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From the perspective of large-scale system, a directed complex dynamic network (DCDN) may be considered as a coupling system of the node subsystem (NS) and the link subsystem (LS). In this paper, by using the outgoing link vector and incoming link vector for DCDN, the dynamics of LS is described by employing the vector differential equation instead of the matrix differential equation. Since the outgoing and incoming link vectors have the stronger geometric intuition, the results in this paper show that this kind model of links can not only reflect the direction of links but also find the dynamic tracking goal of links more easily when the state synchronization of NS emerges. Furthermore, by employing the simple mathematical conditions, the nonlinear controller of NS and the coupling term of LS are proposed to ensure achieving the asymptotical state synchronization for DCDN. Finally, the numerical simulations are given to demonstrate the validity of the results in this paper.

The model of complex dynamical network (CDN) can be represented as the mathematic graph, in which some characteristics may emerge from the dynamic nodes group (NG) and links group (LG). This paper primarily focuses on the feature appearing from the dynamic links. The average link weight (ALW), as a novel quantitative index to describe the characteristic of dynamic links is introduced. Inspired by the Hebb's neuroscience theory, the Hebb complex dynamical network (HCDN) is constructed. The ALW of the HCDN can track a given target via external stimulus signals with adaptive amplifiers' proportional coefficients. In other words, the stationary network implies the ALW is a constant in time. Finally, two simulation examples are performed to validate the proposed adaptive update law's effectiveness.

This paper focuses primarily on the mean square consensus problem of a class of nonlinear multi-agent systems suffering from stochastic impulsive deception attacks. The attacks here are modeled by completely stochastic destabilizing impulses, where their gains and instants satisfy all distributions and the Markovian process. Compared with existing methods, which assume that only gains are stochastic, it is difficult to deal with systems with different types of random variables. Thus, estimating the influence of these different types on the consensus problem is a key point of this paper. Based on the properties of stochastic processes, some sufficient conditions to solve the consensus problem are derived and some special cases are considered. Finally, a numerical example is given to illustrate the main results. Our results show that the consensus can be obtained if impulsive attacks do not occur too frequently, and it can promote system stability if the gains are below the defined threshold.

From the large system perspective, the directed complex dynamic network is considered as being composed of the nodes subsystem (NS) and the links subsystem (LS), which are coupled with together. Different from the previous studies which propose the dynamic model of LS with the matrix differential equations, this paper describes the dynamic behavior of LS with the outgoing links vector at every node, by which the dynamic model of LS can be represented as the vector differential equation to form the outgoing links subsystem (OLS). Since the vectors possess the flexible mathematical operational properties than matrices, this paper proposes the more convenient mathematic method to investigate the double tracking control problems of NS and OLS. Under the state of OLS can be unavailable, the asymptotical state observer of OLS is designed in this paper, by which the tracking controllers of NS and OLS are synthesized to ensure achieving the double tracking goals. Finally, the example simulations for supporting the theoretical results are also provided.

In this paper, the concept of the state domain of the hidden layer neuron is introduced for the three-layer BP (Back Propagation) neural network. By employing the state domain and the partition of unity method to modularize the hidden layer, and a new artificial network called Nested BP neural network is constructed to reduce the computational complexity and improve the learning quality. The Nested BP network is employed to approximate the unknown nonlinear items in controlled systems. Compared to the usual BP network, the control effect in this paper is greatly improved. Finally, an example shows that the proposed method is superior to the usual BP neural network in learning quality and control effect.