Census Dual Graphs: Properties and Random Graph Models

2026-04-03

AI Summary

This study addresses the lack of systematic understanding regarding the structural properties of census dual graphs in political redistricting and the suitability of random graph models for representing them. Through empirical analysis of dual graphs derived from U.S. counties, census tracts, and block groups, we establish their characteristic “nearly planar, nearly triangulated” nature and establish benchmark values for key graph-theoretic metrics. Integrating graph-theoretic analysis, spatial data processing, and multiple random graph generation techniques—including perturbed grids and Delaunay triangulations—we systematically evaluate how closely various synthetic models replicate the structural features of real-world dual graphs. Our findings clarify the core topological characteristics of these graphs and identify the most effective random graph models for faithfully simulating their structure, thereby providing a foundational framework for both theoretical and empirical research in redistricting algorithms.

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.

Problem

• dual graphs
• political redistricting
• graph properties
• random graph models
• census geography

Innovation

• dual graphs
• random graph models
• political redistricting
• graph properties
• Delaunay triangulation