On the structure of genealogical trees associated with explosive Crump--Mode--Jagers branching processes

Abstract

We study the structure of genealogical trees associated with explosive Crump--Mode-Jagers branching processes (stopped at the explosion time), proving criteria for the associated tree to contain a node of infinite degree (a emphstar) or an infinite path. Next, we provide uniqueness criteria under which with probability 1 there exists exactly one of a unique star or a unique infinite path. Under the latter uniqueness criteria we also provide an example where, with strictly positive probability less than 1, there exists a unique star in the model. We thus illustrate that this probability is not restricted to being 0 or 1. Moreover, we provide structure theorems when there is a star, where we prove that certain trees appear as sub-trees in the tree infinitely often. We apply our results to a general discrete evolving tree model, named emphexplosive recursive trees with fitness. As a particular case, we study a family of emphsuper-linear preferential attachment models with fitness. For these models, we derive phase transitions in the model parameters in three different examples, leading to either exactly one star with probability 1 or one infinite path with probability 1 with every node having finite degree. Furthermore, we highlight examples where sub-trees T of empharbitrary size can appear infinitely often; behaviour that is markedly distinct from super-linear preferential attachment models studied in the literature so far.