Implications of Distance over Redistricting Maps: Central and Outlier Maps

TL;DR

This work addresses gerrymandering by introducing a nonpartisan distance framework over redistricting maps. It defines a distance $d_{\Theta}$ between map adjacencies, and uses it to identify a central map (the population medoid) and a reference centroid, enabling outlier detection without relying on election results. The authors prove a Kemeny-inspired justification for the medoid, discuss algorithmic aspects including a linear-time sample medoid method and sample-complexity guarantees for the centroid, and show empirically that gerrymandered maps in NC/PA lie far from the centroid while remedial maps lie near the center. They also prove negative results about estimating the population medoid purely from sampling and demonstrate practical medoid-finding procedures with substantial outlier detection power, highlighting the framework’s potential for transparent, data-driven redistricting assessment and policy-relevant extensions.

Abstract

In representative democracy, a redistricting map is chosen to partition an electorate into districts which each elects a representative. A valid redistricting map must satisfy a collection of constraints such as being compact, contiguous, and of almost-equal population. However, these constraints are loose enough to enable an enormous ensemble of valid redistricting maps. This enables a partisan legislature to gerrymander by choosing a map which unfairly favors it. In this paper, we introduce an interpretable and tractable distance measure over redistricting maps which does not use election results and study its implications over the ensemble of redistricting maps. Specifically, we define a central map which may be considered "most typical" and give a rigorous justification for it by showing that it mirrors the Kemeny ranking in a scenario where we have a committee voting over a collection of redistricting maps to be drawn. We include running time and sample complexity analysis for our algorithms, including some negative results which hold using any algorithm. We further study outlier detection based on this distance measure and show that our framework can detect some gerrymandered maps. More precisely, we show some maps that are widely considered to be gerrymandered that lie very far away from our central maps in comparison to a large ensemble of valid redistricting maps. Since our distance measure does not rely on election results, this gives a significant advantage in gerrymandering detection which is lacking in all previous methods.

Figures (17)

• Figure 1: We are given a hypothetical state consisting of 4 vertices $V=\{v_1,v_2,v_3,v_4\}$ with $M_1$ and $M_2$ being two valid redistricting maps. The adjacency matrices $A_1$,$A_2$, and edit distance interpretation of $\mathop{\mathrm{\mathnormal{d_{\Theta}}}}\nolimits(A_1,A_2)$ are demonstrated. Note that $\mathop{\mathrm{\mathnormal{d_{\Theta}}}}\nolimits(A_1,A_2)=\theta(1,2)+\theta(3,4)+\theta(1,3)+\theta(2,4)$ which is exactly the minimum sum of edge weights that need to be deleted and added to obtain $A_2$ from $A_1$.
• Figure 2: Distance histograms for NC using the unweighted distance measure. Different plots correspond to different seeds. For NC the distances of gerrymandered maps are indicated with red markers whereas the distances of the remedial maps are indicated with green markers (the circle and the X are for 2011 and 2016 enacted maps, respectively).
• Figure 3: NC medoids, each column is for a specific seed. Top row: $A_{\text{closest}}$, Bottom row: $\hat{A}^*$.
• Figure 4: Points $\gamma_1,\gamma_2,\gamma_3$, and $\gamma_4$ all lie on the circle and the angle between any two adjacent points is $90^{\circ}$. Point $\gamma_5$ is a distance $\ell$ from the circle center $c$. The center $c$ is also the origin point of the 2-$d$ plane $(0,0)$.
• Figure 5: The graph shows a hypothetical state. Blue edges indicate that the vertices are adjacent geographically. All vertices have a weight (population) of 1, except for states $\{a_1,b_1,a_4,b_4\}$ which have a weight of $\frac{1}{2}$.

Theorems & Definitions (28)

• Proposition 1
• Proposition 2
• Theorem 5.1
• Remark 1
• Proposition 3
• Proposition 4
• Theorem 5.2
• Theorem 5.3
• Proposition 4
• proof