The Erlang Weighted Tree, A New Branching Process

Abstract

In this paper, we study a new discrete tree and the resulting branching process, which we call the \textbf{E}rlang \textbf{W}eighted \textbf{T}ree(\textbf{EWT}). The EWT appears as the local weak limit of a random graph model proposed in~\cite{La2015}. In contrast to the local weak limit of well-known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi-type branching process with uncountably many types. We derive the main properties of the 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 $\beta_0$ and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if $\beta_0 \leq 1$.