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Persistent hubs in CMJ branching processes with independent increments and preferential attachment trees

Tejas Iyer

Abstract

A sequence of trees $(\mathcal{T}_{n})_{n \in \mathbb{N}}$ contains a \emph{persistent hub}, or displays \emph{degree centrality}, if there is a fixed node of maximal degree for all sufficiently large $n \in \mathbb{N}$. We derive sufficient criteria for the emergence of a persistent hub in genealogical trees associated with Crump-Mode-Jagers branching processes with independent waiting times between births of individuals, and sufficient criteria for the non-emergence of a persistent hub. We also derive criteria for uniqueness of these persistent hubs. As an application, we improve results in the literature concerning the emergence of unique persistent hubs in generalised preferential attachment trees, in particular, allowing for cases where there may not be a \emph{Malthusian parameter} associated with the process. The approach we use is mostly self-contained, and does not rely on prior results about Crump-Mode-Jagers branching processes.

Persistent hubs in CMJ branching processes with independent increments and preferential attachment trees

Abstract

A sequence of trees contains a \emph{persistent hub}, or displays \emph{degree centrality}, if there is a fixed node of maximal degree for all sufficiently large . We derive sufficient criteria for the emergence of a persistent hub in genealogical trees associated with Crump-Mode-Jagers branching processes with independent waiting times between births of individuals, and sufficient criteria for the non-emergence of a persistent hub. We also derive criteria for uniqueness of these persistent hubs. As an application, we improve results in the literature concerning the emergence of unique persistent hubs in generalised preferential attachment trees, in particular, allowing for cases where there may not be a \emph{Malthusian parameter} associated with the process. The approach we use is mostly self-contained, and does not rely on prior results about Crump-Mode-Jagers branching processes.

Paper Structure

This paper contains 14 sections, 14 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of our contributions and structure
  3. Description of the models and statements of results
  4. Description of CMJ processes and related results
  5. Description of preferential attachment trees and related results
  6. Proofs of results
  7. Some auxiliary lemmata
  8. Proofs of Theorem \ref{['thm:persistence']} and Theorem \ref{['thm:unique']}
  9. Proof of Item \ref{['item:non-persistence']} of Theorem \ref{['thm:persistence']}
  10. Proof of Item \ref{['item:persistent-hub']} of Theorem \ref{['thm:persistence']}
  11. Proof of Theorem \ref{['thm:unique']}
  12. Proofs of Theorems \ref{['thm:main']} and \ref{['thm:superlinear']}
  13. Proof of Theorem \ref{['thm:main']}
  14. The proof of Theorem \ref{['thm:superlinear']}

Key Result

Theorem 2.3

Assume $(X_{j})_{j \in \mathbb{N}}, (X'_{j})_{j \in \mathbb{N}}$ are defined as in eq:def-ex and Assumption ass:gen-ass is satisfied. Then, in the process $(\mathscr{T}_{\tau_{n}})_{n \in \mathbb{N}_0}$, we have the following:

Figures (2)

  • Figure 1: A illustration of the way birth times are assigned to individuals in the first three generations of the process. Note that birth times are increasing on paths directed away from the root.
  • Figure 2: A possible sample of the process $(\mathscr{T}_{t})_{t\geq 0}$ at time $t = 1.99$, with birth times labelled. In this case $\tau_{9} = 1.99$.

Theorems & Definitions (48)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 38 more