Week 16.11.2026 – 22.11.2026

We study a random tree model known as the Preferential Attachment tree with Vertex Death. Here, one can both add vertices to the tree as well as kill vertices, mimicking the non-monotone growth of real-world networks, absent in classical preferential attachment models. One initialises the tree with a single root vertex labelled 1 and, at every step n, either a new vertex labelled n+1 is added to the tree and connected to an already present alive vertex selected preferentially, or an already present alive vertex is selected preferentially killed. Killed vertices can make no new connections. We are interested in the behaviour of the richest alive vertex $I_n$ (with the largest degree) and the oldest alive vertex $O_n$ (with the smallest label). We uncover three distinct regimes in which the behaviour of $I_n$ and $O_n$ is different. In this talk we shall discuss how the three regimes can be identified and what drives the behaviour observed in each regime. Joint work with Markus Heydenreich.

TBA