Persistent hubs in CMJ branching processes with independent increments and preferential attachment trees
Authors
- Iyer, Tejas
ORCID: 0000-0002-2005-3164
2020 Mathematics Subject Classification
- 60J80 90B15 05C80
Keywords
- Generalised preferential attachment trees, Crump--Mode--Jagers branching processes, persistent hubs, degree centrality, Malthusian parameter
DOI
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
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