Random Walk Problems On Two Classes Of Weighted Self-similar Networks

This paper takes two classes of weighted self-similar networks(weighted iterative hexagonal networks and weighted hierarchical triangle networks with primary node iteration)as study objects,and finds out the Laplacian spectrum,the average trapping time and the average weighted shortest path respectively.First,this paper constructs a class of weighted iterative hexagonal networks and considers the spectrum of the networks.Based on the self-similar structure,the recursive relationship of eigenvalues and eigenvectors of the standard adjacency matrix between two adjacent generations is explored and the standard Laplacian spectrum is further obtained.Second,this paper also constructs a class of weighted hierarchical triangle networks with primary node iteration,and considers the average trapping time(ATT)and average weighted shortest path(AWSP)on the networks.According to the positions of the primary nodes,the networks are divided.Then they are further obtained.On the one hand,the ATT is solved by the average first passage time of the primary nodes.On the other hand,the AWSP is solved by discussing the shortest paths between same or different modules categorically,where the paths of different modules are related to the primary nodes.The results show that both the network size and weights affect the efficiency of random walk,and the ATT grows linearly or sublinearly with the network size under different weights: when r >1 / 2,it grows linearly;when 0 <r ≤1/ 2,it grows sublinearly.In addition,the ATT of the iterative network with primary node iteration is smaller compared with the original iterative network,i.e.,the efficiency of random walk is higher in the primary node iterative way.As the weight increases,the difference of ATT is larger in two iterative ways,while the difference is not significant when the weight is small.Meanwhile,the weights also have an important impact on the AWSP,and it is larger as the weight increases.However,the effect of network size on AWSP is very different with different weights: when r =1,it grows sublinearly;when 0 <r <1,it is independent of the network size.Therefore,the networks have obvious small-world characteristics.