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Takeover, fixation and identifiability in finite neutral genealogy models

Eric Foxall, Jen Labossiere

TL;DR

By introducing the dual notions of forward and backward neutrality, this work gives a more intuitive implementation of the lookdown rearrangement of particles, and shows that the lookdown arranges subtrees in size-biased order of the number of their descendants.

Abstract

For neutral genealogy models in a finite, possibly non-constant population, there is a convenient ordered rearrangement of the particles, known as the lookdown representation, that greatly simplifies the analysis of the family trees. By introducing the dual notions of forward and backward neutrality, we give a more intuitive implementation of this rearrangement. We also show that the lookdown arranges subtrees in size-biased order of the number of their descendants, a property that is familiar in other settings but appears not to have been previously established in this context. In addition, we use the lookdown to study three properties of finite neutral models, as a function of the sequence of unlabelled litter sizes of the model: uniqueness of the infinite path (fixation), existence of a single lineage to which almost all individuals can trace their ancestry (takeover) and whether or not we can infer the lookdown rearrangement by examining the unlabelled genealogy model (identifiability). Identifiability of the spine path in size-biased Galton-Watson trees was previously studied, so we also discuss connections to those results, by relating the spinal decomposition to the lookdown.

Takeover, fixation and identifiability in finite neutral genealogy models

TL;DR

By introducing the dual notions of forward and backward neutrality, this work gives a more intuitive implementation of the lookdown rearrangement of particles, and shows that the lookdown arranges subtrees in size-biased order of the number of their descendants.

Abstract

For neutral genealogy models in a finite, possibly non-constant population, there is a convenient ordered rearrangement of the particles, known as the lookdown representation, that greatly simplifies the analysis of the family trees. By introducing the dual notions of forward and backward neutrality, we give a more intuitive implementation of this rearrangement. We also show that the lookdown arranges subtrees in size-biased order of the number of their descendants, a property that is familiar in other settings but appears not to have been previously established in this context. In addition, we use the lookdown to study three properties of finite neutral models, as a function of the sequence of unlabelled litter sizes of the model: uniqueness of the infinite path (fixation), existence of a single lineage to which almost all individuals can trace their ancestry (takeover) and whether or not we can infer the lookdown rearrangement by examining the unlabelled genealogy model (identifiability). Identifiability of the spine path in size-biased Galton-Watson trees was previously studied, so we also discuss connections to those results, by relating the spinal decomposition to the lookdown.

Paper Structure

This paper contains 26 sections, 27 theorems, 113 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lookdown representations
  3. Organization of the article
  4. Preliminaries
  5. Graphs and models
  6. Basic examples
  7. Neutral models
  8. Forward neutral
  9. Backward neutral
  10. Forward/backward coupling, the lookdown coupling
  11. Size-biased ordering in the ordered lookdown
  12. Identification probability
  13. The spinal decomposition
  14. Coalescent time scale
  15. Main results
  16. ...and 11 more sections

Key Result

lemma 1

Let $(V,\mathcal{E})$ and $(V,\mathcal{E}')$ be forward neutral models based on the same $(k_n)$ and let $U=(U_n)$ be an i.i.d. $\mathrm{uniform}[0,1]$ sequence independent of $\mathcal{E}$. There is a permutation coupling $\mathcal{E}'=\alpha(\mathcal{E})$ such that for each $n$, $(\alpha_m)_{m\le

Figures (4)

  • Figure 1: Moran model and both versions of the lookdown model, with 5 particles and 4 events corresponding to the same particle pairs in each case. In the Moran model, ordered pairs are red and green, with red replacing green. In the lookdown, both are red, with the replacement rule as specified in the discussion.
  • Figure 2: Example of genealogy graph with $(X_0,\dots,X_4)=(3,5,4,6,4)$, and litter sizes $K_0=(3,0,2)$, $K_1=(1,1,0,0,2)$, $K_2=(2,1,3,0)$ and $K_3=(1,2,0,1,0,0)$.
  • Figure 3: Example of forward neutral model based on $k_0=(3,2,0)$, $k_1=(2,1,1,0,0)$, $k_2=(3,2,1,0)$ and $k_3=(2,1,1,0,0,0)$.
  • Figure 4: Example of the ordered lookdown based on $k_0=(3,2,0)$, $k_1=(2,1,1,0,0)$, $k_2=(3,2,1,0)$ and $k_3=(2,1,1,0,0,0)$.

Theorems & Definitions (64)

  • definition 1: unlabelled litter sizes, model based on $(k_n)$
  • definition 2
  • definition 3: Permutation coupling
  • lemma 1: Rearrangement
  • lemma 2: Frequency
  • proof
  • definition 4
  • definition 5
  • theorem 1: Scrambling
  • corollary 1
  • ...and 54 more