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The Horton-Strahler number of Galton-Watson trees with possibly infinite variance

Robin Khanfir

TL;DR

This work characterizes the Horton–Strahler number $\mathcal{S}(\tau)$ of critical Galton–Watson trees conditioned to be large, when the offspring law lies in the domain of attraction of an $\alpha$-stable law $(\alpha\in[1,2])$. For $\alpha\in(1,2]$, they derive exponential tail decay $-\ln \mathbb{P}(\mathcal{S}(\tau)>n)\sim n\ln\frac{\alpha}{\alpha-1}$ and prove that, under size conditioning, $\mathcal{S}(\tau)$ grows as $\frac{1}{\alpha}\log_{\alpha/(\alpha-1)} n$ in probability; this extends finite-variance results to infinite variance and includes the $\alpha=2$ case. In the $\alpha=1$ regime, the spectrally positive Cauchy case, the behavior is more delicate and is described through the slowly varying function $\Upsilon$ evaluated at the appropriate scaling sequence $1/b_n$, yielding $\mathcal{S}(\tau)/\Upsilon(1/b_n)\to 1$ under various conditioning. The proofs combine probabilistic methods linking $\mathcal{S}$ to the height and to maximal out-degree, with limit theorems for $\alpha$-stable trees (Duquesne) and uniform absolute continuity results for positive excursions (Kortchemski), together with a recursive equation for the distribution of $\mathcal{S}(\tau)$ and a detailed analysis of joint tail laws. These results sharpen our understanding of how branching complexity scales in heavy-tailed Galton–Watson trees and pave the way for studying fluctuations and extensions to more general tree models.

Abstract

The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size $n$ and whose offspring distribution is in the domain of attraction of an $α$-stable law with $α\in [1, 2]$. We give tail estimates and when $α\neq 1$, we prove that it grows as $\frac{1}α\log_{α/(α-1)} n$ in probability. This extends the result in Brandenberger, Devroye \& Reddad [6] dealing with the finite variance case for which $α=2$. We also characterize the cases where $α=1$, namely the spectrally positive Cauchy regime, which exhibits more complex behaviors. Our proofs are new and probabilistic; they relate the Horton-Strahler number with other shape parameters such as the height or largest degree.

The Horton-Strahler number of Galton-Watson trees with possibly infinite variance

TL;DR

This work characterizes the Horton–Strahler number of critical Galton–Watson trees conditioned to be large, when the offspring law lies in the domain of attraction of an -stable law . For , they derive exponential tail decay and prove that, under size conditioning, grows as in probability; this extends finite-variance results to infinite variance and includes the case. In the regime, the spectrally positive Cauchy case, the behavior is more delicate and is described through the slowly varying function evaluated at the appropriate scaling sequence , yielding under various conditioning. The proofs combine probabilistic methods linking to the height and to maximal out-degree, with limit theorems for -stable trees (Duquesne) and uniform absolute continuity results for positive excursions (Kortchemski), together with a recursive equation for the distribution of and a detailed analysis of joint tail laws. These results sharpen our understanding of how branching complexity scales in heavy-tailed Galton–Watson trees and pave the way for studying fluctuations and extensions to more general tree models.

Abstract

The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size and whose offspring distribution is in the domain of attraction of an -stable law with . We give tail estimates and when , we prove that it grows as in probability. This extends the result in Brandenberger, Devroye \& Reddad [6] dealing with the finite variance case for which . We also characterize the cases where , namely the spectrally positive Cauchy regime, which exhibits more complex behaviors. Our proofs are new and probabilistic; they relate the Horton-Strahler number with other shape parameters such as the height or largest degree.
Paper Structure (17 sections, 27 theorems, 138 equations, 1 figure)

This paper contains 17 sections, 27 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Framework and tools
  3. Galton--Watson trees
  4. Limit theorems of the literature
  5. Distribution of the Horton--Strahler number of Galton--Watson trees
  6. Alternative definitions of the Horton--Strahler number and basic properties
  7. Recursive equation for the distribution
  8. Tail estimates of joint laws
  9. Asymptotics of the Horton--Strahler number: the cases where $\alpha\in(1,2]$
  10. Proof of Proposition \ref{['HS_tails_alpha']}
  11. Proof of Theorem \ref{['stable_HS_order']}
  12. Asymptotics of the Horton--Strahler number: the $1$-stable cases
  13. Proof of Proposition \ref{['HS_tails_1']}
  14. Proof of Theorem \ref{['cauchy_HS_order_atleast']}
  15. Proof of Theorem \ref{['cauchy_HS_order_equal']}
  16. ...and 2 more sections

Key Result

Proposition 1.2

Assume that $\mu$ is critical and non-trivial and that it belongs to the domain of attraction of a stable law of index $\alpha\in (1,2]$. Then,

Figures (1)

  • Figure 1: An example of a tree $t$ on the left, and its mirror image $t^\star$ on the right. Two vertices $u,v$ of $t$ are depicted, as well as their mirror images $u^\star,v^\star$ on $t^\star$.

Theorems & Definitions (65)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.3
  • ...and 55 more