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

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. 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 $I_n$ be the label of the alive vertex with the largest degree (among all alive vertices). Persistence occurs when $I_n/O_n$ is tight; lack of persistence occurs when $I_n/O_n$ diverges with $n$. We study lack of persistence and identify two regimes: the old are rich and the rich die young regime. In the rich are old regime, though the oldest alive vertices in the tree typically have the largest degrees, lack of persistence can occur subject to the condition $\sum_{i=0}^\infty 1/(b(i)+d(i))^2=\infty$, under which lucky vertices that are younger than the oldest vertices can attain the largest degrees by step $n$, generalising results by Banerjee and Bhamidi. In contrast, lack of persistence always occurs in the rich die young regime. This regime is novel and cannot be observed in models without death. Here, vertices can survive exceptionally long by obtaining a low degree, whereas vertices with a large degree die much faster, causing lack of persistence. A main technique is an embedding of the discrete tree process into a Crump-Mode-Jagers branching process and a higher-order analysis of the resulting birth-death mechanism based on moderate deviation principles with exponential tilting.