Structural Balance Control For Complex Dynamical Networks

From the point of mathematical graph theory,a complex dynamical network is composed of many dynamic nodes and links,where the nodes represent the "participants" and the links represent the "connection relationships(weights or strengthes)" between the nodes.The dynamic nodes are interconnected by their connection relationships and evolve with time.As a graph theory model of complex dynamical systems,complex dynamical networks include many realistic networks,such as biological neural networks,computer networks,sensor networks,social economic networks and so on.Therefore,the development and application of complex network theory are extensive and far-reaching.In the field of control theory,complex dynamical networks are regarded as a kind of controlled plans,the synchronization or stabilization is one of the research hotspots.From the results we obtained,the existing research mainly focuses on the dynamic behaviors of the controlled nodes group.In other words,the nodes group is the main body of the synchronization and stabilization behaviors,while the connection relationships are regarded only as the auxiliary part and affect the behaviors of the nodes through the coupling mode and strengthes with the nodes.Obviously,if the connection relationships are time-varying in complex dynamical networks,the above research ignores the dynamic behaviors of the connection relationships themselves.From the angle of large-scale system theory,a complex dynamical network with time-varying connection relationships can be considered to be composed of two coupled subsystems,i.e.the nodes subsystem and connection relationships subsystem.This means that the synchronization and stabilization in existing researches can be regarded as the dynamic behaviors of the nodes subsystem.Meanwhile,it is worth noting that the dynamic behaviors of the nodes subsystem may also lead to the emergence of some characteristic phenomena in the connection relationships subsystem due to the existence of coupling.For example,in signed networks or social networks,some special examples show that the connection relationships between individuals sometimes emerge the dynamic characteristic phenomenon,which is called structural balance.Therefore,it is meaningful to discuss the conditions,mechanisms and control strategies for emergence of structural balance in connection relationships subsystem.Especially,it is a new research topic to explore how the nodes subsystem effectively assists the emergence of structural balance under the coupling conditionsFor the above mentioned problems,this thesis uses the matrix differential equations to describe the dynamic behaviors of time-varying connection relationships in complex dynamical networks,and mainly explores and discusses the mechanisms and control strategies of the structural balance for the connection relationships subsystem,which is driven by the coupling effect of the controlled nodes subsystem.The research works are carried out from the following aspects.(1)The complex dynamic networks are regarded as the large-scale systems.As a supplement and extension to the existing research,it is the research topic of this thesis that how the dynamic changes of controlled nodes subsystem affect the dynamic behaviors of connection relationships subsystem through the couplings.At the same time,the signifieance and contents of this thesis are introduced in view of the structural balance mechanism and control of complex dynamic networks(2)Without considering control,it is discussed that the connection relationships subsystem emerges the structural balance with the help of dynamic coupling of nodes Firstly,the concept of equilibrium state for complex dynamic networks is proposed,by which the coupling term in connection relationships subsystem is designed.Then,under some mathematical conditions,it is proved that the dynamic behaviors of nodes subsystem can force the connection relationships subsystem asymptotically achieving the structural balance via the coupling tern.Finally,numerical simulations are given to verify the effectiveness of the proposed method(3)In view of the difficulty to accurately measure the state(weights or strengthes)of the connection relationships subsystem in practical application,three structural balance control methods are proposed based on dynamic coupling of controlled nodes.The common purpose of the control methods is to force the connection relationships subsystem asymptotically approaching the structural balance with the help of the controlled nodes subsystem and the coupling term in the connection relationships subsystem.The validities are verified by the numerical simulations.(4)In view of the difficulty to accurately measure the state(weights or strengthes)of the connection relationships subsystem in practical application,the asymptotical state observer for the connection relationships system is proposed,by which a structural balance control is proposed based on the asymptotical state observer.Firstly,the output information of the connection relationships subsystem and the state of nodes subsystem are employed to synthesize the state asymptotic observer.Then,a structural balance controller is designed for the connection relationships subsystem by using the observer's state.The validity of the proposed methods is verified by the corresponding numerical simulations.