Graphical Construction of Spatial Gibbs Random Graphs

TL;DR

This paper introduces a new graphical construction for spatial Gibbs random graphs on d6^d, proving existence, uniqueness, and properties like mixing and CLT, along with a perfect simulation algorithm.

Contribution

It presents a novel graphical construction for spatial Gibbs random graphs, establishing existence, uniqueness, and sampling methods.

Findings

Existence and uniqueness of the infinite volume measure.

Exponential mixing and CLT for the model.

A perfect simulation algorithm for finite samples.

Abstract

We consider a Random Graph Model on $\mathbb{Z}^{d}$ that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. Based on a graphical construction of the model as the invariant measure of a birth and death process, we prove the existence and uniqueness of a measure defined on graphs with vertices in $\mathbb{Z}^{d}$ which coincides with the limit along the measures over graphs with finite vertex set. As a consequence, theoretical properties such as exponential mixing of the infinite volume measure and central limit theorem for averages of a real-valued function of the graph are obtained. Moreover, a perfect simulation algorithm based on the clan of ancestors is described in order to sample a finite window of the equilibrium measure defined on $\mathbb{Z}^{d}$.