The evolution of random reversal graph

TL;DR

This paper studies the structure of the random reversal graph, revealing a phase transition at a critical reversal rate where the graph shifts from small components to a giant, dense component.

Contribution

It identifies the critical reversal rate at which the random reversal graph undergoes a dramatic structural change, including the emergence of a giant component.

Findings

Below critical rate, components are at most O(n log n) in size.

Above critical rate, a giant component of size ~2^n * n emerges.

The giant component is dense in the reversal graph.

Abstract

The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $\lambda_n=1/\binom{n+1}{2}$. For $\lambda_n=(1-\epsilon)/\binom{n+1}{2}$, the random graph consists of components of size at most $O(n\ln(n))$ a.s. and for $(1+\epsilon)/\binom{n+1}{2}$, there emerges a unique largest component of size $\sim \wp(\epsilon) \cdot 2^n\cdot n$!$ a.s.. This "giant" component is furthermore dense in the reversal graph.