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

TL;DR

This paper studies the eponymous Preferential Attachment with Vertex Death (PAVD) model, focusing on the persistence of the maximum degree by analyzing the ratio of the oldest alive vertex to the vertex with the largest degree. By embedding the discrete process into a Crump-Mode-Jagers branching process, the authors derive regime-dependent behaviors, uncovering a robust 'rich are old' regime where persistence can fail under mild conditions and a novel 'rich die young' regime where lack of persistence always occurs due to death biasing survival. They establish precise asymptotics for the oldest-alive level and for the growth of the largest degree, linking continuous-time embedding results to discrete-time quantities and highlighting how death fundamentally alters persistence in evolving trees. The work extends prior results for PA without death, reveals new mechanisms for survival under death, and lays a foundation for future extensions to broader graph classes and alternative death/birth rules.

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.

Figures (4)

• Figure 1: Population size of the countries Austria, Bosnia and Herzegovina, Croatia, and Portugal, from 1960 until 2023 Pop.
• Figure 2: Proportion of females within the total population Popfem (left) and the proportion of females aged 25 to 29 years old within the total female population Popfem2529 (right) of the countries Austria, Bosnia and Herzegovina, Croatia, and Portugal, from 1960 until 2023.
• Figure 3: The functions $H$ and $\widetilde{H}$, with their unique maximum $u^*$.
• Figure 4: Partition of the interval $[0,t]$. The red interval is the part we omit. The smallest intervals are the $\widetilde{U}_{t,i}$ with $i\in\mathbb{I}$.

Theorems & Definitions (72)

• Definition 1.1: Preferential Attachment with Vertex Death
• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5: Converging death sequences, smaller than $R$
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Remark 2.9