Mean Hitting Time on Recursive Growth Tree Network

TL;DR

This paper introduces combinatorial Mapping Transformation techniques to exactly compute mean hitting times on recursive growth tree networks, offering a more convenient alternative to spectral methods and extending to various stochastic models.

Contribution

The paper develops a new combinatorial approach called Mapping Transformation for exact calculation of mean hitting times on recursive growth trees, surpassing spectral methods in simplicity.

Findings

Exact formulas for mean hitting times on recursive growth trees.

Extension of methods to stochastic models like BA-scale-free trees.

Closed-form solutions for Wiener index extensions.

Abstract

In this paper, we are concerned with mean hitting time $\langle\mathcal{H}\rangle$ for random walks on recursive growth tree networks that are built based on an arbitrary tree as the seed via implementing various primitive graphic operations, and propose a series of combinatorial techniques that are called Mapping Transformation to exactly determine the associated $\langle\mathcal{H}\rangle-$polynomial. Our formulas can be able to completely cover the previously published results in some well-studied and specific cases where a single edge or a star is often chose to serve as seed for creating recursive growth models. The techniques proposed are more convenient than the commonly-used spectral methods mainly because of getting around the operations of matrix inversion and multiplication. Accordingly, our results can be extended for both many other stochastic models including BA-scale-free tree and random uniform tree as well as graphs of great interest consisting of line graph of tree and Vicsek fractal network to derive numerical solutions of related structural parameters. And then, the closed-form solutions of two extensions of Wiener index with respect to multiplicative and additive degrees on an arbitrary tree are conveniently obtained as well. In addition, we discuss some extremal problems of random walks on tree networks and outline the related research directions in the next step.