Influence Research Of Backbone Structure On The Synchronizability And Controllability Of Networks And Identification

With the rapid development of science and technology represented by internet technology,in our real lives,the interdependence and interactions among large numbers of complex systems has become increasingly closer.In order to analyze and explore the characteristics and evolution rules of the interdependence,the scientific community has described this kind of interdependence with the topological structure from the perspective of complex network theory.It is generally known that the topology of complex network determines the function of the network.Taking the interdependent network as a research object,in this paper,we analyze and discuss the spectrum properties of the super-Laplace matrix,which can illustrate the topological structure of the network,the influence of the inter-structure between the subnets(that is,the backbone structure)in the network on the optimization of synchronizability,controllability,as well as the backbone structure identification.The main research contents and innovation points of this paper are as the following.(1)The spectrum of the Laplacian matrix of complex network is a key factor for network function.In this paper,the spectral properties of Supra-Laplacian for partially interdependent networks are investigated.Based on Matrix Perturbation Theory,refined results of the eigenvalue properties of Laplacian matrices are provided,which shows that the size relationship of the first-order approximate solutions of the eigenvalues remains unchanged,even if there is perturbation.Using these results,the theoretical approximate formulae of the minimum non-zero eigenvalue and the maximum eigenvalue of Supra-Laplacian for partially interdependent networks are derived,respectively.Compared with the literature [50-51],the outcomes in this paper are more general and need fewer calculations.Finally,the simulations and applications of synchronizability and diffusion process verify the feasibility and effectiveness of the proposed solutions.The findings can be instructive for networks when linking to the spectral properties of the Supra-Laplacian.(2)Based on a genetic algorithm(GA),combined with two fitness functions,a new method is presented to optimize the inter-structures to enhance the synchronizability of the interdependent networks.For bounded and unbounded synchronous regions,when the number of inter-connections or the inter-coupling strength related to the inter-structures is changed,we apply the proposed method to optimize two interdependent networks.The first network,namely NW-BA network,includes two subnets,which are a small-world subnet generated by Newman-Watts algorithm and a Barabási-Albert scale-free subnet.The second network,ST-BA network,includes a star subnet and a BA subnet.Generally,the analyses and numerical simulations demonstrate the optimization scheme performs well.According to the betweenness correlation defined in this paper,with the number of inter-connections increasing,the optimized inter-structure experiences a process that shifts a nearly positive correlation to one of irrelevance.However,growing the inter-coupling strength gradually makes no difference on optimized inter-structures.Overall,the NW-BA networks are better choice for interdependent networks than the ST-BA networks.(3)In this paper,without changing the number and positions of the controllers,the effect of interdependence between directed subnets(controllable or uncontrollable)on the controllability of complex networks is investigated.According to the controllable criteria of Kalman rank and PBH rank and maximum matching theory,the sufficient conditions and necessary conditions for the controllability or uncontrollability of interdependent networks are presented.The results show that:(i)if the uncontrollable and controllable nodes of a subnet are inter-coupled with the controllable and uncontrollable nodes of the other subnet respectively,and the directions are the controllable nodes pointing to the uncontrollable nodes,then the interdependent network may be controllable;(ii)the interdependent network is certainly controllable when the topological structure of the uncontrollable node-set of the two subnets have a maximum matching which covers all the uncontrollable nodes,separately;(iii)The interdependent networks are controllable no matter what kind of inter-coupling topologies between two controllable subnets.All of these results are adaptive to undirected subnets.The final example verifies the effectiveness and correctness of the results provided in this paper.(4)Based on event-triggered control technology,Lyapunov stability theory and adaptive control method,in this paper,research on the identification of backbone structure of interdependent networks is carried out.The interdependent network considered includes directed and known intra-structures and directed and unknown inter-structures.With designed event-triggered controllers,event threshold and adaptive laws,in this paper,a sufficient condition for event-triggered adaptive asymptotic synchronization of the network is presented,which can ensure the heterogeneous asymptotic synchronization between the drive network and the response network.Meanwhile,it can identify the backbone structure of the network successfully.Finally,numerical simulations show the correctness and effectiveness of the proposed theory.