The dynamic behaviors of nodes driving the structural balance for complex dynamical networks via adaptive decentralized control

Abstract

The structural balance based on the triads structure is used to describe the evolution of the relationships in a social network of humans or animals, where the social network can be abstracted into a complex dynamical network which is composed of the nodes subsystem (NS) and the connection relationships subsystem (CS) coupled with each other. Similar to the synchronization or stabilization in NS with the help of CS, structural balance may be arrived at in CS with the help of NS. In this paper, the CS is described by the Riccati linear matrix differential equation with dynamical coupling term, only including the internal states of the NS. We mainly focus on the dynamic behaviors of NS which can lead to the structural balance in CS. It has been proved under some mathematical conditions that if the NS converges to some nonzero constant targets via the adaptive decentralized control scheme for each node, then the CS will asymptotically track a certain structural balance via the effective coupling. Such a result can be used as a specific explanation for the relationship between the structural balance and the dynamic changes of the nodes’ states. Finally, the simulation example is given to show the validity of the method in this paper.

Initial State Causes the Structural Balance of Complex Networks With Dynamical Models

“…References [28], [29] regarded the complex network as a large-scale system, which was composed of nodes subsystem and links subsystem and proposed two different adaptive control schemes for the complex network so that the link subsystem could asymptotically reach a structural balance. References [30] designed a state observer for the links subsystem and proposed a control scheme based on the information of the observer so that the state of the links subsystem could track a structural balance asymptotically.…”

“…However, the above existing results [28]- [32] considered a linear matrix Riccati differential equation as a model of the links subsystem. In this paper, different from the above literature, we choose a nonlinear matrix Riccati differential equation as a model.…”

“…(2) Model Differences: There are many results about structural balance, such as [28]- [32]. However, these papers discussed the tracking problem of the network with structural balance based on the linear matrix Riccati differential equation.…”

Asymptotical stability for complex dynamical networks via link dynamics

“…[1][2][3][4][5] During the past decades, the dynamical behaviors of various complex networks have attracted increasing attention from the synchronization of nodes [6][7][8][9][10] and the structure balance of links. 5,[11][12][13][14][15] From the angle of large-scale system, a complex dynamical network can be considered as the interconnected system with the node subsystem and the link subsystem. 14,15 Therefore, the synchronization can be regarded as emerged by the node subsystem and the structure balance by the link subsystem.…”

“…5,[11][12][13][14][15] From the angle of large-scale system, a complex dynamical network can be considered as the interconnected system with the node subsystem and the link subsystem. 14,15 Therefore, the synchronization can be regarded as emerged by the node subsystem and the structure balance by the link subsystem. This implies that the node subsystem and the link subsystem are the two main parts of dynamical behavior of complex network.…”