Topological Properties, Spectra Analysis, and Consensus Problems for a Class of Network Models Based on m-Fission Operation

The network model, especially the network model constructed by network operation, is an important tool to study complex networks. Many topological and dynamic properties of complex networks can be studied in this way. In this article, the
$m$
-fission operation is constructed based on the phenomenon of node splitting in the network, which is quite common in complex networks. Many network models, including dual Sierpinski gaskets, are built based on this operation, but it has never been systematically studied. Then, the topological and dynamic properties of the
$m$
-fission operation and the corresponding iterative fission network model are studied, and the influence of the operation on the network structure is revealed. Among them, the topological properties of the network include diameter, degree distribution, clustering coefficient, average distance, and modularity. By studying these properties, it can be concluded that the iterative fission network is a fractal homogeneous network with high clustering and high community characteristics. Since the dynamic properties are closely related to the spectrum of the Laplacian matrix corresponding to the network, the iterative relation of the spectrum in operation is studied, and the complete solution of the spectrum of the iterative fission network is obtained. Based on the above results, we calculate the analytical expressions of the characteristic quantities related to the dynamic properties on the network, including the Kirchhoff index and the average of hitting times. Finally, due to the close connection between the network model and the system, we further analyzed the consensus problems on the system corresponding to the network, including convergence rate, delay robustness, first-order noise coherence, and second-order noise coherence.