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

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.