Associated to any toric ideal are two special generating sets: the universal
Gr\"obner basis and the Graver basis. While the former is a subset of the
typically much larger Graver basis, there are cases for which the two sets
coincide. The most prominent examples among them are toric ideals of unimodular
matrices. Yet, a general classification of all matrices for which both sets
agree is far from known.

There are two seemingly unrelated classical objects associated to a
simplicial complex: a hierarchical model and a Stanley-Reisner ring. A
hierarchical model gives rise to a toric ideal, a relationship that is a staple
of algebraic statistics. In this note, we explore the connection between
degrees of Markov bases elements of the model and the rows of the Betti diagram
of the Stanley-Reisner ideal. We propose a precise conjecture, which we
establish in several cases, most notably for decomposable and
vertex-decomposable complexes.