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Stein's Method for Spatial Random Graphs

Dominic Schuhmacher, Leoni Carla Wirth

Abstract

In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main theorems provide explicit upper bounds for integral probability metrics and, at improved rates, a recently introduced Wasserstein metric for random graph distributions. The bounds are in terms of a vertex error term based on the Papangelou kernels of the vertex point processes and two edge error terms based on conditional edge probabilities. In addition to providing new tools for spatial random graphs along the way, such as a graph-based Georgii--Nguyen--Zessin formula, we also give applications of our bounds to the percolation graph of large balls in a Boolean model and to discretising a generalised random geometric graph.

Stein's Method for Spatial Random Graphs

Abstract

In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main theorems provide explicit upper bounds for integral probability metrics and, at improved rates, a recently introduced Wasserstein metric for random graph distributions. The bounds are in terms of a vertex error term based on the Papangelou kernels of the vertex point processes and two edge error terms based on conditional edge probabilities. In addition to providing new tools for spatial random graphs along the way, such as a graph-based Georgii--Nguyen--Zessin formula, we also give applications of our bounds to the percolation graph of large balls in a Boolean model and to discretising a generalised random geometric graph.

Paper Structure

This paper contains 13 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Setup and Notation
  3. Gibbs processes
  4. Generalised random geometric graphs
  5. Metrics for spatial graphs
  6. Georgii--Nguyen--Zessin Formula for Spatial Random Graphs
  7. Stein's Method
  8. Coupling
  9. Bounds for general IPMs
  10. Wasserstein bounds
  11. Applications
  12. Percolation graph of large balls in a Boolean model
  13. Discrete Graphs

Key Result

Theorem 3.1

For any measurable function $h:\mathfrak{N} \times \mathfrak{N}_2 \times \mathcal{X} \times \mathfrak{N}_2 \to \mathbb{R}\xspace_+$ we have

Theorems & Definitions (41)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • Corollary 3.6
  • Remark 3.7
  • ...and 31 more