Preferential Attachment When Stable
Svante Janson, Subhabrata Sen, Joel Spencer
TL;DR
The paper analyzes a two-urn preferential attachment model with α>1, focusing on the rare event that the urns remain balanced at time 2n. By embedding the discrete process in continuous time and connecting it to a Brownian framework, the authors derive precise asymptotics for balance probabilities and establish a lower-tail Large Deviation Principle for the quadratic functional L_n = ∑_{i=1}^n S_i^2/i^2 of a simple random walk, with rate function Λ^*(x) = (x−1)^2/(8x). They then study conditional trajectories given balance, proving a functional limit theorem: the scaled imbalance converges to a distorted Brownian bridge G_α on [0,1], and under logarithmic time, to a stationary Ornstein–Uhlenbeck process. A second Brownian-analytic route via Gaussian Hilbert space and Gartner–Ellis theory corroborates the LDP, and the results yield a rich link between discrete preferential attachment and continuous Gaussian processes, including explicit MGFs, tightness criteria, and conditional trajectory limits.
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 $α^{th}$ power $(α>1)$ 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.