Fine Structure for Markov Random Fields
A Markov Random Field is based on a set of variables that form the vertices of an adjacency graph. The Gibbs energy is decomposed in terms of the energies of configurations of maximal cliques of the adjacency graph.
We introduce a finer structure given by a simplicial complex whose 1-simplices are the edges of the adjacency graph. Every simplex is a clique of the adjacency graph. However, not all cliques need be simplices. We require that the Gibbs energy be decomposable in terms of the energies of configurations of simplices of the complex.