Spectral analysis and consensus problems for a class of fractal network models

Abstract

Abstract Because of the application of fractal networks and their spectral properties in various fields of science and engineering, they have become a hot topic in network science. In this paper, a class of fractal network models G m ( t ) based on the fractal cactus model is studied whose structure is controlled by the positive integer coefficient m and the number of iterations t . After analyzing the structure and construction of network model G m ( t ), the iterative relation of spectrum of Laplacian matrix corresponding to the network is obtained by using the decimation procedure. Then, due to the important role of consensus problems in network and system science, we use the spectrum of Laplacian matrix to analysis the convergence speed, delay robustness and noise robustness of the system built according to the network model G m ( t ). Since the spectrum of the Laplacian matrix of the network model G m ( t ) is completely determined by the number of iterations t and the coefficient m , and the convergence speed, delay robustness and noise robustness of the system are determined by the second smallest eigenvalue, the largest eigenvalue and all non-zero eigenvalues, respectively, we can finally obtain the analytical expressions and approximate expressions between the variables t , m and the above three characteristic quantity. The above results not only help to analyze the topology and dynamic properties of the network model, but also can accurately regulate the performance of the system, so it has potential application prospects.