Study Of The Synchronizability Of Two Kinds Of Multilayer Networks

The multilayer network is composed of several networks that evolve and interact with each other,and there have many examples in our daily life.Such as,financial market networks,infrastructure networks,climate networks,ecology networks,brain networks,and so on.Synchronization is a kind of spontaneous phenomenon that is common in the real world.In order to make full use of the positive side of synchronization and avoide the negative side,and greatly facilitate our daily life.The research on the synchronizability of multilayer networks is not only helpful for us to understand the real-world network,but also can provide strong theoretical support for improving the synchronization performance of the real-world network.In this paper,the synchronizability of the multilayer K-nearest neighbor network is studied by using the master stability function method.The effects of various topological parameters(such as network size,node degree,number of layers,intra-layer coupling strength,and inter-layer coupling strength)on the synchronizability of multilayer K-nearest neighbor networks are analyzed.Firstly,the network models of two-layer and multilayer K-nearest neighbor networks are given,and corresponding expressions of supra-Laplacian matrices are given.Secondly,the analytical expression of the eigenvalues of the supra-Laplacian matrices are solved,and the influence of the topological parameters on the synchronizability of the network are analyzed.Finally,the correctness of the theoretical derivation is verified by the simulation experiment.The results show that when the synchronization region is bounded and the coupling strength between layers is less than a certain threshold,the synchronizability of the network can be enhanced by increasing the intra-layer coupling strength.When the inter-layer coupling strength is greater than a certain threshold,the synchronizability of the network can be increased by increasing the intra-layer coupling strength and the number of adjacent nodes of the network.When the synchronization region is unbounded,the results show that the conclusion can be adopted to enhance the synchronizability of the network.When the intra-layer coupling strength varies according to direct proportion,inverse proportion,exponent,logarithm,and cosine function,the variation trend of network synchronizability in the case of unbounded synchronization region is consistent with the variation trend of intra-layer coupling strength.When the synchronization region is bounded,the variation of the network's synchronizability is consistent with the variation of the coupling strength of the remaining part of the layer.Besides,the synchronizability of multilayer networks with windmill structure are also discussed.Firstly,the topological structure models of multilayer French and Dutch windmill networks are given.Secondly,the supra-Laplacian matrices corresponding to the multilayer generalized French and Dutch windmill networks are written,and the analytical expression of its corresponding eigenvalues are strictly deduced.Finally,when the synchronization region is bounded or unbounded,we discuss how the layers number,nodes number,the number of cycles c_n,the inter and intra-layer coupling strength of the windmill network influence the synchronizability of multilayer windmill networks by using the master stability function method.The correctness of the theoretical results were verified by simulation experiments.The results show that when the synchronization region is unbounded,the synchronizability of the windmill network can be enhanced by increasing the intra-layer coupling strength,and the number of network layers.When the synchronization region is bounded,the intra-layer and inter-layer coupling strength,and the increase of the number of network layers can enhance the synchronizability of the network,while too many nodes or too many cycles'number that forming the windmill network will weaken the synchronizability of the network.It is worth noting that for the multilayer French windmill network,the synchronizability of the network can only be enhanced by increasing the intra-layer coupling strength when the intra-layer coupling strength is weak.In addition,the simulation results of multilayer generalized French windmill network show that the synchronizability of the network can be enhanced by increasing the coupling strength within and between layers and the number of layers in the network.The paper has 41 pictures,4 tables,and 50 references.