Fluctuation Bounds for Continuous Time Branching Processes and Evolution of Growing Trees With a Change Point

TL;DR

This paper studies the evolution of random trees with a change point in attachment rules, providing deterministic degree distribution approximations, change point estimators, and analyzing effects of change points in different time scales.

Contribution

It introduces new methods for analyzing inhomogeneous continuous-time branching processes and applies them to understand change points in growing random trees.

Findings

Deterministic approximations for degree distributions in the standard model.

Consistent non-parametric estimator for the change point.

Effect of pre-change dynamics diminishes in degree distribution but persists in maximal degree.

Abstract

We consider dynamic random trees constructed using an attachment function $f : \mathbb{N} \to \mathbb{R}_+$ where, at each step of the evolution, a new vertex attaches to an existing vertex $v$ in the current tree with probability proportional to $f$(degree(v)). We explore the effect of a change point in the system; the dynamics are initially driven by a function f until the tree reaches size $\tau(n) \in (0,n)$, at which point the attachment function switches to another function, $g$, until the tree reaches size $n$. Two change point time scales are considered, namely the standard model where $\tau(n) = \gamma n$, and the quick big bang model where $\tau(n) = n^\gamma$, for some $0 < \gamma < 1$. In the former case, we obtain deterministic approximations for the evolution of the empirical degree distribution (EDF) in sup-norm and use these to devise a provably consistent non-parametric estimator for the change point $\gamma$. In the latter case, we show that the effect of pre-change point dynamics asymptotically vanishes in the EDF, although this effect persists in functionals such as the maximal degree. Our proofs rely on embedding the discrete time tree dynamics in an associated (time) inhomogeneous continuous time branching process (CTBP). In the course of proving the above results, we develop novel mathematical techniques to analyze both homogeneous and inhomogeneous CTBPs and obtain rates of convergence for functionals of such processes, which are of independent interest.