Measuring Segregation via Analysis on Graphs

TL;DR

This paper analyzes how graph structures influence the interpretation of Moran's I, a measure of segregation, using theoretical insights and real-world geographic data.

Contribution

It introduces a comprehensive analysis of Moran's I on graphs, highlighting the impact of graph structure and exploring alternative matrices for better interpretation.

Findings

Graph structure significantly affects Moran's I interpretation

Laplacian and doubly-stochastic matrices connect to Fourier analysis and random walks

Theoretical insights are supported by synthetic and real data examples

Abstract

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, which in our language is a score associated to a real-valued function on a graph, computed with respect to a spatial weight matrix such as the adjacency matrix associated to the geographic units that tile a city. Our results characterizing the extremal behavior of I illustrate the important role of the underlying graph structure, especially the degree distribution, in interpreting the score. In addition to the standard spatial weight matrices encoding unit adjacency, we consider the Laplacian L and a doubly-stochastic approximation M. These alternatives allow us to connect I to ideas from Fourier analysis and random walks. We offer illustrations of our theoretical results with a mix of stylized synthetic examples and real geographic/demographic data.