Exact calculations of first-passage properties on the pseudofractal scale-free web

Abstract

In this paper, we consider discrete time random walks on the pseudofractal scale-free web (PSFW) and we study analytically the related first passage properties. First, we classify the nodes of the PSFW into different levels and propose a method to derive the generation function of the first passage probability from an arbitrary starting node to the absorbing domain, which is located at one or more nodes of low-level (i.e., nodes with large degree). Then, we calculate exactly the first passage probability, the survival probability, the mean and the variance of first passage time by using the generating functions as a tool. Finally, for some illustrative examples corresponding to given choices of starting node and absorbing domain, we derive exact and explicit results for such first passage properties. The method we propose can as well address the cases where the absorbing domain is located at one or more nodes of high-level on the PSFW, and it can also be used to calculate the first passage properties on other networks with self-similar structure, such as $(u,v)$ flowers and recursive scale-free trees.

arXiv comment12 pages, 4 figures

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• Scaling laws for diffusion on (trans)fractal scale-free networks
• Exact results for the first-passage properties in a class of fractal networks
• Multitarget search on complex networks: A logarithmic growth of global mean random cover time
• Analysis of fluctuations in the first return times of random walks on regular branched networks
• The normalized Laplacian spectrum of $n$-polygon graphs and its applications
• Constructions and properties of a class of random scale-free networks
• Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web