Synchronization Of Complex Duplex Networks With Different Inner Coupling Matrices

This paper discusses synchronization of multiplex networks.For the multiplex net-works with identical inner coupling matrices of intra-and inter-layer,we investigate the synchronizability from eigenvalues of the networks' Laplacian matrices based on the master stability function(MSF)method.For the multiplex networks with different inner coupling matrices of intra-and inter-layer,we take duplex networks for example to discuss the relationship between the coupling mode and the synchronized region.The coupling mode is represented by the inner coupling matrices,and the synchronized region is the range of the coupling strength that make the network synchronize.The size and other quantitative characteristics of the synchronized region change with the structure of the network,the dynamics of oscillators and the coupling mode.Consid-ering that we cannot get the analytical result right now,we start with the qualitative characteristics of the synchronized region.Our concern is concentrated on the type of synchronized region including bounded,unbounded and empty.We select three typical inner coupling matrices which represent the bounded,unbounded and empty type in single-layer networks.There are three kind of inner coupling matrix in duplex networks:the inner coupling matrix of 1st layer,the inner coupling matrix of 2nd layer and the inner coupling matrix of inter-layer.They form an inner coupling matrix combina-tion in duplex networks.We assign those three special matrices to this inner coupling matrix combination.There are 27 ways in total,and 18 ways left after a few repeated simulations are excluded.We get the types of intra-layer,inter-layer and global synchro-nized region in duplex networks under 18 different inner coupling matrix combinations.The simulations are carried out in a duplex network with identical intra-layer structure first.Then the results are extended to general duplex networks with different intra-layer structure,and the effects of structure on the type of synchronized region is studied.In the simulations,when both intra-layer structures and coupling modes in each layer are identical,inter-layer synchronization is not affected by intra-layer coupling;on the con-trary,when either intra-layer structures or coupling modes in each layer are different,inter-layer synchronization is affected by intra-layer coupling.The types of synchro-nized region change with the inner coupling matrices.We give the law of change in duplex networks.