Census Dual Graphs: Properties and Random Graph Models

TL;DR

This paper investigates the properties of dual graphs used in political redistricting, characterizes their structure across the US, and compares them with various random graph models to inform future algorithm development.

Contribution

It provides the first comprehensive analysis of dual graph properties in US political redistricting and evaluates which random graph models best resemble these real-world graphs.

Findings

Dual graphs are nearly planar and nearly triangulated.

Certain random graph models closely match the properties of dual graphs.

This foundational work aids in developing algorithms for political redistricting.

Abstract

In the computational study of political redistricting, feasibility necessitates the use of a discretization of regions such as states, counties, and towns. In nearly all cases, researchers use a dual graph, whose vertices represent small geographic units (such as census blocks or voting precincts) with edges for geographic adjacency. A political districting plan is a partition of this graph into connected subgraphs that satisfy certain additional properties, such as connectedness, compactness, and equal population. Though dual graphs underlie nearly all computational studies of political redistricting, little is known about their properties. This is a unique graph class that has been described colloquially as `nearly planar, nearly triangulated,' but thus far there has been a lack of evidence to support this description. In this paper we study dual graphs for counties, census tracts, and census block groups across the United States in order to understand and characterize this graph class. We also consider several random graph models (most based on randomly perturbing grids or Delauney triangulations of random point sets), and determine which most closely resemble dual graphs under key metrics. This work lays an initial foundation for understanding and modeling the properties of dual graphs; this will provide invaluable insight to researchers developing algorithms using them to understand, assess, and quantify the properties of political districting plans.