Reciprocal properties of random fields on undirected graphs

We clarify and refine the definition of a reciprocal random field on an undirected graph, with the reciprocal chain as a special case, by introducing four new properties: the factorizing, global, local, and pairwise reciprocal properties, in decreasing order of strength, with respect to a set of nodes delta. They reduce to the better-known Markov properties if 8 is the empty set, or, with the exception of the local property, if delta is a complete set. Conditions for each reciprocal property to imply the next stronger property are derived, and it is shown that, conditionally on the values at a set of nodes delta(0), all four properties are preserved for the subgraph induced by the remaining nodes, with respect to the node set delta \ delta(0). We note that many of the above results are new even for reciprocal chains.

Automated summary

This paper clarifies and refines the definition of a reciprocal random field on an undirected graph, introducing four new properties (factorizing, global, local, and pairwise reciprocal properties) and their relationships. It shows that these properties reduce to the well-known Markov properties for specific cases and derives conditions for each reciprocal property to imply the next stronger property. Furthermore, it demonstrates that the subgraph induced by the remaining nodes preserves all four properties with respect to the node set delta \ delta(0).