| Complex networks in the real world are typically composed of multiple simple networks interacting with each other,exhibiting characteristics such as node diversity,edge heterogeneity,nested structures,and multilevel properties,which require a multilayer network model to describe.Various emergent phenomena and dynamical behaviors on multilayer networks,as well as the mechanisms that give rise to them,are important research directions in the study of complex networks.Synchronization is a typical collective behavior,and research on synchronization in single-layer networks is already well-developed.However,research on synchronization in multilayer networks is still in its early stages.Due to the complexity of multilayer network structures,theoretical research is quite challenging,and there are few related findings.Therefore,this article focuses on the network topology and uses the theory of the main stable function(MSF),combined with matrix analysis and numerical simulation methods,to study the eigenvalue spectra and synchronization capabilities of several types of complex multilayer networks,as well as the parameter values when the network synchronization capability reaches its optimal structure.The main work is as follows:(1)The synchronizability of four types of double-layer ring networks is studied:doublelayer inter-layer directed ring networks(Networks-A),double-layer intra-layer inversed ring networks(Networks-B),double-layer undirected ring networks(Networks-C),and doublelayer undirected interlayer random edge-added ring networks(Networks-D).For the first three types of networks,the research results show that the parity of the sub-layer size will affect the synchronizability of the network when the synchronization domain is bounded;the synchronizability of Networks-A reaches its optimum when d=4aπ/N2;a larger a and smaller d and N are more conducive to the synchronizability of Networks-B;and the synchronizability of Networks-C reaches its optimum when d=2aπ2/N2.For Networks-D,the eigenvalue spectrum is calculated through numerical simulation methods,and the influence of structural parameters on synchronizability is analyzed in conjunction with the MSF theory.The findings reveal that the larger the probability of randomly adding edges between layers p,the more favorable the synchronizability of the network,and Networks-D have the strongest synchronizability among the four types of two-layer toroidal networks with the same structural parameters.(2)The synchronizability of two types of multi-layer directed nearest-neighbor coupling networks is studied:nearest-neighbor coupling networks with directed star connections between multiple layers(Networks I)and nearest-neighbor coupling networks with directed chain connections between multiple layers(Networks Ⅱ).For each of these two types of multilayer networks,their super Laplace matrices and eigenvalue spectrum were rigorously derived,and the effect of changes in the structural parameters of the networks on the synchronizability of the networks was further analyzed.It is shown That the synchronizability of Network Ⅱ is not affected by changes in the number of layers.When the synchronization domain is unbounded,networks Ⅰ and networks Ⅱ have the same pattern with the change in the structural parameters of the network.When the synchronization domain is bounded,the synchronizability of Networks Ⅱ is better than that of networks Ⅰ with the same structural parameters.(3)The synchronizability of a specific type of double-layer self-similar networks is studied,which comprises three network models with distinct interlayer connections.Firstly,the eigenvalue spectrum of a double-layer self-similar network with one-to-one coupling of interlayer nodes is theoretically derived,and the impact of variation in network structural parameters on the network’s synchronization capacity is analyzed using MSF theory.Secondly,numerical simulation experiments are conducted to examine the effects of different interlayer connection methods on the synchronizability of the network.The findings indicate that the central node of the interlayer is more favorably connected to the synchronization ability of the network in one-sided coupling mode.In the mode of two cross-linked edges of the interlayer,selecting the node with a large degree to be cross-linked or connected to the central node is more conducive to the synchronization of the network. |
