On a uniformly random chord diagram and its intersection graph

TL;DR

This paper studies the properties of a random chord diagram and its intersection graph, analyzing degree distribution, core structure, and connectivity as the number of chords grows large, and explores evolution processes leading to monolithic diagrams.

Contribution

It introduces new probabilistic analyses of random chord diagrams and their intersection graphs, including evolution models and asymptotic properties.

Findings

Degree distribution of a random vertex in the intersection graph

Behavior of the k-core in the intersection graph

Connectivity and monolithic structure emergence in evolution models

Abstract

A chord diagram refers to a set of chords with distinct endpoints on a circle. The intersection graph of a chord diagram $\cal C$ is defined by substituting the chords of $\cal C$ with vertices and by adding edges between two vertices whenever the corresponding two chords cross each other. Let $C_n$ and $G_n$ denote the chord diagram chosen uniformly at random from all chord diagrams with $n$ chords and the corresponding intersection graph, respectively. We analyze $C_n$ and $G_n$ as $n$ tends to infinity. In particular, we study the degree of a random vertex in $G_n$, the $k$-core of $G_n$, and the number of strong components of the directed graph obtained from $G_n$ by orienting edges by flipping a fair coin for each edge. We also give two equivalent evolutions of a random chord diagram and show that, with probability approaching $1$, a chord diagram produced after $m$ steps of these evolutions becomes monolithic as $m$ tends to infinity and stays monolithic afterward forever.