Study On Spectra And Biased Walk Of Two Kinds Of Weighted Iterative Networks

This paper mainly studies the spectra and biased walk of single-weighted claw network as well as the biased walk on double-weighted cherry network and derive the expression of the average weighted receiving time on single-weighted network and double-weighted network.The full paper is mainly composed of three parts,which are the biased walk on weighted claw network,spetra and its application of weighted claw network and the biased walk on weighted cherry network.Chapter 1 mainly introduces the research background and current situation of complex network and weighted network,gives the development status of the research content of this paper,and lists the main work of this paper.Chapter 2 gives the related concepts and definitions that need to be used in the following research.Chapter 3 studies the topological properties and the biased walk of the weighted claw netwrok.A new method is proposed to derive the exact solution of the average weighted shortest path.The sublinear relationship between the average weighted shortest path and the network size is analyzed by numerical simulation.Similarly,we study the biased walk problem on weighted claw network.The expression of average weighted receiving time is derived by block method,and the power-law growth relationship between the average weighted receiving time and the network size is analyzed.In chapter 4,the normalized Laplacian spectrum of weighted claw network and its application are studied.According to the structure characteristics of the network and the definitions of eigenvalue and eigenvector,we deduce the recursive relationship of two successive generation eigenvalues on the weighted claw network and and then determine the spectra of the network.Using the spectra of the standardized Laplacian matrix and Vieta's formulas,the expressions of the eigentime identity,the degree-Kirchhoff index and the Kemeny's constant are calculated and the growth relationship between them and the network size are found.In chapter 5,we study the basied walk on double-weighted cherry network.Two different weighted factors are selected to construct a double-weighted cherry network,and the exact expressions of average weighted first-passing time and the average weighted receiving time are derived by the definitions of weighted time and weighte-dependent walk.The results show that the average weighted receiving time tends to be a constant with the increase of network size;and when the restriction condition is given,there is a power-law relationship between the average weighted receiving time and network size.Consequently,the average weighted receiving time on double-weighted network is more consistent with the actual situation by analyzing the results of the wight-dependent walk on single-weighted network and double-weighted network.In other words,it is more meaningful to study double-weighted network than single-weighted network.