Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon

TL;DR

This paper derives asymptotic formulas for the degree distribution and homomorphism densities in large dense random graphs generated from a graphon, using precise binomial asymptotics and smoothing techniques.

Contribution

It introduces a unified approach to asymptotics of degree distributions and homomorphism densities in graphon-sampled graphs, extending previous results.

Findings

Asymptotic formulas for degree CDF in large dense graphs.

General asymptotics for homomorphism densities with smooth functions.

Recovery of recent homomorphism density asymptotics.

Abstract

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs with smoother functions. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.