Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Binary Galton-Watson trees with mutations

Qiao Huang, Nicolas Privault

TL;DR

This work analyzes a multitype Galton–Watson process with mutation and reversion on marked binary trees, in both discrete and continuous time. By conditioning on the total progeny and developing recursive descriptions, the authors obtain exact expressions for the joint distribution and moments of type counts, as well as generating-function characterizations tied to Fuss–Catalan numbers. The approach yields explicit formulas for mean type proportions, first-occurrence times, and integrability of product functionals, without relying on approximations. The results are validated with simulations and equipped with computational tools, providing precise insight into mutation dynamics in branching structures with potential applications in population genetics and evolutionary biology.

Abstract

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and distributions of different types. This allows us in particular to estimate the proportions of different types in the long run, as well as the distribution of the first time of occurrence of a given type as the tree size or time increases. Our approach relies on the recursive computation of the joint distribution of types conditionally to the value of the total progeny. In comparison with the literature on related multitype models, we do not rely on approximations.

Binary Galton-Watson trees with mutations

TL;DR

This work analyzes a multitype Galton–Watson process with mutation and reversion on marked binary trees, in both discrete and continuous time. By conditioning on the total progeny and developing recursive descriptions, the authors obtain exact expressions for the joint distribution and moments of type counts, as well as generating-function characterizations tied to Fuss–Catalan numbers. The approach yields explicit formulas for mean type proportions, first-occurrence times, and integrability of product functionals, without relying on approximations. The results are validated with simulations and equipped with computational tools, providing precise insight into mutation dynamics in branching structures with potential applications in population genetics and evolutionary biology.

Abstract

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and distributions of different types. This allows us in particular to estimate the proportions of different types in the long run, as well as the distribution of the first time of occurrence of a given type as the tree size or time increases. Our approach relies on the recursive computation of the joint distribution of types conditionally to the value of the total progeny. In comparison with the literature on related multitype models, we do not rely on approximations.
Paper Structure (12 sections, 14 theorems, 129 equations, 9 figures)

This paper contains 12 sections, 14 theorems, 129 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Discrete-time setting
  3. Marked Galton--Watson process
  4. Conditional type distribution
  5. Generating functions
  6. Continuous-time setting
  7. Marked binary branching process
  8. Conditional type distribution
  9. Generating functions
  10. Proofs - discrete-time setting
  11. Proofs - continuous-time setting
  12. Acknowledgement.

Key Result

Proposition 2.1

The distribution of the count $S^{\scaleto{\neq 0}{6pt}}_\infty$ of nodes with non-zero types is given by with the probability generating function and we have $\mathbb{P}(S^{\scaleto{\neq 0}{6pt}}_\infty < \infty ) = 1$ if $p\leq 1/2$.

Figures (9)

  • Figure 1: Marked random tree sample started from the initial type $j=3$.
  • Figure 2: Conditional average type proportions \ref{['fjkld3-1']} given the values of $S^{\scaleto{\neq 0}{6pt}}_\infty$ in abscissa.
  • Figure 3: Average type proportions \ref{['fjkld3-2']} as functions of $p\in [0,1/2)$.
  • Figure 4: Limiting distributions \ref{['jkld2']} for $p=1/2$.
  • Figure 5: Sample of the marked random tree ${\cal T}_t$, $t >0$, started from the initial type $j=3$.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Proposition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Proposition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • ...and 4 more