Persistent hubs in CMJ branching processes with independent increments and preferential attachment trees

Abstract

A sequence of trees (Tn) n ∈ N contains a emphpersistent hub, or displays emphdegree centrality, if there is a fixed node of maximal degree for all sufficiently large n ∈ N. We derive sufficient criteria for the emergence of a persistent hub in genealogical trees associated with Crump-Mode-Jagers branching processes with independent waiting times between births of individuals, and sufficient criteria for the non-emergence of a persistent hub. We also derive criteria for uniqueness of these persistent hubs. As an application, we improve results in the l iterature concerning the emergence of unique persistent hubs in generalised preferential attachment trees, in particular, allowing for cases where there may not be a emphMalthusian parameter associated with the process. The approach we use is mostly self-contained, and does not rely on prior results about Crump-Mode-Jagers branching processes