Chain graph models: topological sorting of meta-arrows and efficient construction of $${\mathcal{B}}$$ -essential graphs

Abstract

Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of directed acyclic graphs and chain graphs respectively, especially useful in the context of model selection. Recently, the notion of a labelled block ordering of vertices\({\mathcal{B}}\) was introduced as a flexible tool for specifying subfamilies of chain graphs. In particular, both the family of directed acyclic graphs and the family of “unconstrained” chain graphs can be specified in this way, for the appropriate choice of \({\mathcal{B}}\) . The family of chain graphs identified by a labelled block ordering of vertices \({\mathcal{B}}\) is partitioned into equivalence classes each represented by means of a \({\mathcal{B}}\) -essential graph. In this paper, we introduce a topological ordering of meta-arrows and use this concept to devise an efficient procedure for the construction of \({\mathcal{B}}\) -essential graphs. In this way we also provide an efficient procedure for the construction of both largest chain graphs and essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once.