The nodes and their connection relationships are the two main bodies for dynamic complex networks. In existing theoretical researches, the phenomena of stabilization and synchronization for complex dynamical networks are generally regarded as the dynamic characteristic behaviors of the nodes, which are mainly caused by coupling effect of connection relationships between nodes. However, the connection relationships between nodes are also one main body of a time-varying dynamic complex network, and thus they may evolve with time and maybe show certain characteristic phenomena. For example, the structural balance in the social networks and the synaptic facilitation in the biological neural networks. Therefore, it is important to investigate theoretically the reasons in dynamics for the occurrence. Especially, from the angle of large-scale systems, how the dynamic behaviors of nodes (such as the individuals, neurons) contribute to the connection relationships is one of worthy research directions. In this paper, according to the structural balance theory of triad proposed by F. Heider, we mainly focus on the connection relationships body, which is regarded as one of the two subsystems (another is the nodes body), and try to find the dynamic mechanism of the structural balance with the internal state behaviors of the nodes. By using the Riccati linear matrix differential equation as the dynamic model of connection relationships subsystem, it is proved under some mathematic conditions that the connection relationships subsystem is asymptotical structural balance via the effects of the coupling roles with the internal state of nodes. Finally, the simulation example is given to show the validity of the method in this paper.

This paper investigates the observer-based structural balance control for a class of complex dynamical networks. Generally speaking, a complete complex dynamical network is composed of two coupled subsystems, which are called node subsystem (NS) and connection relationship subsystem (CS), respectively. Similar to synchronization and stabilization of networks, the structural balance is another phenomenon of networks and determined by the state of connection relationships. However, it is not feasible to design the controller for the CS directly because the states of the connection relationships are difficult to be measured accurately in practical applications. In order to solve this problem, a state observer for the CS has been designed. Thus, the structural balance controller in the CS can be directly designed by using the estimation information of the state observer. Then, with the help of the Lyapunov stability theory, it is proved that the CS can asymptotically track a given structural balance matrix under the influence of the observer-based controller. Finally, the results derived from this paper are demonstrated by performing a numerical example.