Normalized Laplacian Spectra And Their Applications Of Weighted Level-3 And Level-4 Sierpiński Networks

The spectral analysis of complex networks has attracted increasing attention from scholars in various scientific fields,such as physics,chemistry,computer science,engineering,and economics,due to its wide-ranging applications.In recent years,the study of eigenvalues and eigenvectors of the normalized Laplacian matrix has gained particular interest as many properties of network structure and dynamics are closely related to them,especially in terms of random walks.It is noteworthy that current work mainly focuses on unweighted networks,and research on weighted networks is still in its nascent stage.However,the behavior of real networks is significantly different,which is evident not only in terms of node strength distribution but also in terms of edge weight distribution.In this paper,we focus on weighted networks and investigate the normalized Laplacian spectra and their applications of a class of iteratively constructed weighted level-3 and level-4 Sierpiński networks,where both node degree and edge weight distributions follow power-law distributions.Firstly,we introduced the research background and significance of the normalized Laplacian spectra of networks,reviewed the current research status at home and abroad,and proposed the main research content of this paper.Secondly,we presented some fundamental knowledge involved in the study.Then,using spectral decimation techniques and matrix theory analysis supported by symbolic and numerical calculations,we respectively obtained the recursive relationships between the normalized Laplacian spectra of weighted level-3 and level-4 Sierpiński networks after two consecutive iterations.This means that the normalized Laplacian spectra of each generation of networks can be iteratively obtained from the initial network’s normalized Laplacian spectra.As applications of the normalized Laplacian spectrum,we derived closed-form expressions for the Kemeny’s constant and the number of spanning trees of the two networks separately.By analyzing the Kemeny’s constant expression,we found that the Kemeny’s constant of these two networks increase with superlinear growth in network size,and this relationship is not affected by the weight factor.This is different from existing research results on the Kemeny’s constant of weighted graphs,which enriches research findings on the normalized Laplacian spectra of weighted networks.Moreover,we calculated the asymptotic complexity constants of the two networks with unit weight using the obtained closed expressions for the number of spanning trees,to verify the effectiveness of our results.Finally,we summarized the research work of this paper and proposed future research directions.The work presented in this paper advances the study of the normalized Laplacian spectral analysis of weighted Sierpiński networks,and develops new weighting schemes,research methods,and techniques,enriching related research achievements.These efforts provide a new perspective for ultimately solving the normalized Laplacian spectra and applications of general weighted Sierpiński networks,as well as spectral analysis of various weighted fractal networks.Additionally,our research also lays a theoretical foundation for the application of Sierpiński networks in various scientific fields.