arXiv subjects

Ellen Veomett

Publications and source records attributed to Ellen Veomett.

16 recordsLinked to original sources

arXiv2026

Connectivity of Districting Metagraphs

In this article, we prove irreducibility results for a family of Markov chains arising in the study of redistricting and detecting gerrymandering. These chains use ReCom moves as their transition mechanism and are commonly employed in Markov chain Monte Carlo methods to generate ensembles of districting plans. Such ensembles are frequently used for outlier analysis, in which a proposed districting map is compared against the ensemble to determine whether it behaves atypically; this methodology often appears in expert testimony in redistricting litigation. We show that when the underlying dual graph is a triangular subset of the triangular lattice with side length 5 or larger, and each district consists of two merged geographic regions, the associated ReCom chain is irreducible. This provides another entry in the very small list of known classes of ReCom chains for which irreducibility has been established. We also demonstrate the fragility of this phenomenon by constructing an infinite family of maps for which the corresponding ReCom chain is not irreducible. Indeed, we produce a districting map that, after implementing a single ReCom move, always yields the same original map. These examples remain structurally close to the triangular lattice: they arise as subdivisions of the triangular lattice, and the resulting graphs have maximum degree at most 8. Finally, we prove irreducibility for a further special case: the ReCom chain on a 3 x n grid graph partitioned into three districts of size n.

math.CO↗

arXiv2025

States of Disarray: Cleaning Data for Gerrymandering Analysis

The mathematics of redistricting is an area of study that has exploded in recent years. In particular, many different research groups and expert witnesses in court cases have used outlier analysis to argue that a proposed map is a gerrymander. This outlier analysis relies on having an ensemble of potential redistricting maps against which the proposed map is compared. Arguably the most widely-accepted method of creating such an ensemble is to use a Markov Chain Monte Carlo (MCMC) process. This process requires that various pieces of data be gathered, cleaned, and coalesced into a single file that can be used as the seed of the MCMC process. In this article, we describe how we have begun this cleaning process for each state, and made the resulting data available for the public at https://github.com/eveomett-states . At the time of submission, we have data for 22 states available for researchers, students, and the general public to easily access and analyze. We will continue the data cleaning process for each state, and we hope that the availability of these datasets will both further research in this area, and increase the public's interest in and understanding of modern techniques to detect gerrymandering.

cs.DB↗

arXiv2024

The Intersectionality Problem for Algorithmic Fairness

A yet unmet challenge in algorithmic fairness is the problem of intersectionality, that is, achieving fairness across the intersection of multiple groups -- and verifying that such fairness has been attained. Because intersectional groups tend to be small, verifying whether a model is fair raises statistical as well as moral-methodological challenges. This paper (1) elucidates the problem of intersectionality in algorithmic fairness, (2) develops desiderata to clarify the challenges underlying the problem and guide the search for potential solutions, (3) illustrates the desiderata and potential solutions by sketching a proposal using simple hypothesis testing, and (4) evaluates, partly empirically, this proposal against the proposed desiderata.

cs.LG↗

arXiv2024

Don't Trust A Single Gerrymandering Metric

In recent years, in an effort to promote fairness in the election process, a wide variety of techniques and metrics have been proposed to determine whether a map is a partisan gerrymander. The most accessible measures, requiring easily obtained data, are metrics such as the Mean-Median Difference, Efficiency Gap, Declination, and GEO metric. But for most of these metrics, researchers have struggled to describe, given no additional information, how a value of that metric on a single map indicates the presence or absence of gerrymandering. Our main result is that each of these metrics is gameable when used as a single, isolated quantity to detect gerrymandering (or the lack thereof). That is, for each of the four metrics, we can find district plans for a given state with an extremely large number of Democratic-won (or Republican-won) districts while the metric value of that plan falls within a reasonable, predetermined bound. We do this by using a hill-climbing method to generate district plans that are constrained by the bounds on the metric but also maximize or nearly maximize the number of districts won by a party. In addition, extreme values of the Mean-Median Difference do not necessarily correspond to maps with an extreme number of districts won. Thus, the Mean- Median Difference metric is particularly misleading, as it cannot distinguish more extreme maps from less extreme maps. The other metrics are more nuanced, but when assessed on an ensemble, none perform substantially differently from simply measuring number of districts won by a fixed party. One clear consequence of these results is that they demonstrate the folly of specifying a priori bounds on a metric that a redistricting commission must meet in order to avoid gerrymandering.

physics.soc-ph↗

arXiv2024

Bounds and Bugs: The Limits of Symmetry Metrics to Detect Partisan Gerrymandering

We consider two symmetry metrics commonly used to analyze partisan gerrymandering: the Mean-Median Difference (MM) and Partisan Bias (PB). Our main results compare, for combinations of seats and votes achievable in districted elections, the number of districts won by each party to the extent of potential deviation from the ideal metric values, taking into account the political geography of the state. These comparisons are motivated by examples where the MM and PB have been used in efforts to detect when a districting plan awards extreme number of districts won by some party. These examples include expert testimony, public-facing apps, recommendations by experts to redistricting commissions, and public policy proposals. To achieve this goal we perform both theoretical and empirical analyses of the MM and PB. In our theoretical analysis, we consider vote-share, seat-share pairs (V, S) for which one can construct election data having vote share V and seat share S, and turnout is equal in each district. We calculate the range of values that MM and PB can achieve on that constructed election data. In the process, we find the range of (V,S) pairs that achieve MM = 0, and see that the corresponding range for PB is the same set of (V,S) pairs. We show how the set of such (V,S) pairs allowing for MM = 0 (and PB = 0) changes when turnout in each district is allowed to vary. By observing the results of this theoretical analysis, we can show that the values taken on by these metrics do not necessarily attain more extreme values in plans with more extreme numbers of districts won. We also analyze specific example elections, showing how these metrics can return unintuitive results. We follow this with an empirical study, where we show that on 18 different U.S. maps these metrics can fail to detect extreme seats outcomes.

cs.CY↗

arXiv2021

The Geography and Election Outcome (GEO) Metric: An Introduction

We introduce the Geography and Election Outcome (GEO) metric, a new method for identifying potential partisan gerrymanders. In contrast with currently popular methods, the GEO metric uses both geographic information about a districting plan as well as district-level partisan data, rather than just one or the other. We motivate and define the GEO metric, which gives a count (a non-negative integer) to each political party. The count indicates the number of previously lost districts which that party potentially could have had a 50% chance of winning, without risking any currently won districts, by making reasonable changes to the input map. We then analyze GEO metric scores for each party in several recent elections. We show that this relatively easy to understand and compute metric can encapsulate the results from more elaborate analyses.

physics.soc-ph↗

arXiv2018

Declination as a Metric to Detect Partisan Gerrymandering

We explore the Declination, a new metric intended to detect partisan gerrymandering. We consider instances in which each district has equal turnout, the maximum turnout to minimum turnout is bounded, and turnout is unrestricted. For each of these cases, we show exactly which vote-share, seat-share pairs $(V,S)$ have an election outcome with Declination equal to 0. We also show how our analyses can be applied to finding vote-share, seat-share pairs that are possible for nonzero Declination. Within our analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples of that occurring.

physics.soc-ph↗

arXiv2018

The Efficiency Gap, Voter Turnout, and the Efficiency Principle

Recently, scholars from law and political science have introduced metrics which use only election outcomes (and not district geometry) to assess the presence of partisan gerrymandering. The most high-profile example of such a tool is the efficiency gap. Some scholars have suggested that such tools should be sensitive enough to alert us when two election outcomes have the same percentage of votes going to political party $A$, but one of the two awards party $A$ more seats. When a metric is able to distinguish election outcomes in this way, that metric is said to satisfy the efficiency principle. In this article, we show that the efficiency gap fails to satisfy the efficiency principle. We show precisely how the efficiency principle breaks down in the presence of unequal voter turnout. To do this, we first present a construction that, given any rationals $1/4< V<3/4$ and $0<S<1$, constructs an election outcome with vote share $V$, seat share $S$, and EG = 0. (For instance, one party can get 26% of the vote and anywhere from 1% to 99% of the seats while the efficiency gap remains zero.) Then, for any election with vote share $1/4<V<3/4$, seat share $S$, and EG= 0, we express the ratio $\rho$ of average turnout in districts party $A$ lost to average turnout in districts party $A$ won as a function in only $V$ and $S$. It is well known that when all districts have equal turnout, EG can be expressed as a simple formula in $V$ and $S$; we express the efficiency gap of any election as an equation only in $V, S,$ and $\rho$. We also report on the values of $\rho$ that can be observed in actual elections.

physics.soc-ph↗

arXiv2016

A General Method to Determine Limiting Optimal Shapes for Edge-Isoperimetric Inequalities

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski's theorem on Mixed Volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in the discrete problem approach the shape of the optimal set in the continuous problem as the size of the set grows. The solution is the zonotope defined as the Minkowski sum of the edges of the original graph. We demonstrate the efficacy of this method by revisiting some previously solved classical edge-isoperimetric problems. We then apply our method to some discrete isoperimetric problems which had not previously been solved. The complexity of those solutions suggest that it would be quite difficult to find them using discrete methods only.

math.CO↗

arXiv2016

A Simple Proof of Cauchy's Surface Area Formula

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

math.DG↗

arXiv2013

Edge Boundaries for a Family of Graphs on $\mathbb{Z}^n$

We consider the family of graphs whose vertex set is $\mathbb{Z}^n$ where two vertices are connected by an edge when their $\ell_\infty$-distance is 1. Towards an edge isoperimetric inequality for this graph, we calculate the edge boundary of any finite set $S \subset \mathbb{Z}^n$. This boundary calculation leads to a desire to show that a set with optimal edge boundary has no ``gaps'' in any direction $\epsilon \in \{-1,0,1\}^n, \epsilon \not=0$. We show that one can find a set with optimal edge boundary that does not have gaps in any direction $e_i$ (or $-e_i$) where $e_i$ is the standard basis vector.

math.CO↗

arXiv2012

Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k

We consider the family of graphs whose vertex set is Z^k where two vertices are connected by an edge when their l\infty-distance is 1. We prove the optimal vertex isoperimetric inequality for this family of graphs. That is, given a positive integer n, we find a set A \subset Z^k of size n such that the number of vertices who share an edge with some vertex in A is minimized. These sets of minimal boundary are nested, and the proof uses the technique of compression. We also show a method of calculating the vertex boundary for certain subsets in this family of graphs. This calculation and the isoperimetric inequality allow us to indirectly find the sets which minimize the function calculating the boundary.

math.CO↗

arXiv2010

Spaces of small metric cotype

Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result.

math.MG↗

arXiv2006

An Efficient Approximation of the Traveling Salesman Polytope Using Lifting Methods

For the Traveling Salesman Polytope on n cities T_n, we construct its approximation Q_k, k=1, 2, . . ., n^(1/3) using a projection of a polytope whose number of facets is polynomial in n (of degree linear in k). We show that T_n is contained in Q_k for each k, and that the scaling of Q_k by k/n+O(1/n) is contained in T_n for each k. We show that certain facets of T_n lie on the boundary of Q_k.

math.CO↗

arXiv2006

The computational complexity of convex bodies

We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinatorial optimization.

math.MG↗

arXiv2006

A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope

For a convex body B in a vector space V, we construct its approximation P_k, k=1, 2, . . . using an intersection of a cone of positive semidefinite quadratic forms with an affine subspace. We show that P_k is contained in B for each k. When B is the Symmetric Traveling Salesman Polytope on n cities T_n, we show that the scaling of P_k by n/k+ O(1/n) contains T_n for k no more than n/2. Membership for P_k is computable in time polynomial in n (of degree linear in k). We discuss facets of T_n that lie on the boundary of P_k. We introduce a new measure on each facet defining inequality for T_n in terms of the eigenvalues of a quadratic form. Using these eigenvalues of facets, we show that the scaling of P_1 by n^(1/2) has all of the facets of T_n defined by the subtour elimination constraints either in its interior or lying on its boundary.

math.CO↗