Synchronizability and eigenvalues of multilayer star networks through unidirectionally coupling

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Previous studies on multilayer networks have found that properties of multilayer networks show great differences from those of the traditional complex networks. In this paper, we derive strictly the spectra of the Supra-Laplace matrix of three-layer star networks and multilayer star networks through unidirectionally coupling by using the master stability method to analyze the synchronizability of these two networks. Through mathematical analyses of the eigenvalues of the Supra-Laplace matrix, we explore how the node number, the intra-layer coupling strength the inter-layer coupling strength, and the layer number influence the synchronizability of multilayer star networks through unidirectionally coupling in two different ways. In particular, we focus on the layer number and the inter-layer coupling strength between the hub nodes, and then we conclude that the synchronizability of networks is greatly affected by the layer number. We find that when the synchronous region is unbounded, the synchronizability of the two different coupling multilayer star networks is related to not only the intra-layer coupling strength or the inter-layer coupling strength between the leaf nodes of the entire network, but also the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the intra-layer coupling strength is weak, the synchronizability of the two different coupling multilayer star networks is different with the changing of the intra-layer coupling strength and the inter-layer coupling strength between the leaf nodes and the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the inter-layer coupling strength between the hub nodes is weak, the two different coupling multilayer star networks are consistent with the changing of the intra-layer coupling strength and the layer number while different from the inter-layer coupling strength between the leaf nodes and the inter-layer coupling strength between the hub nodes. We find that the node number has no effect on the synchronizability of multilayer star networks through coupling from the hub node to the leaf node. The synchronizability of the network is directly proportional to the layer number, while inversely proportional to the inter-layer coupling strength between the hub nodes. Finally, the effects of the coupling strength, the layer number and the node number on the synchronizability of the two different coupling star networks can be extended from three-layer network to multilayer networks.