Generalisation of the Hammersley-Clifford Theorem on Bipartite Graphs

TL;DR

This paper extends the Hammersley-Clifford theorem to bipartite graphs, demonstrating that certain Markov random fields supported on these graphs are Gibbs states with nearest neighbor interactions, under new conditions.

Contribution

It generalizes the theorem by introducing strong config-folds and strong config-unfolds, expanding the class of graphs where the theorem applies.

Findings

The theorem holds for bipartite graphs with the new conditions.

Supported Markov random fields are Gibbs with nearest neighbor interactions.

Introduces the concept of strong config-folding for configuration spaces.

Abstract

The Hammersley-Clifford theorem states that if the support of a Markov random field has a safe symbol then it is a Gibbs state with some nearest neighbour interaction. In this paper we generalise the theorem with an added condition that the underlying graph is bipartite. Taking inspiration from "Gibbs Measures and Dismantlable Graphs" by Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong config-folding proving that if all Markov random fields supported on $X$ are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the 'strong config-folds' and 'strong config-unfolds' of $X$.