Synchronization Mechanism for Controlled Complex Networks under Auxiliary Effect of Dynamic Edges

State synchronization of controlled nodes via the dynamics of links for complex dynamical networks

Stabilization and synchronization control for discrete-time complex networks via the auxiliary role of edges subsystem

In the existing research results of the complex dynamical networks controlled, the controllers are mainly used to guarantee the synchronization or stabilization of the nodes’ state, and the terms coupled with connection relationships may affect the behaviors of nodes, this obviously ignores the dynamic common behavior of the connection relationships between the nodes. In fact, from the point of view of large-scale system, a complex dynamical network can be regarded to be composed of two time-varying dynamic subsystems, which can be called the nodes subsystem and the connection relationships subsystem, respectively. Similar to the synchronization or stabilization of the nodes subsystem, some characteristic phenomena can be also emerged in the connection relationships subsystem. For example, the structural balance in the social networks and the synaptic facilitation in the biological neural networks. This paper focuses on the structural balance in dynamic complex networks. Generally speaking, the state of the connection relationships subsystem is difficult to be measured accurately in practical applications, and thus it is not easy to implant the controller directly into the connection relationships subsystem. It is noted that the nodes subsystem and the relationships subsystem are mutually coupled, which implies that the state of the connection relationships subsystem can be affected by the controllable state of nodes subsystem. Inspired by this observation, by using the structural balance theory of triad, the controller with the parameter adaptive law is proposed for the nodes subsystem in this paper, which may ensure the connection relationship matrix to approximate a given structural balance matrix in the sense of the uniformly ultimately bounded (UUB). That is, the structural balance may be obtained by employing the controlling state of the nodes subsystem. Finally, the simulations are used to show the validity of the method in this paper.

This paper utilises the synthesised link observers to investigate the asymptotic outer synchronisation for two complex dynamical networks (CDNs) with time-varying links, in which one is called as the drive network (DN), another is the response network (RN). Meanwhile, each CDN is regarded to be composed of two mutually coupled subsystems: the node subsystem (NS) and the link subsystem (LS). In each CDN, the dynamics of NS is modelled as the vector differential equation with the unknown interaction caused by the nodes state of another network, and the LS is modelled as the matrix differential equation with link output. Two kinds of differential equations are regarded to be coupled with each other due to the coupling relation between nodes and their connections. By employing the above dynamical models of NSs and the outputs of LSs, the asymptotic state observers of two LSs are constructed, respectively, by which the adaptive controller of NS is synthesised for the DN under some mild conditions. The results in this paper show that the synthesised adaptive controller ensures not only the asymptotic outer synchronisation is achieved for DN and RN but also the corresponding states of LSs are asymptotically the same. Compared with the existing literature, the most prominent highlight of this paper is that two link observers are synthesised under the unavailable links to assist the adaptive controller of NS in RN to achieve the asymptotic outer synchronisation. Finally, the effectiveness of the theoretical results in this paper is proved by the numerical simulation.

In this article, the cluster synchronization control for a complex dynamic network (CDN) with mutually coupled nodes subsystem (NS) and connection relations subsystem is investigated. The process of cluster synchronization is segmented into three dynamic continuous stages: first, the free movement, then the task‐oriented clustering, and finally the cluster synchronization. The control scheme, founded on the clustering nodes of CDN, is proposed, such that the cluster synchronization arises when clustering all the nodes into two different sub‐networks (SNs). This approach originates from the flight motion for a group of autonomous aircrafts. At a certain moment, free flight is interrupted, and all the aircrafts split into two SNs according to the given clustering rules. Then the two controlled SNs are required to achieve cluster synchronization, that is, every SN converges to the same in speed. In order to split all the nodes into two different SNs, a clustering algorithm according to the “central nodes” is proposed. Based on Lyapunov stability theory, by designing the suitable inner cluster links, the adaptive controller of NS for each SN is synthesized to guarantee the stability of the error system. It is shown that the eventual inner topology of each SN is displayed in the pre‐given form when the cluster synchronization achieves. Finally, the merits and effectiveness of the proposed control scheme are verified by a simulation example.

This paper is concerned with the optimal output synchronization control problem for a class of heterogeneous complex dynamical networks (CDNs) with partially unknown system dynamics. By defining a new quadratic performance index including the information of state couplings, an optimal controller consisting of a feedback control and a compensated feedforward control is then designed. A necessary and sufficient condition for the optimal control minimizing the quadratic performance index is established. Especially, an algebraic Riccati equation (ARE) associated with a discount factor is obtained, and a novel online iteration algorithm by iteratively solving the ARE is proposed to compute the optimal controller gain for the case that the CDNs are with partially unknown system dynamics. Moreover, it is proved that the output synchronization errors converge to zero asymptotically by applying the optimal control laws. Finally, two simulation examples are provided to verify the effectiveness of the proposed control method.

In this paper, the finite-time stochastic outer synchronization between two different complex dynamical networks with noise perturbation is investigated. By using suitable controllers, sufficient conditions for finite-time stochastic outer synchronization are derived based on the finite-time stability theory of stochastic differential equations. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the settling time is also numerically demonstrated.

This paper deals with the function projective synchronization of two complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences. Based on Lyapunov stability theory and inequality transformation method, the robust adaptive synchronization controller is designed, by which the drive and response systems can achieve synchronization according to the function scaling factor. Different from some existing studies on nonlinear system with sector nonlinear input, this paper studies the synchronization of two complex dynamic networks when the boundary of sector nonlinear input is unknown. The controller does not include the boundary value of the sector nonlinear input and the time delay term, so it is more practical and relatively easy to implement. The corresponding simulation examples demonstrate the effectiveness of the proposed scheme.

In this paper, in the case of quantized communication and coupling delays, a novel quantized pinning control (QPC) law is designed to deal with the synchronization problem of complex dynamic networks (CDN). Utilizing the Lyapunov method, non-smooth analysis tools and convex combination techniques, some sufficient criteria for achieving synchronization of CDNs with coupling delays are given. Finally, the effectiveness of the main results is demonstrated by presenting a simulation example.

Complex dynamical networks consist of a large set of interconnected nodes with each node being a fundamental unit with detailed contents. A great number of natural and man-made complex networks such as social networks, food networks, neural networks, the World Wide Web, computer networks, electrical power grid and so on, can be effectively modelled by complex dynamical networks. Synchronization of a group of dynamical nodes in a complex network topology is one of the most interesting and significant phenomenon in complex dynamical networks. In this section, we shall discuss impulsive synchronization of the discrete delay networks. By employing the impulsive control method and the Razumikhin-type results obtained in previous chapter, two criteria for exponential synchronization will be derived.

Robust synchronization in finite time for fractional-order hybrid coupling discontinuous complex dynamic networks with nonlinear growth

The robustness of controlling complex networks is significant in network science. In this paper, we focus on evaluating and analyzing the robustness of controlling edge dynamics in complex networks against node failure. Using three categories of all nodes to quantify the robustness, we find that the percentages of the three types of nodes are mainly related to the degree distribution of networks. The simulation results of model networks and analytic calculations show that the sparse inhomogeneous networks, which emerge in many real complex networks, have strong control robustness from the point of the number of ordinary nodes, but the strong positive correlation between in and out degrees reduces the control robustness. Evaluation of real-world networks indicates that most of them have few or no critical nodes, that is, they do not need to increase driver nodes to maintain control for most of node failures. Then an adding circuit-link strategy is proposed to optimize the robustness of edge controllability.

This paper is devoted to the study of synchronization problems in uncertain dynamical networks with time-delayed nodes and coupling delays. First, a complex dynamical network model with time-delayed nodes and coupling delays is given. Second, for a complex dynamical network with known or unknown but bounded nonlinear couplings, an adaptive controller is designed, which can ensure that the state of a dynamical network asymptotically synchronizes at the individual node state locally or globally in an arbitrary specified network. Then, the Lyapunov-Krasovskii stability theory is employed to estimate the network coupling parameters. The main results provide sufficient conditions for synchronization under local or global circumstances, respectively. Finally, two typical examples are given, using the M-G system as the nodes of the ring dynamical network and second-order nodes in the dynamical network with time-varying communication delays and switching communication topologies, which illustrate the effectiveness of the proposed controller design methods.

To study the underlying organizing principles of various complex systems, designing an efficient graph-based model for data representation, is a fundamental aspect. As the topological structure of the network changes over time, it is a challenging task to design a communication system having ability to respond to randomly changing traffic. We are interested to find out the suitable and fair traffic flow rates to each system for getting optimal system utility using dynamic complex network model. In this context, we design and simulate a growth model of the data communication network based on the dynamics of in-flowing links which is motivated by the concept that newly added node will connect to the most influential nodes already present in the system. The connectivity distribution of the evolved communication networks follows power law form, free from network scale. We analyze Kelly’s optimization framework for a rate allocation problem in communication networks at different time instants, and optimal rates are obtained with the consideration of arbitrary communication delays.

This paper deals with the synchronization problem of complex dynamical networks with interval time-varying coupling delays. A simple local linear feedback controller is introduced to guarantee the synchronizability of the networks. Some delay-dependent synchronization conditions for the controlled complex dynamical networks are presented by using the Lyapunov–Krasovskii functional method and the reciprocally convex combination approach. Theoretical analysis and numerical examples show that the obtained conditions have less computational complexity and less conservatism than some recently reported ones.