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Graphical Construction of Spatial Gibbs Random Graphs

Andressa Cerqueira, Nancy L. Garcia

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

Graphical Construction of Spatial Gibbs Random Graphs

Abstract

We consider a Random Graph Model on 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 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 .

Paper Structure

This paper contains 16 sections, 11 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Spatial Gibbs Random Graphs
  4. Dependent graph process
  5. Main Results
  6. Graphical Representation
  7. Construction of the independent multigraph process
  8. Finite volume construction of the dependent process
  9. Infinite volume construction of the dependent model
  10. Perfect simulation of $\mu$
  11. Proofs
  12. Proof of Theorem \ref{['theo:clan_finite']}
  13. Properties of the clan of ancestors
  14. Time length and space diameter of a clan of ancestors
  15. Proofs of Existence and Uniqueness
  16. ...and 1 more sections

Key Result

Theorem \oldthetheorem

If $\alpha(\beta)<\infty$, then for any infinite $V \subset \mathbb{Z}^{d}$ the Markov process with generator $A^V$ exists and admits at least one invariant measure.

Figures (1)

  • Figure 1: Function $\alpha(\beta)$ for the model that penalizes 2-stars in the graph.

Theorems & Definitions (27)

  • Example 2.2
  • Example 2.3
  • Example
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Example
  • Example
  • Theorem \oldthetheorem: Exponential space convergence
  • Example 3.1: Expectation of the restricted degree
  • Theorem \oldthetheorem: Exponential mixing
  • ...and 17 more