Residential segregation in urban areas refers to the spatial clustering of households based on characteristics such as ethnicity, income, language, or others. While often linked to adverse social outcomes, this phenomenon can also strengthen community ties, which is especially crucial in vulnerable contexts like migration or poverty, as the emergence of social identities and trust within homogeneous communities enhances social capital and group cohesion. However, when regions become overly homogeneous, the resulting territorial separation limits interactions between groups, hindering broader social integration,  and exacerbates inequalities in resource and opportunity access, reinforcing urban divisions.

In a previous work [1], we proposed that housing location quality influences individuals' preference for similar neighbors through a modified Schelling segregation model [2]. We examined how different weight functions representing territory valuations affect the scale of emerging segregation patterns, finding that smooth variations lead to large-scale segregation, while highly oscillating weights produce fragmented, smaller clusters.

This work focuses on the characterization of ethnicity-based segregation scales in urban environments. We construct a spatial city-network from census data, where nodes represent census tracts and links connect adjacent tracts, each identified by its majority ethnicity. Using random walkers traversing this network, we define metrics capturing progressive scales of diversity—from homogeneous to fully diverse encounters. Unlike previous approaches that relied on simulations of the walkers [3], our analytical methods allow for the numerical assessment of the probability distribution of classes encountered by a random walker at each time step.

As the walker moves through the network, it encounters different ethnicities until it eventually visits all the classes present, denoted by C. For each node, we define tk​ as the k-th expected encounter time, indicating the expected time when the walker first encounters a k-th new ethnicity, achieving a k/C diversity level. We focus on two key scales: t2​, the homogeneity scope index, which measures the effort in time to meet a second class and leave the homogeneous origin ethnic group; and tC​, the complete diversity scope index, that of the encounter of all ethnic groups. Additionally, we introduce an effectiveness metric for distinguishing locations with similar expected encounter times but differing mean numbers of classes encountered at those times.

Our metrics enable the analysis of segregation patterns and the diversity levels experienced by different ethnic groups and areas within the city. We apply these metrics in lattice toy models, comparing configurations with different spatial patterns and the same number of nodes per class, and real urban networks of the Rio de Janeiro area, using 2010 Brazilian census data. Future work will extend this analysis to different cities and refine the metrics to consider the full ethnic distribution at each node, rather than the majority ethnicity.

[1] V. Arcon, I. Caridi , J.P. Pinasco , and P. Schiaffino, Communications in Nonlinear Science and Numerical Simulation 120, 107140 (2023).

[2] T. Schelling, Journal of Mathematical Sociology 1(2) 143-186 (1971).

[3] S. Sousa, V. Nicosia, Quantifying ethnic segregation in cities through random walks, Nature Communications 13 (2022)