-time stability theory and the adaptive control method.
It is often difficult to find a suitable model in the practical application, due to model's parameters partially known or completely unknown, especially the selection of structure. The issues of parameters or topology identification are urgent and important. It mainly depends on information of the known system to identify and estimate the unknown models, further to achieve the purpose of application. To the best of our knowledge, few works focus on finite-time topology identification and synchronization and the existing researches are not considered the effect of external environment and multiple time delays, which have practical significance in the real system. Motivated by above discussions, in this paper we take fully into account stochastic noise perturbation, uncertain topological structure and multiple delays to build a new dynamic model. In order to overcome these difficulties of the above factors interaction, two approaches are proposed to achieve the topological identification and stochastic synchronization for complex network in the finite-time. Different from the previous works, the effective controllers are designed, the finite-time topological identification and stochastic synchronization for complex networks with the same topological structure and different topological structure are considered. The former uses the response system with an unknown structure to track the known drive system to achieve the finite-time topological identification, and the topological structure of the drive system can be tracked down and identified based on the adaptive update law of a novel controller in the latter. Several novel identification and synchronization criteria for complex network are then obtained.
The paper is organized as follows. In Section 2, the model of complex network and preliminaries are given. Two main results about the finite-time topological structure and synchronization are respectively given in 3 Finite-time stochastic synchronization and topology identification for complex networks with the same topological structure, 4 Finite-time stochastic synchronization and topology identification for complex network with different topological structure. Two numerical examples are given to show the effectiveness of our results in Section 5. Finally, conclusion and prospect are given in Section 6.
Section snippets
Network model and preliminaries
In the paper, according to the property of multiple time delays, we consider a model of stochastic complex dynamical network as follows:where is the state vector of the ith node, is a nonlinear function, which describes the dynamics of node i in the absence of interactions with other nodes. and
Finite-time stochastic synchronization and topology identification for complex networks with the same topological structure
In this section, the following Theorem is presented to achieve the finite-time topological identification and stochastic synchronization simultaneously for complex network with multiple time delays. Theorem 1 If Assumption 1, Assumption 2 are satisfied, and suppose that there exists a sufficient large positive constant r, which satisfieswhere , and δ is a nonnegative constant, which is defined in Assumption 1. Then the drive system
Finite-time stochastic synchronization and topology identification for complex network with different topological structure
We consider the following drive and response networks with multiple time delays and noise perturbation:where and are the coupling configuration matrices of both networks, cijl and dijl are defined as follows: if node i and j are linked by an edge, then and ; otherwise, and , and the
Numerical simulations
Example 1 We consider a network with six different sub-networks which have different time-delays. Assuming the network is composed of four nodes, we can easily get the weight configuration matrixes A0, A1, A2, A3, A4, A5, in which matrix A0 has no time-delay, and matrix A5 has the maximal time-delay. The weighted configuration matrixes are described by
We
Conclusion and prospect
In this paper, we give two control methods to deal with topological identification and stochastic synchronization of complex network with multiple time-delays in the finite-time. In main results, we take fully account the complex networks with the same and different topological structures, and design two different kinds of effective controllers to obtain effective results of identification and stochastic synchronization in the finite time. Furthermore, based on finite-time control theory,
Hui Zhao received the B.S. degree in Computer Science and Technology from Hebei University of Engineering, HanDan, China in 2009, and the M.S. degree in Computer Application Technology from Taiyuan University of Science and Technology, TaiYuan, China, in 2013. Since 2013, she is going on study as a Ph.D. student in Beijing University of Posts and Telecommunications, Beijing, China. Her current research interests include complex dynamical network, neural network, memristors and neural network
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2021, Physica A: Statistical Mechanics and its ApplicationsCitation Excerpt :In many real situations, networks cannot realize synchronization by themselves. Therefore, some effective methods have been discovered, such as linear feedback control [9], adaptive control [10,11], periodically intermittent control [12–14], pinning control [15,16], etc. Although the performance and robustness of the above methods are acceptable, but we can notice that the drive–response networks mainly focus on networks with identical topological of sub-networks, inner coupling, number of node and dynamics.
Fixed-time synchronization of delayed complex dynamical systems with stochastic perturbation via impulsive pinning control
2020, Journal of the Franklin InstituteCitation Excerpt :Unlike asymptotic synchronization and exponential synchronization, finite-time synchronization greatly increases the efficiency and has better robustness and disturbance rejection properties. Over the years much work has been dedicated to this concept [12,13]. The paper [12] studied finite-time synchronization of Markov jumping complex networks with partially unknown transition rates, time-delay and stochastic noise.
Finite-time synchronization and parameter identification of uncertain fractional-order complex networks
2019, Physica A: Statistical Mechanics and its ApplicationsCitation Excerpt :Recently, some researchers have studied FTSPI problem of uncertain integer-order complex networks [21,22,37–39]. As we know, the results in [21,22,37–39] cannot be directly extended to study FTSPI problem for uncertain fractional-order complex networks, because of those results are obtained under the framework of integer-order finite-time stability theorem. In this paper, FTSPI criteria are obtained based on the properties of fractional calculus.
Finite-time synchronization of a class of nonlinear complex-valued networks with time-varying delays
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Hui Zhao received the B.S. degree in Computer Science and Technology from Hebei University of Engineering, HanDan, China in 2009, and the M.S. degree in Computer Application Technology from Taiyuan University of Science and Technology, TaiYuan, China, in 2013. Since 2013, she is going on study as a Ph.D. student in Beijing University of Posts and Telecommunications, Beijing, China. Her current research interests include complex dynamical network, neural network, memristors and neural network based on memristor.
Lixiang Li received the M.S. degree in circuit and system from Yanshan University, Qinhuangdao, China, in 2003, and the Ph.D. degree in signal and information processing from Beijing University of Posts and Telecommunications, Beijing, China, in 2006. The National Excellent Doctoral theses winner, the New Century Excellent Talents in university, Henry Fok education foundation, Hong Kong Scholar Award winner, Beijing higher education program for young talents winner. Visiting Potsdam Germany in 2011. Engaged in research of swarm intelligence and network security; having more than 80 papers and a monograph published and being supported by 10 national foundations in recent five years. She is currently a professor at the School of Computer Science and Technology, Beijing University of Posts and Telecommunications, China. Her research interests include swarm intelligence, information security and network security. Dr. L. Li is the co-author of 70 scientific papers and 10 Chinese patents.
Haipeng Peng received the M.S. degree in system engineering from Shenyang University of Technology, Shenyang, China, in 2006, and the Ph.D. degree in signal and information processing from Beijing University of Posts and Telecommunications, Beijing, China, in 2010. He is currently an associate professor at the School of Computer Science and Technology, Beijing University of Posts and Telecommunications, China. His research interests include information security, network security, complex networks and control of dynamical systems. Dr. H. Peng is the co-author of 50 scientific papers and over 10 Chinese patents.
Jinghua Xiao received the B.Sc, M.Sc and Ph.D. degrees in Physics from Beijing Normal University, Beijing, China in 1987, 1990 and 1999. He is currently a professor at Beijing University of Posts and Telecommunications, Beijing, China. His research interests include nonlinear dynamics, complex networks. He has authored over 70 refereed papers in international journals and five of them were published in PRL.
Yixian Yang received the M.S. degree in applied mathematics in 1986 and the Ph.D. degree in electronics and communication systems in 1988 from Beijing University of Posts and Telecommunications, Beijing, China. He is the Managing Director of information security center, Beijing University of Posts and Telecommunications, Beijing, China. The Yangtze River scholar Program professor, National Outstanding Youth Fund winners, the National Teaching Masters. Major in coding and cryptography, information and network security, signal and information processing; having authored more than 40 national and provincial key scientific research project, published more than 300 high-level papers and 20 monographs.
Mingwen Zheng received the B.S. degree in School of Science from Shan Dong University of Technology, ZiBo, China in 2002, and the M.S. degree in School of Computer and Communication Engineering from China University of Petroleum, Dong Ying, China, in 2009. Since 2014, he is going on study as a Ph.D. student Beijing University of Posts and Telecommunications, Beijing, China. His current research interests include complex dynamical network, neural network, and memristors.
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Foundation item: Supported by the National Natural Science Foundation of China (Grant nos. 61472045 and 61573067), the National Key Research and Development Program (Grant nos. 2016YFB0800602 and 2016YFB0800604), the Beijing City Board of Education Science and Technology Key Project (Grant no. KZ201510015015), and the Beijing City Board of Education Science and Technology Project (Grant no. KM201510015009).