Some Studies Of Weighted Networks With Biased Walks

In this paper,we mainly study the spectrum properties and its related applications of weighted iterated triangulations of graphs and weighted tree-like fractals,meanwhile,the multifractal analysis of one-dimensional biased walk is also concerned.This paper is divided into four chapters.In Chapter 1,the development processes and the basic perceptions of complex networks and weighted networks are displayed on the whole.Furthermore,we simply present the basic problems of biased walks on the weighted networks.In Chapter 2,a kind of weighted iterated triangulations of graphs,whose neighbor vertexes are still neighbors,is introduced.In this chapter,an important performance index,the normalized Laplacian spectrum,is studied,through years of our painstaking efforts,we have at last deduced the specific expressions for the every generation’s spectrum of the weighted iterated triangulations of graphs and they are applied to determine the graphs’ multiplicative Kirchhoff index,Kemeny’s constant and the number of weighted spanning trees,then it shows that some relevant invariants of these graphs are related to the number of iteration generation,the weight factor and the property of initial graph.In Chapter 3,we focus on a kind of specific weighted tree-like fractals whose initial graph is defined.Unlike the weighted iterated triangulations of graphs above,this kind of specific weighted tree-like fractals have no loops,and because of its different way of iterations,the neighbor vertexes of these graphs are no longer neighbors.In this chapter,we firstly study the the spectral relationship of every two successive generations of the graphs,then every generation’s spectrum of the weighted tree-like fractals is obtained,they can contribute to the properties of other relevant invariants related to its structure.The study reveals that these relevant invariants corresponding to its structure are all influenced only by the number of iteration generation and the weight factor.In Chapter 4,for researching more comprehensive properties of weighted networks,we firstly study the problems of biased walks on the weighted networksto have its property of fractal dimensions.As the first exploration of this work,we put emphasis on making the multifractal analysis of one-dimensional biased walks,as a result,we obtain some relevant fractal dimensions of the level set with one-dimensional biased walks.