Topology Identification Of Complex Dynamical Networks: From Single-layer Networks To Multilayer Networks

The topological structure of a complex dynamical network indicates the connected relationship between the nodes, it plays an important role in determining the network' s evolutionary mechanisms and functional behaviors, and it is a prerequisite to analyze and predict the dynamic behavior of the complex networks. However, for complex net-works of the real world, the exact topology is often unknown or partly unknown, thus how to recognize and infer the network structure from dynamical variables have been detected is both of theoretical and practical significance. This is the problem of topol-ogy identification of complex dynamical networks with extensive practical background, and it is also a challenging problem in the research of complex network science develop-ment. In recent years, topology identification of complex dynamical networks gradually attracted the attention of many domestic and foreign scholars, who have started a large amount of research work, and achieved good results in the topology identification of the relatively ideal single-layer network.This paper mainly studies the problem of topology identification of complex dy-namical networks with stochastic perturbation and coupling delay, and attempts to extend the results from single-layer networks to multilayer networks. Compared with the single-layer networks, the multilayer complex dynamical networks can simulate real network systems and describe the real network scenarios more clearly. Thus with the development of complex network science, the research on single-layer networks has been not satisfied the actual requirements of the complex system, and the description and research on multilayer networks is urgently needed. This can provide the foundation for exploring the dynamical evolution mechanism of large scale networks and reshape the network structure and other issues, and also provides new perspectives and methods for the research and development of many disciplines, such as biology, social information and many others. This paper has six major chapters. Chapter 1 briefly introduces the research background and research status of this paper. Chapter 2 gives the basic knowledge related to the following content. Chapters 3 to 5 introduce the main work of the paper, and the last chapter gives the summary and the prospect of future work on above basis. The main contents and innovations of the article are as follows.Firstly, Chapter 3 studies the topology identification of single-layer complex dy-namical networks with stochastic perturbations and coupling delay based on complete synchronization. Take the original network with unknown topology as drive network, by constructing a response network without noise and designing appropriate controllers, and combining the stability theory of stochastic differential equations, the topology structure of the drive network can be derived. It is worth noting that the network model considered in this chapter contains practical stochastic perturbations, while the constructed auxiliary network on receives the noisy observations as control inputs, which greatly simplifies the identification process to a certain extent, and improves the iden-tification efficiency. In addition, the proposed control method can be effectively used for the detection of hidden sources or hidden information, which is a new discovery. The approach can provide some theoretical guidance and method foundation for the localization of network topology parameters and hidden sources in engineering practice.Chapter 4 gives the network topology identification based on generalized synchro-nization. An adaptive control technique is proposed so that the drive network achieves generalized synchronization with the response network and the unknown topology of the drive network can be successfully recovered. And the structure of the response net-work can be any form, even isolated nodes that are not connected. It is worth pointing out that this method can be not only used to detect the partial structure information of complex systems, but also locate the hidden sources. Moreover, when the considered network with unknown topology is composed of nodes carrying high node dimensions or complicated node dynamics, one can design a response network using nodes with low dimensions or simpler dynamics, this is an unprecedented advantage.Chapter 5 discusses the identification of two-layer networks based on the auxiliary system method. For multilayer complex dynamical networks, one usually only can get the information of one layer or a portion of the whole network, thus here focuses on the topology identification of a two-layer network with peer-to-peer unidirectional inter-layer connections. Take the output layer as drive layer, the input layer as response layer, and construct an auxiliary layer that is a duplicate of the response layer to identify the topological structure of the response layer. The most important feature of this method is that the controller is relatively simple, which can greatly reduce the amount of control input information, and improve the efficiency of identification. Simulation results are given to illustrate the effectiveness of the theoretical results. And the impact of the inter-layer information transmission speed on the topology identification performance is further investigated. We hope to provide a theoretical basis for the location of the route and sources of rumors or pseudo news spreading.