Measuring Segregation via Analysis on Graphs

In this paper, we use analysis on graphs to study quantitative measures of segregation. We focus on a classical statistic from the geography and urban sociology literature known as Moran's I score. Our results characterizing the extremal behavior of I illustrate the important role of the underlying graph geometry and degree distribution in interpreting the score. This leads to caveats for users about the usefulness of I for making comparisons across different localities. We present a novel random walk interpretation of I, connecting the measured level of segregation to the rate of variance reduction via diffusion. Concerns about interpretability of I lead us to propose an H^1-norm on graphs as an alternative measure of segregation, enabling connections with the literature on community detection in networks and high- and low-frequency graph Fourier modes. Our methods outline a new program for the study of geographic segregation that is motivated by time-frequency analysis on graphs. We offer illustrations of our theoretical results with a mix of stylized synthetic examples and graphs derived from real geographic and demographic data.