Exact solution for mean first-passage time on a pseudofractal scale-free web

TL;DR

This paper derives an exact analytical solution for the mean first-passage time of random walks on a pseudofractal scale-free web, revealing insights into transport efficiency in complex networks.

Contribution

It provides the first exact solution for MFPT on PSFW, a model with real-world network properties, using recurrence relations based on the network's structure.

Findings

MFPT scales as a power-law with the number of nodes, exponent less than 1.

The PSFW structure enhances transport efficiency compared to regular lattices and fractals.

The analytical approach can be extended to other deterministic networks.

Abstract

The explicit determinations of the mean first-passage time (MFPT) for trapping problem are limited to some simple structure, e.g., regular lattices and regular geometrical fractals, and determining MFPT for random walks on other media, especially complex real networks, is a theoretical challenge. In this paper, we investigate a simple random walk on the the pseudofractal scale-free web (PSFW) with a perfect trap located at a node with the highest degree, which simultaneously exhibits the remarkable scale-free and small-world properties observed in real networks. We obtain the exact solution for the MFPT that is calculated through the recurrence relations derived from the structure of PSFW. The rigorous solution exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We confirm the closed-form solution by direct numerical calculations. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure, such as regular lattices, Sierpinski fractals, and T-graph. The analytical method can be applied to other deterministic networks, making the accurate computation of MFPT possible.