Rigid Components of Random Graphs

TL;DR

This paper investigates the emergence and size of rigid components in Erdős-Rényi random graphs, revealing thresholds for the sizes of these components as the graph density varies.

Contribution

It provides the first probabilistic analysis of rigid component sizes in G(n,p) graphs, identifying conditions under which large rigid structures appear.

Findings

Rigid components are typically of size 2, 3, or linear in n.

For c>4, the largest rigid components are at least n/10 in size.

Results are the first to analyze rigidity in all graphs with O(n) edges.

Abstract

The planar rigidity problem asks, given a set of m pairwise distances among a set P of n unknown points, whether it is possible to reconstruct P, up to a finite set of possibilities (modulo rigid motions of the plane). The celebrated Maxwell-Laman Theorem from Rigidity Theory says that, generically, the rigidity problem has a combinatorial answer: the underlying combinatorial structure must contain a spanning minimally-rigid graph (Laman graph). In the case where the system is not rigid, its inclusion-wise maximal rigid substructures (rigid components) are also combinatorially characterized via the Maxwell-Laman theorem, and may be found efficiently. Physicists have used planar combinatorial rigidity has been used to study the phase transition between liquid and solid in network glasses. The approach has been to generate a graph via a stochastic process and then experimentally analyze its rigidity properties. Of particular interest is the size of the largest rigid components. In this paper, we study the emergence of rigid components in an Erdos-Renyi random graph G(n,p), using the parameterization p=c/n for a fixed constant c>0. Our first result is that for all c>0, almost surely all rigid components have size 2, 3 or Omega(n). We also show that for c>4, almost surely the largest rigid components have size at least n/10. While the G(n,p) model is simpler than those appearing in the physics literature, these results are the first of this type where the distribution is over all graphs on n vertices and the expected number of edges is O(n).