Graphical construction of spatial Gibbs random graphs

Abstract

We consider a random graph model on Zd 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 Zd, which coincides with the limit along the measures over graphs with the 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 Zd.