The Erlang Weighted Tree, A New Branching Process
Abstract: In this paper, we study a new branching process that we call Erlang Weighted Tree(EWT). EWT appears as the local weak limit of a random graph model proposed in the work of Richard La and Maya Kabkab. In contrast to the local weak limit of well-known random graph models, EWT has an interdependent structure, and its vertices follow a Markovian behavior. In particular, each vertex has a type that depends on the type of its parent.
We derive the main properties of EWT, such as the probability of extinction, growth rate, etc. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein-Rutman eigenvalue β and the corresponding eigenfunctions and show that the probability of extinction equals one if and only if β <= 1.
The preprint of the paper is available on arXiv: https://arxiv.org/abs/2002.03993, and it has been recommended for publication on RSA with minor edits.
