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

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 Gm(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 Gm(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 Gm(t). Since the spectrum of the Laplacian matrix of the network model Gm(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.

Spectral analysis for weighted extended Vicsek polygons

“…Spectral analysis of graphs has become increasingly popular with researchers from various scientific fields owing to its extensive applications in physics, communication, transportation, chemistry, biology, and so on. Over the past decades, there has been particular interest in the study of the eigenvalues and eigenvectors of the normalized Laplacian matrix, since it can unveil structural properties and the influences of various topological structures on the dynamic processes of a graph [1][2][3][4][5][6]. The normalized Laplacian matrix, which is closely related to the structure and dynamic process of the graph, was initiated by Chung [7].…”

“…And a variety of network models based on the fractal network model have been extensively studied [29][30][31][32]. Furthermore, the fractal cactus model and its variations are also important fractal network models that have been studied extensively in recent years and can be used to study the topology and dynamic properties of molecular polymer networks [5,33,34].…”

Topology design for leader-follower coherence in noisy asymmetric networks

“…Existing studies have demonstrated that calculations of the Laplacian eigenvalues face an enormous challenge because the eigenvalues are dominated by the network topology [9][10][11]. It is of interest to explore diverse methods to reveal the correlation between the topology and the coherence [12,13]. Particularly, for some fractal networks [14,15], such as Koch networks [16], Vicsek fractals [1], Sierpiński graphs [17], are all good candidate network models to derive an exact scaling of network coherence regarding the network size.…”