Degree asymptotics with rates for preferential attachment random graphs

TL;DR

This paper establishes optimal convergence rates for the degree distribution of fixed vertices in preferential attachment graphs, introducing a novel Stein's method approach and providing new distributional properties.

Contribution

It introduces a new Stein's method variation to obtain convergence rates and characterizes the limiting degree distributions with explicit density formulas.

Findings

Optimal convergence rates for degree distributions are derived.

Explicit density expressions involve confluent hypergeometric functions.

New properties and representations of the limiting distributions are provided.

Abstract

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Stein's method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.