Homogenization scales in residential segregation through random walkers

This paper proposes metrics to characterize and quantify the spatial scale of ethnic-based residential segregation using random walkers on city networks built from census data. We analyze the diversity of encounters experienced by the walkers over time. We provide a probabilistic framework and numerical methods to compute the probability of encountering a certain number of ethnic groups at each time, and use it to define two indices: the homogeneity scope, which represents the time when it becomes more likely or expected to leave the homogeneous area of the starting ethnic group, and the complete diversity scope, the time when it becomes more likely or expected to encounter all ethnic groups. These indices are also generalized to reach specific intermediate levels of diversity. We apply the methodology to lattice toy models and a case study in Rio de Janeiro. Our approach identifies areas and groups of high homogeneity, highlighting opportunities to enhance social interaction through improved connectivity. The proposed framework can be adapted for alternative definitions of city networks, broadening its applicability to various research interests.