Date of Award

2014

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

McDonald, Patrick

Keywords

Algebra, Graphs, Algebraic Statistics

Area of Concentration

Mathematics

Abstract

A fundamental theorem of algebraic statistics is that a very general class of sampling problems is equivalent to the problem of computing a presentation for a (not necessarily normal) toric ideal. A special case of this problem is that of generating a random multigraph on a fixed vertex set with fixed degrees; algebraically, this problem is formulated in the setting of toric graph ideals. A basic deficiency of many presentations obtained in solving the algebraic problem is that they do not transfer to the generalized problem in which certain edges are restricted. This deficiency is avoided in the case of a robust graph - one whose corresponding toric ideal is minimally generated by its universal Grobner basis. Such toric ideals are interesting in an algebro-geometric setting, as their zeroth Betti numbers are preserved under atdeformation. This thesis provides a review of commutative algebra and Grobner basis theory in the context of deformations and algebraic statistics, as well as a classification of robust graph ideals. The classification is given in terms of combinatorial conditions on a special class of elementary walks. As an application, we show that the operation of barycentric subdivision from simplicial topology always produces a robust graph, and discuss the implications for certain independence problems in algebraic statistics. We conclude with a discussion of some possible generalizations and open problems.

Download

Share

COinS