The rigidity transition in random graphs

TL;DR

This paper investigates the emergence of a giant rigid component in random graphs, establishing a sharp threshold at c ≈ 3.588 for the phase transition, and quantifying the size of the component in the (3+2)-core.

Contribution

It proves the existence of a sharp threshold for a giant rigid component in Erdős-Rényi graphs and provides bounds on its size within the (3+2)-core structure.

Findings

Giant rigid component appears when c > 3.588

Threshold matches the 2-orientability constant

Giant component spans nearly all vertices in the (3+2)-core

Abstract

As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. The constant c_2 \approx 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.