Preferential Attachment Trees with Vertex Death: Persistence of the Maximum Degree

TL;DR

The paper investigates persistence phenomena for the Preferential Attachment with Vertex Death (PAVD) model by framing it within a Crump-Mode-Jagers (CMJ) branching process. It establishes a two-regime picture: in the infinite lifetime regime, it provides precise criteria for the existence or nonexistence of persistent m-hubs (and a persistent 1-hub as a special case), with results extending prior work to multiple hubs and non-constant death rates; in the finite lifetime regime, it proves persistence under a 'rich are old' scenario using refined CMJ analysis and a detailed lifetime/offspring decomposition. The methodology hinges on a sharp CMJ embedding, Laplace-transform criteria for the offspring process, and careful control of lifetime tails and growth via $\mathcal K_\alpha$-type corrections. Collectively, the results illuminate how death coupling alters persistence, clarifying when a single late-appearing hub can monopolize degree growth and when aging and death prevent lasting domination, with implications for network archaeology and root-finding in evolving trees.

Abstract

We consider an evolving random discrete tree model called Preferential Attachment with Vertex Death, as introduced by Deijfen. Initialised with an alive root labelled $1$, at each step $n\geq1$ either a new vertex with label $n+1$ is introduced that attaches to an existing alive vertex selected preferentially according to a function $b$, or an alive vertex is selected preferentially according to a function $d$ and killed. In this article we introduce a generalised concept of persistence for evolving random graph models. Let $O_n$ be the smallest label among all alive vertices (the oldest alive vertex), and let $I_n^m$ be the label of the alive vertex with the $m^{\mathrm{th}}$ largest degree. We say a persistent $m$-hub exists if $I_n^m$ converges almost surely, we say that persistence occurs when $I_n^1/O_n$ is tight, and that lack of persistence occurs when $I_n^1/O_n$ tends to infinity. We identify two regimes called the infinite lifetime and finite lifetime regimes. In the infinite lifetime regime, vertices are never killed with positive probability. Here, we provide conditions under which we prove the (non-)existence of persistent $m$-hubs for any $m\in\mathbb N$. This expands and generalises recent work of Iyer, which covers the case $d\equiv 0$ and $m=1$. In the finite lifetime regime, vertices are killed after a finite number of steps almost surely. Here we provide conditions under which we prove the occurrence of persistence, which complements recent work of Heydenreich and the author, where lack of persistence is studied for preferential attachment with vertex death.

Theorems & Definitions (49)

• Definition 1.1: Preferential Attachment with Vertex Death
• Theorem 2.1: Infinite lifetime regime
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6: Finite lifetime regime
• Remark 2.7
• Remark 2.8
• Proposition 3.1: Embedding of PAVD in CTPAVD