Sharp threshold for rigidity of random graphs

TL;DR

This paper establishes precise thresholds for when random graphs become rigid and globally rigid in Euclidean space as edges are added, linking these properties to minimum degree conditions.

Contribution

It proves sharp thresholds for rigidity and global rigidity in Erdős-Rényi random graphs based on minimum degree, clarifying the evolution of structural properties.

Findings

Graph becomes rigid in ^d at minimum degree d

Graph becomes globally rigid in ^d at minimum degree d+1

High probability thresholds for rigidity properties

Abstract

We consider the Erd\H{o}s-R\'enyi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed $d\ge 1$, we show that with high probability, the graph becomes rigid in $\mathbb R^d$ at the very moment its minimum degree becomes $d$, and it becomes globally rigid in $\mathbb R^d$ at the very moment its minimum degree becomes $d+1$.