Preferential Attachment When Stable

TL;DR

This paper analyzes a preferential attachment urn process with superlinear attachment, deriving precise probabilities for balanced configurations, establishing a large deviation principle, and describing the process's limiting behavior under conditioning.

Contribution

It introduces a detailed asymptotic analysis of a superlinear urn process, including probability estimates, a large deviation principle, and a functional limit law under conditioning.

Findings

Precise asymptotics for balanced urn configurations.

A large deviation principle for a related sum involving the process.

A functional limit law for the urn trajectory conditioned on balance.

Abstract

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $\alpha^{th}$ power $(\alpha >1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, a fine control on this probability may be leveraged to derive a lower tail Large Deviation Principle (LDP) for $L = \sum_{i=1}^{n} \frac{S_i^2}{i^2}$, where $\{S_n : n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternate proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous time analogue of $L$. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.