Global rigidity of random graphs in $\mathbb{R}$

TL;DR

We study reconstructing a point set $P\subseteq \mathbb{R}$ from partial distance data encoded by a random graph on $P$. We prove that the hitting time $\tau$ when the graph first achieves minimum degree $2$ yields global rigidity in $\mathbb{R}$ with high probability, meaning any injective embedding compatible with edge lengths is determined up to isometry, resolving a conjecture of Benjamini and Tzalik. Beyond full reconstruction, we show near-full reconstruction in sparser regimes: for some $C$ every $\varepsilon>0$ admits a $G\sim G(n, C/n)$ containing a subset $V'$ of size $(1-\varepsilon)n$ with $G[V']$ globally rigid in $\mathbb{R}$, while there is a lower bound below $p\approx 1.1/n$ preventing large globally rigid subsets. The work introduces a simple two-property criterion for global rigidity and leverages random-graph machinery to derive both the full and partial reconstruction results, and it outlines openness questions on random regular graphs, phase transitions, algorithmic aspects, and higher-dimensional extensions.

Abstract

We investigate the problem of reconstructing a set $P\subseteq \mathbb{R}$ of distinct points, where the only information available about $P$ consists of the distances between some of the pairs of points. More precisely, we examine which properties of the graph $G$ of known distances, defined on the vertex set $P$, ensure that $P$ can be uniquely reconstructed up to isometry. We prove that as soon as the random graph process has minimum degree 2, with high probability it can reconstruct all distances within any point set in $\mathbb{R}$. This resolves a conjecture of Benjamini and Tzalik. We also study the feasibility and limitations of reconstructing the distances within almost all points using much sparser random graphs. In doing so, we resolve a question posed by Girão, Illingworth, Michel, Powierski, and Scott.

Theorems & Definitions (23)

• Theorem 1.1: girao2023reconstructing
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:main']}