Quantifying the Spatial and Demographic Scales of Segregation

TL;DR

This paper tackles the problem of identifying at which spatial and demographic scales segregation operates. It develops a joint spatial-demographic decomposition of the Divergence Index, $I(X,S)$, grounded in Shannon entropy and KL divergence, to quantify between- and within-scale contributions. Through a case study of England and Wales using 2021 census data and a Croydon example, it demonstrates how different scales capture different segregation patterns and how within-group segregation emerges. The framework offers a flexible, general tool for determining relevant scales and informing multi-scale urban models and policy analyses. Overall, it enables objective scale selection and cross-domain applications in networks beyond residential geography.

Abstract

Cities around the world exhibit residential segregation along ethnic, religious, socioeconomic, and other social divides. As high-resolution demographic and spatial data become widely available, decomposable measures have become essential tools for understanding the multi-scale structure of segregation. In this paper, we introduce a framework that quantifies how much segregation is expressed at different spatial and demographic scales. Extending existing spatial decompositions, our approach also measures the internal segregation of broad demographic groups, enabling a joint assessment of geographic and demographic structure. We illustrate the usefulness of this framework with a case study of ethnic residential segregation in England and Wales. Our methods provide a flexible, general tool for identifying the scales at which segregation operates and for guiding multi-scale models of urban systems.

Figures (15)

• Figure 1: Method schematic. This fictional setting has four cities (A-D) aggregated into two districts. Cities A and B form the North district and C and D form the South. There are four demographic identities at the fine scale (green), aggregated into two broad groups (orange; hatched and solid). The bar charts plot the demographic compositions of each city and district. Fine-graining geography from the (fine, district) scale adds 22.7% of the total information at the (fine, city) scale. Dis-aggregating the North and South districts account for $1.8\%$ and $20.9\%$ respectively, capturing that the South district combines cities with very different demographic compositions. Fine-graining demography from the (broad, city) scale also adds around 22.7% of the total information. Splitting the hatched and solid groups contribute $18.5\%$ and $4.2\%$ respectively, indicating that the hatched group is more internally segregated. That we lose roughly the same amount of information in both coarse-grainings is peculiar to this simple example and is not generally the case.
• Figure 2: Segregation at different scales in England and Wales. The information captured as a fraction of $I(\textnormal{fine}, \textnormal{OA})$ (y-axis) at different spatial scales (x-axis). The colour indicates demographic scale (green for fine and orange for broad). For either demographic scale, as we fine-grain spatially (left to right), we annotate the dashed lines with the information added at each successive stage as a percentage of the maximum. For instance, going from MSOAs to LSOAs at the broad demographic scale increases the information by 7% of $I(\textnormal{broad}, \textnormal{OA})$. The 'E&W' spatial scale aggregates all the OAs into one region and contains no information. For each spatial scale $\tilde{\mathcal{S}}$, as we fine-grain demographically (bottom to top), we annotate the dashed lines with the information added as a fraction of $I(\textnormal{fine}, \tilde{S})$. The maximally coarse-grained demographic scale with no information corresponds to considering all ethnicities to be identical.
• Figure 3: Within-district segregation against diversity. The within-district segregation $\bar{I}_{\tilde{s}}^{\textnormal{broad}}$ (y-axis) against diversity (x-axis) at the broad demographic scale. Each point corresponds to a district. The wide range of segregation for relatively diverse districts shows that diversity does not necessarily lead to segregation. The plot is qualitatively identical at the fine demographic scale (Fig. \ref{['fig:scatter_segregation_vs_diversity_fine_lads']}).
• Figure 4: Spatial distribution of ethnic groups in Harrow and Luton. Each subplot depicts the distribution of a broad ethnic group within Harrow (top) and Luton (bottom) at the OA scale. The colour of an OA indicates the fraction of its population that belongs to the ethnic group. While they have similar demographic compositions, Harrow and Luton show very different levels of segregation.
• Figure 5: Spatial distribution of ethnic groups in Croydon. Each subplot depicts the distribution of an ethnic group at the OA scale. The colour of an OA indicates the fraction of its population that belongs to the ethnic group. We annotate each subplot with the population share of the ethnic group in Croydon. The box on the left has the distribution of the broad Asian, Black, and White groups, with the colour scale at the bottom. Each row depicts the distribution of the largest fine-scale ethnicities of the corresponding group with colour scales on the right. We truncate colour scales to make variation in the distributions of the minorities visible.