Crossings in Randomly Embedded Graphs

TL;DR

This paper analyzes the distribution of crossings in randomly embedded graphs, providing a normal approximation with a convergence rate of order n^{-1/2} for various graph families, including chord diagrams and cycles.

Contribution

It offers a new probabilistic estimate for the distribution of crossings in random graph embeddings, with explicit convergence rates.

Findings

Normal distribution approximation for crossings

Convergence rate of order n^{-1/2}

Applicable to chord diagrams and cycles

Abstract

We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various families of graphs, including random chord diagrams or full cycles.