Graphical Construction of Spatial Gibbs Random Graphs
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Publication:6324096
DOI10.1063/5.0073834zbMath1507.05084arXiv1908.08880WikidataQ101496563 ScholiaQ101496563 📅 23 August 2019
👤 Andressa Cerqueira 👤 Nancy L. Garcia
Publication date: 23 August 2019
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 $mathbb{Z}^{d}$.
Social networks; opinion dynamics (91D30) Random graphs (graph-theoretic aspects) (05C80) Interacting random processes; statistical mechanics type models; percolation theory (60K35) Combinatorial probability (60C05) Point processes (e.g., Poisson, Cox, Hawkes processes) (60G55)
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