Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galled Tree-Child Networks

Yu-Sheng Chang, Michael Fuchs, Guan-Ru Yu

TL;DR

The class of galled tree-child networks is proposed, which is obtained as intersection of the classes of galled networks and tree-child networks and it is shown that the numbers of reticulation nodes of random galled tree-child networks are asymptotically normal distributed.

Abstract

We propose the class of galled tree-child networks which is obtained as intersection of the classes of galled networks and tree-child networks. For the latter two classes, (asymptotic) counting results and stochastic results have been proved with very different methods. We show that a counting result for the class of galled tree-child networks follows with similar tools as used for galled networks, however, the result has a similar pattern as the one for tree-child networks. In addition, we also consider the (suitably scaled) numbers of reticulation nodes of random galled tree-child networks and show that they are asymptotically normal distributed. This is in contrast to the limit laws of the corresponding quantities for galled networks and tree-child networks which have been both shown to be discrete.

Galled Tree-Child Networks

TL;DR

The class of galled tree-child networks is proposed, which is obtained as intersection of the classes of galled networks and tree-child networks and it is shown that the numbers of reticulation nodes of random galled tree-child networks are asymptotically normal distributed.

Abstract

We propose the class of galled tree-child networks which is obtained as intersection of the classes of galled networks and tree-child networks. For the latter two classes, (asymptotic) counting results and stochastic results have been proved with very different methods. We show that a counting result for the class of galled tree-child networks follows with similar tools as used for galled networks, however, the result has a similar pattern as the one for tree-child networks. In addition, we also consider the (suitably scaled) numbers of reticulation nodes of random galled tree-child networks and show that they are asymptotically normal distributed. This is in contrast to the limit laws of the corresponding quantities for galled networks and tree-child networks which have been both shown to be discrete.
Paper Structure (6 sections, 18 theorems, 36 equations, 1 figure, 1 table)

This paper contains 6 sections, 18 theorems, 36 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Component Graph Method
  3. Networks with Few and Many Reticulation Nodes
  4. Proof of the Main Results
  5. Conclusion
  6. Acknowledgments.

Key Result

Theorem 1.6

For the number of galled tree-child networks, we have, as $n\rightarrow\infty$,

Figures (1)

  • Figure 1: A galled network $N$ and its component graph $C(N)$ which is a phylogenetic tree.

Theorems & Definitions (35)

  • Definition 1.1: Phylogenetic Network
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5: Galled Tree-Child Network
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 2.1: GuRaZh
  • Definition 2.2: One-component Network
  • ...and 25 more