Tokunaga and Horton self-similarity for level set trees of Markov chains

TL;DR

This paper demonstrates that level set trees derived from finite symmetric homogeneous Markov chains exhibit Horton and Tokunaga self-similarity, extending these concepts to infinite trees and proposing conjectures for fractional Brownian motions.

Contribution

It establishes Horton and Tokunaga self-similarity in trees from Markov chains and extends these concepts to infinite trees, including Brownian motion.

Findings

Horton and Tokunaga self-similarity proven for Markov chain trees

Self-similarity extended to infinite trees of Brownian motion

Conjecture on self-similarity of fractional Brownian motions

Abstract

The Horton and Tokunaga branching laws provide a convenient framework for studying self-similarity in random trees. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for elements of a branching system. The stronger Tokunaga self-similarity addresses so-called side branching. The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems, and proven for two paradigmatic models: the critical Galton-Watson branching process with finite progeny and the finite-tree representation of a regular Brownian excursion. This study establishes the Tokunaga and Horton self-similarity for a tree representation of a finite symmetric homogeneous Markov chain. We also extend the concept of Horton and Tokunaga self-similarity to infinite trees and establish self-similarity for an infinite-tree representation of a regular Brownian motion. We conjecture that fractional Brownian motions are also Tokunaga and Horton self-similar, with self-similarity parameters depending on the Hurst exponent.