# Unimodular hierarchical models and their Graver bases

## TL;DR

This paper classifies vertex-weighted simplicial complexes that produce unimodular vector configurations and characterizes their Graver bases, advancing understanding of hierarchical models in algebraic statistics.

## Contribution

It provides a complete classification of complexes leading to unimodular configurations and offers a combinatorial description of their Graver bases, a novel theoretical result.

## Key findings

- Classified all vertex-weighted complexes with unimodular configurations
- Provided a combinatorial characterization of Graver bases
- Enhanced understanding of hierarchical models in algebraic geometry

## Abstract

Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted simplicial complexes that give rise to unimodular vector configurations. We also provide a combinatorial characterization of their Graver bases.