Normal approximation of the numbers of isolated edges and isolated 2-stars in uniform simple graphs with given vertex degrees

TL;DR

This paper develops new probabilistic approximation methods to accurately estimate the distribution of isolated subgraphs in random graphs with fixed degrees, providing the first finite sample normal approximation results for such graphs.

Contribution

It introduces a novel Stein's method and coupling approach for joint normal-Poisson approximation in the context of random graphs with given degrees.

Findings

Derived bounds for approximation errors in the configuration model.

Established finite sample normal approximation results for uniform simple graphs.

Developed new coupling techniques for sums of indicators.

Abstract

We consider the configuration model and the uniform simple graph with given degree sequence $\boldsymbol{d}=\left(d_i\right)_{i=1}^n$. We derive quantitative bounds for the errors in (i) joint normal-Poisson approximation to the numbers of isolated edges, isolated 2-stars, self-loops and double edges in the configuration model, and (ii) normal approximation to the numbers of isolated edges and isolated 2-stars conditioned on that the configuration model is simple. The latter provides the first finite sample normal approximation results for the uniform simple graph with given vertex degrees. To achieve this, we develop a new Stein's method for joint normal-Poisson approximation and a new coupling approach to sums of indicators, which may be of independent interest.