Consistent Markov Edge Processes and Random Graphs

Donatas Surgailis ()
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Donatas Surgailis: Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania

Mathematics, 2025, vol. 13, issue 21, 1-24

Abstract: We discuss Markov edge processes { Y e ; e ∈ E } defined on edges of a directed acyclic graph ( V , E ) with the consistency property P E ′ ( Y e ; e ∈ E ′ ) = P E ( Y e ; e ∈ E ′ ) for a large class of subgraphs ( V ′ , E ′ ) of ( V , E ) obtained through a mesh dismantling algorithm. The probability distribution P E 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 Z 2 , representing a far-reaching extension of the latter class. A particular case of binary consistent edge process on Z 3 was disclosed by Arak in a private communication. We prove that the symmetric binary Pickard model generates the Arak model on Z 2 as a contour model.

Keywords: directed acyclic graph; Markov edge process; clique distribution; consistency criterion; mesh dismantling algorithm; Bayesian network; Pickard random field; regular lattice; Arak model; evolution of particle system; broken line process; contour edge process (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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