A Method for Geodesic Distance on Subdivision of Trees with Arbitrary Orders and Their Applications

TL;DR

This paper introduces a novel method to analytically compute exact geodesic distances on recursively constructed trees, providing closed-form solutions that are easier to implement than existing spectral techniques, and also derives mean first-passage times for random walks.

Contribution

The paper presents a new, general approach for deriving closed-form solutions for geodesic distances on growth trees created by subdivision and star-fractal operations, surpassing previous methods in simplicity and applicability.

Findings

Closed-form solutions for geodesic distances on specific growth trees.

Analytical expressions for mean first-passage time in these trees.

Distinct effects of topological operations on random walk metrics.

Abstract

Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree $\mathcal{T}$ using two types of well-known operations, first-order subdivision and ($1,m$)-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth trees using a single edge as seed in the literature, we propose a novel method for deriving closed-form solutions on the presented trees completely. Meanwhile, our technique is more general and convenient to implement compared to those previous methods mainly because there are not complicated calculations needed. In addition, the closed-form expression of mean first-passage time ($MFPT$) for random walk on each member in tree families is also readily obtained according to connection of our obtained results to effective resistance of corresponding electric networks. The results suggest that the two topological operations above are sharply different from each other due to $MFPT$ for random walks, and, however, have likely to show the similar performance, at least, on geodesic distance.