Random walks on stochastic uniform growth trees: Analytical formula for mean first-passage time

TL;DR

This paper introduces an analytical method to compute mean first-passage time for random walks on stochastic uniform growth trees, linking it to the Wiener index, and offers a more manageable alternative to spectral techniques.

Contribution

The paper develops a general formula connecting Wiener index and mean first-passage time on trees, extending it to stochastic growth models, and simplifies calculations compared to spectral methods.

Findings

Derived a formula relating Wiener index and mean first-passage time.

Extended deterministic tree results to stochastic growth trees.

Provided a more manageable approach than spectral techniques.

Abstract

As known, the commonly-utilized ways to determine mean first-passage time $\overline{\mathcal{F}}$ for random walk on networks are mainly based on Laplacian spectra. However, methods of this type can become prohibitively complicated and even fail to work when the Laplacian matrix of network under consideration is difficult to describe in the first place. In this paper, we propose an effective approach to determining quantity $\overline{\mathcal{F}}$ on some widely-studied tree networks. To this end, we first build up a general formula between Wiener index $\mathcal{W}$ and $\overline{\mathcal{F}}$ on a tree. This enables us to convert issues to answer into calculation of $\mathcal{W}$ on networks in question. As opposed to most of previous work focusing on deterministic growth trees, our goal is to consider stochastic case. Towards this end, we establish a principled framework where randomness is introduced into the process of growing trees. As an immediate consequence, the previously published results upon deterministic cases are thoroughly covered by formulas established in this paper. Additionally, it is also straightforward to obtain Kirchhoff index on our tree networks using the proposed approach. Most importantly, our approach is more manageable than many other methods including spectral technique in situations considered herein.