Graphical Construction of Spatial Gibbs Random Graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866929389198901248 |
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| author | Cerqueira, Andressa Garcia, Nancy L. |
| author_facet | Cerqueira, Andressa Garcia, Nancy L. |
| contents | 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 $\mathbb{Z}^{d}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_08880 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Graphical Construction of Spatial Gibbs Random Graphs Cerqueira, Andressa Garcia, Nancy L. Statistics Theory 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 $\mathbb{Z}^{d}$. |
| title | Graphical Construction of Spatial Gibbs Random Graphs |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/1908.08880 |