On stochastic generation of ultrametrics in high-dimension Euclidean spaces

TL;DR

This paper proves that Euclidean distances between random points in high-dimensional spaces converge to ultrametrics and introduces a probabilistic algorithm for generating ultrametric structures with various topologies.

Contribution

It provides a new proof of ultrametric convergence in high dimensions and presents an algorithm for generating ultrametrics in Euclidean spaces.

Findings

Euclidean distances converge to ultrametrics as dimension increases

The ultrametric distance matrix is determined by coordinate variances

The algorithm successfully generates ultrametric structures in high dimensions

Abstract

The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the ultrametric is presented. The values of the ultrametric distance matrix is completely determined by variances of point coordinates. Probabilistic algorithm for the generation of finite ultrametric structures of any topology in high-dimensional Euclidean space is presented. The validity of the algorithm is demonstrated by explicit calculations of distance matrices with fixed dimensions and ultrametricity indexes for various dimensions of Euclidean space.