Abstract
In this study, we studied the eigenvalue spectrum and synchronizability of two types of double-layer hybrid directionally coupled star-ring networks, namely, the double-layer star-ring networks with the leaf node pointing to the hub node (Network I) and the double-layer star-ring networks with the hub node pointing to the leaf node (Network II). We strictly derived the eigenvalue spectrum of the supra-Laplacian matrix of these two types of networks and analyzed the relationship between the synchronizability and the structural parameters of networks based on the master stability function theory. Furthermore, the correctness of the theoretical results was verified through numerical simulations, and the optimum structural parameters were obtained to achieve the optimal synchronizability.
Highlights
Complex networks are common in daily life, such as social networks [1], urban transportation networks, and virus spread networks
For Network I, in the case that the synchronous region is unbounded, no matter whether the number of subnetwork nodes is odd or even, the synchronizability is enhanced with the increase of the intralayer coupling strength from the leaf nodes to the hub nodes and the interlayer coupling strength between the hub nodes
In the case that the synchronous region is bounded, when the intralayer coupling strength from the leaf nodes to the hub nodes is weaker, the synchronizability strengthens with the increase of the intralayer coupling strength from the leaf nodes to the hub nodes but weakens with the increase of the interlayer coupling strength between the leaf nodes
Summary
Complex networks are common in daily life, such as social networks [1], urban transportation networks, and virus spread networks. In [32], Deng et al researched the relationship between the eigenvalue spectrum, structure parameters, and synchronizability of multilayer star-ring networks with no weight and no direction, especially analyzing the change of synchronizability under different interlayer coupling strengths. For the eigenvalues of the multilayer hybrid directional coupling network, it is difficult to strictly derive the theoretical solution, mostly directly through the numerical simulation to explore the synchronization, and little previous research was conducted on the multilayer hybrid directionally coupling network synchronizability; a better understanding of the two layers of hybrid directionally coupling network to the topology of the synchronizability is needed. We studied the double-layer hybrid directionally coupling star-ring networks from leaf nodes to hub nodes and from hub nodes to leaf nodes, respectively.
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