Consistent Markov Edge Processes and Random Graphs

We discuss Markov edge processes {Ye;e∈E} defined on edges of a directed acyclic graph (V,E) with the consistency property PE′(Ye;e∈E′)=PE(Ye;e∈E′) for a large class of subgraphs (V′,E′) of (V,E) obtained through a mesh dismantling algorithm. The probability distribution PE of such edge process is a discrete version of consistent polygonal Markov graphs. The class of Markov edge processes is related to the class of Bayesian networks and may be of interest to causal inference and decision theory. On regular ν-dimensional lattices, consistent Markov edge processes have similar properties to Pickard random fields on Z2, representing a far-reaching extension of the latter class. A particular case of binary consistent edge process on Z3 was disclosed by Arak in a private communication. We prove that the symmetric binary Pickard model generates the Arak model on Z2 as a contour model.