Toric algebra of hypergraphs

TL;DR

This paper explores the algebraic structure of hypergraphs through their edge subrings, generalizing known graph results, with applications in algebraic statistics and insights into model geometry.

Contribution

It characterizes the generators of the presentation ideals of hypergraph edge subrings using balanced walks, extending graph-based results to hypergraphs.

Findings

Describes generators of hypergraph edge subrings in terms of balanced walks

Generalizes graph results to hypergraphs in algebraic context

Provides insights into statistical model geometry

Abstract

The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.