Partial Pinning Control for the Synchronization of Fractional-Order Directed Complex Networks
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Partial Pinning Control for the Synchronization of Fractional-Order Directed Complex Networks Fengyi Liu1 · Yongqing Yang1 · Aihua Hu1 · Li Li1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper mainly studies the synchronization problem of fractional-order directed complex networks through partial pinning control. Unlike other papers, the network studied in this paper is neither strongly connected nor contains directed spanning trees. By utilizing the directed acyclic graph condensation and Layering theory, the network is decomposed into several strong connected components and then divided into layers. It proved that all or part of nodes in the network can achieve synchronization with the pinner’s trajectory, only when the root strong connected components in the upstream of these nodes are pinned and satisfy some sufficient conditions. In addition, according to the ControlRank algorithm, an optimized strategy is designed to solve the problem of the optimal selection of the pinning nodes to ensure the specific nodes of the network can achieve synchronization eventually. Meanwhile the amount of control energy cost will also be given in this paper. Finally, two simulation examples are given to verify the reliability and feasibility of the optimized algorithm. Keywords Fractional-order · Partial pinning control · DAG condensation · Layering theory · ControlRank algorithm
1 Introduction In recent years, complex network has become a research hotspot, attracting more and more scholars from all over the world to conduct research. The research results have been widely used in the fields of the WWW, economics, society and politics, life and DNA, and network marketing, etc. [1–4]. The study of complex networks dates back to the eighteenth century. Euler proposed the famous “Königsberg Seven Bridges Problem”, arguing that the structure of a graph or network is the key to understanding the complex networks around us. Hereafter
This work was jointly supported by the Natural Science Foundation of Jiangsu Province of China under Grant Nos. BK20161126, BK20170171, BK20181342, and the Postgraduate Research & Practice Innovation Program of Jiangnan University under Grant No. JNKY19_042.
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Yongqing Yang [email protected] School of Science, Wuxi Engineering Research Center for Biocomputing, Jiangnan University, Wuxi 214122, People’s Republic of China
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in the 1960s, Paul Erdös and A Rényi proposed the random network model [5] to imitate the complexity of the networks, but ignored the regularity behind the complex network. Afterwards, Watts and Strogatz published an article in Nature magazine and the small world model (SW) arose [6] followed. Harvard professor Stanley Milgram brought forward the theory of “six degrees separation” that we are now familiar with [7], which is viewed as the theoretical origin of social networks. With the maturity of the theory and the improvement of computing power, the researchers found that the nodes in the network were distribu
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