Graphical Construction of Spatial Gibbs Random Graphs

08/23/2019

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by   Andressa Cerqueira, et al.

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We present a Spatial Gibbs Random Graphs Model that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. We propose a graphical construction of a model with vertices located in a finite subset of Z^2 that penalizes edges between distant vertices as well as other structures such as stars or triangles. We prove the existence and uniqueness of a measure defined on graphs with vertices in Z^2 as the limit along the measures over graphs with finite vertex set. Moreover, a perfect simulation algorithm is obtained in order to sample a subgraph from the measure defined on graphs with vertex set Z^2.