Abstract
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.
Highlights
In 1946, the structural balance concept is proposed originally by F
In order to use the structural balance theory in this paper, we consider a complete timevarying complex dynamical network with continuous time values for connection strengths, where the connection relationships subsystem is described by the Riccati matrix differential equation possessing the coupling matrix composed of the internal state of the nodes
In Theorem 1, λmin(Q) and lminðQ" Þ represent the minimum eigenvalues of given positive matrices Q and Q", and the matrices K and M can be obtained by solving the Lyapunov Eqs (19) and (20), respectively; The given constant c represents the common connection relationship strength in the network dynamical model (2); The parameters h, δ and L can be obtained by Assumptions 1, 2 and 3, respectively; the equilibrium matrix PÃ can obtained by Eqs (6) and (8)
Summary
In 1946, the structural balance concept is proposed originally by F. In order to use the structural balance theory in this paper, we consider a complete timevarying complex dynamical network with continuous time values for connection strengths, where the connection relationships subsystem is described by the Riccati matrix differential equation possessing the coupling matrix composed of the internal state of the nodes.
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