Research On Weighted-dependent Walks And Correlation Distances On Two Kinds Of Weighted Networks

The application of complex networks in real life is very extensive,involving science,information,medical,ecology,economy and other fields.Weighted networks constitute an indispensable branch in the study of complex networks,they are closer to the real system than unweighted networks.Therefore,considering the practical factors,the weighted networks have high research value.In this paper,two kinds of weighted networks are constructed and their structural properties are described.On the basis of the structural properties and weight factors of the networks,the related distance and weight-dependent walking problems on the networks are studied.In Chapter 1,the research background and current situation of complex networks,weighted networks and fractal theories are briefly introduced.Then,the related concepts of weighted average geodesic distance,average trapping time,eigentime identity and average weighted shortest path on weighted networks are given,and the basic framework of the full text is outlined.In Chapter 2,Vicsek fractal and Vicsek skeleton network with weighted nodes in threedimensional space are constructed,which are the natural generalization of Vicsek fractal and network.Then,we calculate the integral of geodesic distance with respect to self-similarity measure by using the self-similarity property of fractal,and obtain the weighted average geodesic distance on Vicsek fractal in three-dimensional space with n = 5 as an example,and then obtain the asymptotic formula of weighted average geodesic distance on skeleton network,where n represents the number of copies in each coordinate direction in three-dimensional space.Finally,we derive the general formula of weighted average geodesic distance on the model obtained when n is odd.The results show that when n is large enough,the weighted average geodesic distance of the networks tends to be a constant less than 1.This outcome provides a powerful tool for calculating the weighted average geodesic distance of the large self-similar weighted networks in three-dimensional space.In Chapter 3,taking the classical friendship graph as the initial graph,the weighted iterated friendship graphs are constructed by iterative method.According to the characteristics of network structure and the scaling law of weight,we deduce the sum of trapping time of all non-trap nodes in the weighted iterated friendship graphs,and then obtain the specific expression of average trapping time.Through theoretical analysis,it can be found that the size of the initial graph has no effect on the trapping efficiency,and in the network with the same scale,the value of average trapping time decrease with the decrease of the weight factor,and the corresponding trapping efficiency raises.Furthermore,we also find that the trapping process on weighted iterated friendship graphs is more effective than that on scale-free treelike networks.In Chapter 4,the similar matrix of transition probability matrix on weighted iterated friendship graphs is written by node classification method.By solving the characteristic polynomial of the similar matrix,the eigenvalues of the matrix are obtained,that is,all eigenvalues of the transition probability matrix are obtained.Using the eigenvalues of the transition probability matrix,we derive the normalized Laplacian spectrum and obtain the eigentime identity of the weighted iterated friendship graphs.It can be seen from the expression that when the network scale is large enough,the eigentime identity is only related to the weight factor.In Chapter 5,the parameters of the control weighted iterated friendship graphs are changed and the self-similar weighted iterated friendship graphs are obtained.Using the self-similar structure of the model,we derive the sum of all the weighted shortest paths in the self-similar weighted iterated friendship graphs whose endpoints are not in the same branch,and then obtain the exact expression of the average weighted shortest path.The results show that when the weight factor is between 0and 1,the average weighted shortest path is a bounded function of the weight factor,and its value increases with the increase of the weight factor.When the weight factor is 1,the average weighted shortest path has a linear relationship with the order of the networks,which is independent of the weight factor.