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

TL;DR

This paper establishes a clear mathematical connection between Gaussian Markov random fields on metric graphs and Gaussian graphical models, revealing conditions under which these models are faithful to their independence structures.

Contribution

It provides an explicit link between Gaussian Markov random fields and graphical models on metric graphs, clarifying when these fields are faithful to their pairwise independence graphs.

Findings

Markov random fields on vertices form Gaussian graphical models under mild conditions

No Gaussian fields are both Markov and isotropic on general metric graphs with regular metrics

The work clarifies the structure of Gaussian fields on metric graphs

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.