Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An explicit link between graphical models and Gaussian Markov random fields on metric graphs

David Bolin, Alexandre B. Simas, Jonas Wallin

TL;DR

The paper addresses the problem of relating continuously indexed Gaussian Markov random fields on compact metric graphs to discrete Gaussian graphical models defined on finite point sets. It proves an explicit link: for order-1 GMRFs with continuous positive-definite covariance and an admissible finite point set, the induced finite-dimensional distribution is MTP$_2$ and faithful to its pairwise independence graph, with the vertex-neighborhood structure matching the metric-graph adjacency. This yields a sharp incompatibility result: under a mild homogeneity condition, there are no GMRFs that are simultaneously Markov and isotropic on general metric graphs (except in special cases like trees or cycles under certain metrics). Practically, the result enables applying graphical-model inference to GMRFs on networks and informs modeling decisions by clarifying when isotropy and Markov properties can co-occur, and it provides concrete structure for Whittle–Matérn fields on lattices.

Abstract

We derive an explicit link between Gaussian Markov random fields on metric graphs and graphical models, and in particular show that a Markov random field restricted to the vertices of the graph is, under mild regularity conditions, a Gaussian graphical model with a distribution which is faithful to its pairwise independence graph, which coincides with the neighbor structure of the metric graph. This is used to show that there are no Gaussian random fields on general metric graphs which are both Markov and isotropic in some suitably regular metric on the graph, such as the geodesic or resistance metrics.

An explicit link between graphical models and Gaussian Markov random fields on metric graphs

TL;DR

The paper addresses the problem of relating continuously indexed Gaussian Markov random fields on compact metric graphs to discrete Gaussian graphical models defined on finite point sets. It proves an explicit link: for order-1 GMRFs with continuous positive-definite covariance and an admissible finite point set, the induced finite-dimensional distribution is MTP and faithful to its pairwise independence graph, with the vertex-neighborhood structure matching the metric-graph adjacency. This yields a sharp incompatibility result: under a mild homogeneity condition, there are no GMRFs that are simultaneously Markov and isotropic on general metric graphs (except in special cases like trees or cycles under certain metrics). Practically, the result enables applying graphical-model inference to GMRFs on networks and informs modeling decisions by clarifying when isotropy and Markov properties can co-occur, and it provides concrete structure for Whittle–Matérn fields on lattices.

Abstract

We derive an explicit link between Gaussian Markov random fields on metric graphs and graphical models, and in particular show that a Markov random field restricted to the vertices of the graph is, under mild regularity conditions, a Gaussian graphical model with a distribution which is faithful to its pairwise independence graph, which coincides with the neighbor structure of the metric graph. This is used to show that there are no Gaussian random fields on general metric graphs which are both Markov and isotropic in some suitably regular metric on the graph, such as the geodesic or resistance metrics.
Paper Structure (10 sections, 10 theorems, 53 equations, 3 figures)

This paper contains 10 sections, 10 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gaussian graphical models and GMRFs on metric graphs
  3. Gaussian graphical models
  4. GMRFs on metric graphs
  5. Explicit link with graphical models
  6. Additional results and the proof of Theorem \ref{['thm:explicit_link']}
  7. Examples and consequences
  8. Whittle--Matérn fields on regular lattices
  9. Non-Markovianity of isotropic models
  10. Discussion

Key Result

Theorem 1

Let $U$ be a Gaussian random vector indexed by a finite set $S\neq\emptyset$ with an invertible covariance matrix $\Sigma$ and let $P$ be the distribution of $U$. If $P$ is MTP$_2$, then $P$ is faithful to its pairwise independence graph $(S, \mathcal{E}(S))$.

Figures (3)

  • Figure 1: Examples of graphs where no random fields exist which are both Markov of order 1 and isotropic for the geodesic metric.
  • Figure 2: A 2-cycle graph with 7 vertices and 8 edges, each of length 1. This graph also represents the pairwise independence graph of the vertex process for the Whittle--Matérn model with $\alpha=1$ defined on this metric graph.
  • Figure 3: The pairwise independence graph of the isotropic models on the 2-cycle graph.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 2: Theorem 1, BSW2022
  • ...and 24 more