High-degree vertices in uniform recursive directed acyclic graphs with freezing

Abstract

We study uniform recursive directed acyclic graphs with freezing. Here, a graph is built by adding vertices one-by-one and connecting a new vertex to $m\in\mathbb N$ uniformly selected vertices already present. At certain steps vertices can also be frozen, and arriving vertices are not allowed to connect to frozen vertices. This model generalises the uniform attachment tree with freezing, introduced by Bellin et. al (which corresponds to the case $m=1$) as well as the uniform recursive directed acyclic graph model (where no vertices are frozen). Under mild assumptions on when vertices are frozen, we study the empirical degree distribution, large degrees in the graph, and other properties of large-degree vertices such as their label and distance to the first vertex in the graph. Our work improves and/or extends various results from the literature on uniform attachment trees (with freezing) and uniform recursive directed acyclic graphs without freezing. In particular, our results show that statistics that are determined `locally' (e.g. the empirical degree distribution and maximum degree) are essentially unaffected by the freezing of vertices, whereas statistics that are determined `globally' (e.g. the length of paths between vertices) are highly affected by introducing freezing. The analysis relies on adapting the Kingman coalescent construction for uniform attachment trees to the non-tree setting.