Synchronizability And Eigenvalues Of Multilayer Networks

An important aspect of the development of complex network theory is from single-layer network to multilayer network.Multilayer network can reflect the interaction among a variety of networks in the actual complex system,and can better understand that the network topology have an impact on the physical process of the complex system.Synchronicity is a very common and important non-linear phenomenon,which is prevalent in the fields of biology,physics,chemistry,medicine,communication,economy,electric power,social system and so on.It is closely related to our life.Therefore,the research on the synchronization of multilayer networks has attracted the attention of researchers,and has also become the focus of this paper.This paper breaks through the limit of the number of layers of the two-layer network,and studies the eigenvalue spectrum and the synchronizability of multilayer networks.This paper extends the three-layer network to the multilayer networks based on the two-layer network.The multilayer star networks through bi-directionally coupling,the multilayer star networks through unidirectionally coupling,and the multilayer networks based on Scale-free network,whose parameters such as the number of layers,the coupling strength and the number of nodes have an influence on the synchronizability of network and the range of the coupling strength in the case of ideal synchronization are discussed.Firstly,the structure model of multilayer networks is constructed,and the super-Laplacian matrix of multilayer network is constructed according to its connection coupling method,and its eigenvalues are deduced strictly.Secondly,under the different synchronization domain,the influence of the dynamic parameters of the network on the synchronizability of the network is determined by using the main stability function method.The mathematical simulation is carried out to verify and analyze the influence of each dynamic parameter on the synchronizability.The results show that the influence of the number of layers on the synchronizability of the network is very important in addition to the coupling strength.The influence relationship is different according to the coupling mode of the network,the difference of the synchronization domain and the weak side of the coupling strength.The influence of parameters such as coupling strengths and number of nodes on network synchronization can be extended from three-layer network to multilayer networks.Finally,the synchronization of the membrane potential of the two-layer neural network is studied,some conclusions are verified and the range of the parameters of the coupling strength is obtained in the case of ideal synchronization.