Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution

TL;DR

This paper establishes convergence rates for degree distributions in preferential attachment graphs to a power law, using a novel Stein's method approach for the negative binomial distribution.

Contribution

It introduces a new Stein's method formulation for the negative binomial distribution based on a distributional transformation, applied to preferential attachment graphs.

Findings

Provides explicit convergence rates in total variation distance.

Connects degree distributions to negative binomial via Stein's method.

Demonstrates the effectiveness of the new Stein's approach.

Abstract

For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to infinity. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.