Central limit theorem for crossings in randomly embedded graphs

TL;DR

This paper establishes a central limit theorem for the number of crossings in randomly embedded graphs, providing explicit formulas and convergence rates using Stein's method, applicable to various graph classes.

Contribution

It introduces explicit formulas for mean and variance of crossings and applies Stein's method to prove CLTs with convergence rates for multiple graph types.

Findings

Explicit formulas for mean and variance of crossings.

Upper bounds on the distance to normal distribution.

Central limit theorems with convergence rates for various graphs.

Abstract

We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$. Using Stein's method and size-bias coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where $G$ is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random $d$-regular graphs, and mixtures of random graphs.