A note on the distribution of the extreme degrees of a random graph via the Stein-Chen method

TL;DR

This paper presents an alternative proof using the Stein-Chen method for Bollobás' theorem on the distribution of extreme degrees in random graphs, also providing convergence rates and extending to more general models.

Contribution

It introduces a new proof technique for the distribution of extreme degrees in random graphs and extends the results to more general connection probability models.

Findings

Provides a rate of convergence for the extreme degree distribution.

Extends the theorem to models with variable connection probabilities.

Offers an alternative proof approach using the Stein-Chen method.

Abstract

We offer an alternative proof, using the Stein-Chen method, of Bollob\'{a}s' theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.