Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon. Issue 1 (3rd October 2020)

• Record Type: Journal Article
• Authors: Delmas, Jean‐François; Dhersin, Jean‐Stéphane; Sciauveau, Marion
• Is Part Of: Random structures & algorithms. Volume 58:Issue 1(2021)
• Page Start: 94
• Page End: 149
• Publication Date: 2020-10-03
• Subjects: binomial distribution -- cumulative distribution function of degrees -- dense graph -- graphon -- homomorphism density -- partially labeled graph -- random measure; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20965
• Languages: English
• ISSNs: 1042-9832

Abstract

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. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. 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. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.