Tangled Paths: A Random Graph Model from Mallows Permutations

TL;DR

This paper introduces a new random graph model called the tangled path, formed by union of paths with vertices relabeled via Mallows permutations, and analyzes its structural properties as the permutation parameter varies.

Contribution

It defines the tangled path model and provides bounds on treewidth, cutwidth, diameter, and other properties across different permutation parameters.

Findings

Treewidth and cutwidth are determined up to log factors for all q.

A sharp threshold for the property of having a separator of size one.

Bounds on diameter and vertex isoperimetric number for specific q values.

Abstract

We introduce the random graph $\mathcal{P}(n,q)$ which results from taking the union of two paths of length $n\geq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with parameter $0<q(n)\leq 1$. This random graph model, the tangled path, goes through an evolution: if $q$ is close to $0$ the graph bears resemblance to a path, and as $q$ tends to $1$ it becomes an expander. In an effort to understand the evolution of $\mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $\mathcal{P}(n,q)$ up to log factors for all $q$. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of $q$.