两层星形网络的特征值谱及同步能力

多层网络是当今网络科学研究的一个前沿方向.本文深入研究了两层星形网络的特征值谱及其同步能力的问题.通过严格导出的两层星形网络特征值的解析表达式,分析了网络的同步能力与节点数、层间耦合强度和层内耦合强度的关系.当同步域无界时,网络的同步能力只与叶子节点之间的层间耦合强度和网络的层内耦合强度有关;当叶子节点之间的层间耦合强度比较弱时,同步能力仅依赖于叶子节点之间的层间耦合强度;而当层内耦合强度比较弱时,同步能力依赖于层内耦合强度;当同步域有界时,节点数、层间耦合强度和层内耦合强度对网络的同步能力都有影响.当叶子节点之间的层间耦合强度比较弱时,增大叶子节点之间的层间耦合强度会增强网络的同步能力,而节点数、中心节点之间的层间耦合强度和层内耦合强度的增大反而会减弱网络的同步能力;而当层内耦合强度比较弱时,增大层内耦合强度会增强网络的同步能力,而节点数、层间耦合强度的增大会减弱网络的同步能力.进一步,在层间和层内耦合强度都相同的基础上,讨论了如何改变耦合强度更有利于同步.最后,对两层BA无标度网络进行数值仿真,得到了与两层星形网络非常类似的结论.

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.