Synchronizability of Multilayer Directed Dutch Windmill Networks

This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered.