Graphical Construction of Spatial Gibbs Random Graphs

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 ${\mathbb{Z}}^d$.

References in corpus (2)

• Estimating and understanding exponential random graph models
• Consistency under sampling of exponential random graph models