Synchronizability and eigenvalues of two-layer star networks

From the study of multilayer networks, scientists have found that the properties of the multilayer networks show great difference from those of the traditional complex networks. In this paper, we derive strictly the spectrum of the super-Laplacian matrix and the synchronizability of two-layer star networks by applying the master stabi- lity method. Through mathematical analysis of the eigenvalues of the super-Laplacian matrix, we study how the node number, the inter-layer and the intra-layer coupling strengths influence the synchronizability of a two-layer star net-work. We find that when the synchronous region is unbounded, the synchronizability of a two-layer star network is only related to the intra-layer coupling strength between the leaf nodes or the inter-layer coupling strength of the entire network. If the synchronous region of a two-layer star network is bounded, not only the inter-layer coupling strength of the network and the intra-layer coupling strength between the leaf nodes, but also the intra-layer coupling strength between the hub nodes and the network size have influence on the synchronizability of the networks. Provided that the same inter-layer and intra-layer coupling strengths are concerned, we would further discuss the opti-mal ways of strengthening the synchronizability of a two-layer star network. If the inter-layer and intra-layer coupling strengths are far less than unity, changing the intra-layer coupling strength is the best way to enhance the synchronizability no matter what the synchronous region is. While if the coupling strengths are the same as, less than or more than unity, there will be different scenarios for the network with bounded and unbounded synchronous regions. Besides, we also discuss the synchronizability of the multilayer network with more than two layers. And then, we carry out numerical simulations and theoretical analysis of the two-layer BA scale-free networks coupled with 200 nodes and obtain very similar conclusions to that of the two-layer star networks. Finally, conclusion and discussion are given to summarize the main results and our future research interests.