Unimodular binary hierarchical models

Abstract

Abstract. Associated to each simplicial complex is a binary hierarchical model. We classify the simplicial complexes that yield unimodular binary hierarchical models. Our main theorem provides both a construction of all unimodular binary hierarchical models, together with a characterization in terms of excluded minors, where our definition of a minor allows the taking of links and induced complexes. A key tool in the proof is the lemma that the class of unimodular binary hierarchical models is closed under the Alexander duality operation on simplicial complexes.

Unimodular hierarchical models and their Graver bases

“…See [5] for an explanation as to why the conditions in Definition 1 are equivalent. Definition 1 can be seen as a kernel-invariant generalization of the familiar definition of unimodularity for square integer matrices.…”

“…We denote by C \ v the complex obtained by deleting v from V and each face of C containing v. The link of C about v, denoted link v (C), has ground set V \ v and a face F whenever F ∪ v is a face of C. The Alexander dual of C, denoted C * , is the complex on the same ground set as C whose facets are the complements of the minimal non-faces of C. Proof. See Propositions 7.1 and 7.5 in [5].…”

“…Sullivant and the first author gave a complete characterization of the set of simplicial complexes C such that the binary HM pair (C, 2) is unimodular [5]. This characterization can be expressed constructively (Theorem 1) or in terms of forbidden minors (Theorem 2).…”

“…Just as in the characterization of unimodular binary HM pairs that appeared in [5], our classification comes in two forms: a recipe for constructing any unimodular HM pair, and a list of forbidden minors. Remark 1.…”

“…In light of Remark 1, we see that the matrix for (3) has the same integer kernel as the matrix for the HM pair ({12, 23, 13}, (4, 4, 3)) and the matrix for (4) has the same integer kernel as the matrix for the HM pair ({12, 23, 13), (4, 3, 3)} which both have (2) as a minor. Minimal nonunimodularity of the remaining HM pairs are given by [5,Proposition 7.9]. If C has nucleus D m,n , then minors (5) and (7) ensure that (c) is satisfied, so assume C has nucleus ∆ m ∆ n and that C has a Lawrence vertex v such that d v ≥ 3.…”