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

TL;DR

This paper establishes criteria for the emergence and uniqueness of persistent hubs in genealogical trees from Crump-Mode-Jagers branching processes and applies these results to generalized preferential attachment trees, even without a Malthusian parameter.

Contribution

It provides new sufficient and necessary criteria for persistent hub emergence and uniqueness in branching and attachment trees, extending previous results to cases lacking a Malthusian parameter.

Findings

Criteria for persistent hub emergence in branching processes

Criteria for non-emergence of persistent hubs

Improved understanding of hub behavior in preferential attachment trees

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.