Stein's Method for Spatial Random Graphs

TL;DR

This paper develops Stein's method for approximating spatial random graphs with generalized models, providing explicit bounds and applications to percolation and discretization, advancing tools for spatial graph analysis.

Contribution

It introduces Stein's method for spatial random graphs with explicit bounds, including a graph-based Georgii--Nguyen--Zessin formula, and applies these to percolation and discretization problems.

Findings

Derived explicit bounds for integral probability metrics.

Provided improved rates for Wasserstein metric bounds.

Applied bounds to percolation in Boolean models and graph discretization.

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.