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Reconstructing the Geometry of Random Geometric Graphs

Han Huang, Pakawut Jiradilok, Elchanan Mossel

TL;DR

The paper addresses reconstructing the geometry of an unknown $d$-dimensional manifold $M\subset {\bf R}^N$ from a graph $G(n,M,\mu,{\rm p})$ in which edges occur with probability ${\rm p}(\|X_i-X_j\|)$. It develops a cluster-net framework that builds a covering of $M$ by clusters extracted from batches of graph vertices, using a key common-neighbor functional $K(x,y)$ to link graph structure with latent distances. The main contributions are a polynomial-time net-building algorithm BuildNet that output a cluster-net approximating $M$ with quantitative error bounds in both intrinsic (geodesic) and extrinsic (Euclidean) distances, and a metric-measure recovery result illustrating a diffeomorphic manifold close to the original in Gromov–Hausdorff terms. The approach combines new probabilistic concentration arguments, regularity properties of averaged distance-dependent functions, and navigation between almost-orthogonal clusters to propagate local geometry into a global net. This work advances the understanding of geometry recovery from graph data and connects manifold learning with structural graph inference, enabling downstream tasks such as identifying latent distances and constructing near-isometric metric-measure representations from topological data.

Abstract

Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work, we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the manifold assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in $\mathbb{R}^N$. Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.

Reconstructing the Geometry of Random Geometric Graphs

TL;DR

The paper addresses reconstructing the geometry of an unknown -dimensional manifold from a graph in which edges occur with probability . It develops a cluster-net framework that builds a covering of by clusters extracted from batches of graph vertices, using a key common-neighbor functional to link graph structure with latent distances. The main contributions are a polynomial-time net-building algorithm BuildNet that output a cluster-net approximating with quantitative error bounds in both intrinsic (geodesic) and extrinsic (Euclidean) distances, and a metric-measure recovery result illustrating a diffeomorphic manifold close to the original in Gromov–Hausdorff terms. The approach combines new probabilistic concentration arguments, regularity properties of averaged distance-dependent functions, and navigation between almost-orthogonal clusters to propagate local geometry into a global net. This work advances the understanding of geometry recovery from graph data and connects manifold learning with structural graph inference, enabling downstream tasks such as identifying latent distances and constructing near-isometric metric-measure representations from topological data.

Abstract

Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work, we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the manifold assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in . Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.
Paper Structure (44 sections, 52 theorems, 497 equations, 1 figure, 4 algorithms)

This paper contains 44 sections, 52 theorems, 497 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works in the Literature
  3. Some Proof Ideas
  4. Open Problems
  5. Structure of the paper
  6. Random Graph Model
  7. Notations
  8. Random Graph Model
  9. Manifold and probability measure
  10. Distance-probability function
  11. Graph Model
  12. Remark on the choice of $\ell_{\rm p}$ with dependency on $M$.
  13. Parameter Assumptions
  14. Additional Notation and Tools
  15. Common neighbor probability ${\rm K}(x,y)$
  16. ...and 29 more sections

Key Result

Theorem 1.1

There exists $c>0$ so that the following holds. Suppose $(M,\mu,{\rm p})$ satisfies Assumptions assump: M-and-pm and assump: pdf. Then there exists a deterministic polynomial-in-$n$ time algorithm that takes $G=G(n,M,\mu,{\rm p})$ as input, and outputs a weighted graph $\Gamma$ on $[n]$ and a metric Here, $C(M,\mu,{\rm p}) \ge 1$ is a constant which depends on $M$, $\mu$, and ${\rm p}$.

Figures (1)

  • Figure 1: Two closed curves.

Theorems & Definitions (122)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 112 more