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.