Community modulated recursive trees and population dependent branching processes

We consider random recursive trees that are grown via community modulated schemes that involve random attachment or degree based attachment. The aim of this article is to derive general techniques based on continuous time embedding to study such models. The associated continuous time embeddings are not branching processes: individual reproductive rates at each time t depend on the composition of the entire population at that time, and hence vertices do not reproduce independently. Using stochastic analytic techniques we show that various key macroscopic statistics of the continuous time embedding stabilize, allowing asymptotics for a host of functionals of the original models to be derived.

Automated summary

The article focuses on random recursive trees grown by community modulated schemes involving random attachment or degree based attachment. General techniques based on continuous time embedding are derived to study these models. Through stochastic analytic techniques, it is shown that key macroscopic statistics of the continuous time embeddings stabilize, allowing asymptotics for various characteristics of the original models to be obtained.