The Structural Properties Of Complex Networks With Random Walk

Complex networks have seen much interest from all research circles and have found many potential applications in a variety of fields including engineering technology, society, politics, communications, medicine, neural networks, economics and management. Yet despite the importance and pervasiveness of networks, scientists have had little understanding of their structure and dynamics properties. This paper focuses on the aspects of topological characteristics and dynamics of some particular and important networks in real world, such as expansion of uniform recusive tree, small-world networks and scale-free networks.On the one hand, from the perspective of complex networks, we investigate analytically the main structural characteristics of networks, and obtain the accurate solutions for these properties, which include degree distribution, average path length, distribution of node betweenness and degree correlations. Then we determine the complete eigenvalues of the Laplacian matrix for networks. These interesting quantities can be determined through the recurrence relations derived from the structure of the networks. Beginning from the rigorous relations one can obtain the complete spectra for the networks of arbitrary sizes. Compared with traditional methods, our recursive algorithm can greatly reduce the computational time complexity and storage space requirements, and is applicable to solve the spectral properties of large-scale networks.On the other hand, according to the known connections between mean first passage time (MFPT), effective resistance, and the eigenvalues of graph Laplacian, we study analytically the MFPT between all node pairs of networks. The interesting quantity is determined exactly through the recursive relation of the Laplacian spectra obtained from the special construction of networks. And we also determined exactly the mean trapping time (MTT) that is calculated through the recurrence relations derived from the structure of networks. The rigorous solution of Pseudofractal Scale-free Web(PSFW) exhibits that the MTT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure. It is of great theoretical significance to design a network with high transmission efficiency.