Mean first-passage time for random walks on the T-graph

TL;DR

This paper derives an exact formula for the mean first-passage time of random walks on the T-graph, revealing how it scales with network size and linking it to resistance distance.

Contribution

It provides a novel exact expression for MFPT on the T-graph using its self-similar structure, connecting random walks to electrical network theory.

Findings

MFPT scales approximately as a power-law with network size

Derived an exact formula confirmed by numerical calculations

Linked MFPT to resistance distance in electronic networks

Abstract

For random walks on networks (graphs), it is a theoretical challenge to explicitly determine the mean first-passage time (MFPT) between two nodes averaged over all pairs. In this paper, we study the MFPT of random walks in the famous T-graph, linking this important quantity to the resistance distance in electronic networks. We obtain an exact formula for the MFPT that is confirmed by extensive numerical calculations. The interesting quantity is derived through the recurrence relations resulting from the self-similar structure of the T-graph. The obtained closed-form expression exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent lying between 1 and 2. Our research may shed light on the deeper understanding of random walks on the T-graph.