Characterization of random walks on space of unordered trees using efficient metric simulation

TL;DR

This paper investigates the behavior of random walks on the space of unordered trees using an efficient algorithm to estimate escape rates, revealing structural insights through numerical simulations.

Contribution

It introduces a novel, efficient algorithm for computing the edit distance on trees, enabling precise estimation of the random walk's escape rate in a complex combinatorial space.

Findings

The algorithm is approximately 100 times faster than Zhang's method on average-sized trees.

The escape rate of the random walk is accurately estimated through extensive simulations.

The study demonstrates how random walk behavior reflects the structure of the space of trees.

Abstract

The simple random walk on $\mathbb{Z}^p$ shows two drastically different behaviours depending on the value of $p$: it is recurrent when $p\in\{1,2\}$ while it escapes (with a rate increasing with $p$) as soon as $p\geq3$. This classical example illustrates that the asymptotic properties of a random walk provides some information on the structure of its state space. This paper aims to explore analogous questions on space made up of combinatorial objects with no algebraic structure. We take as a model for this problem the space of unordered unlabeled rooted trees endowed with Zhang edit distance. To this end, it defines the canonical unbiased random walk on the space of trees and provides an efficient algorithm to evaluate its escape rate. Compared to Zhang algorithm, it is incremental and computes the edit distance along the random walk approximately 100 times faster on trees of size $500$ on average. The escape rate of the random walk on trees is precisely estimated using intensive numerical simulations, out of reasonable reach without the incremental algorithm.