Understanding U.S. Racial Segregation Through Persistent Homology

TL;DR

This paper applies persistent homology from algebraic topology to analyze and classify the unique spatial patterns of racial segregation across 112 U.S. cities, revealing common structural shapes.

Contribution

It introduces a novel application of persistent homology to urban racial segregation, enabling the identification of shared segregation patterns among cities.

Findings

Cities can be grouped based on similar segregation shapes.

Persistent homology captures meaningful structural differences in segregation.

Clusters correlate with demographic and geographic characteristics.

Abstract

Racial segregation is a widespread social and physical phenomenon present in every city across the United States. Although prevalent nationwide, each city has a unique history of racial segregation, resulting in distinct "shapes" of segregation. We use persistent homology, a technique from applied algebraic topology, to investigate whether common patterns of racial segregation exist among U.S. cities. We explore two methods of constructing simplicial complexes that preserve geospatial data, applying them to White, Black, Asian, and Hispanic demographic data from the U.S. census for 112 U.S. cities. Using these methods, we cluster the cities based on their persistence to identify groups with similar segregation "shapes". Finally, we apply cluster analysis techniques to explore the characteristics of our clusters. This includes calculating the mean cluster statistics to gain insights into the demographics of each cluster and using the Adjusted Rand Index to compare our results with other clustering methods.