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

## TL;DR

This paper analytically derives exact first passage properties for random walks on the pseudofractal scale-free web, providing explicit formulas and a versatile method applicable to various nodes and other self-similar networks.

## Contribution

It introduces a novel analytical approach to compute first passage probabilities and times on the PSFW, including for nodes of different levels and for various absorbing domains.

## Key findings

- Exact first passage probability and mean first passage time formulas derived.
- Method applicable to high-level and low-level absorbing nodes.
- Results extend to other self-similar networks like $(u, v)$ flowers.

## 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.