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Socioeconomic segregation is considered one of the main factors behind theemergence of large-scale inequalities in urban areas, and its characterisationis an active area of research in urban studies. There are currently manyavailable measures of spatial segregation, but almost all of them either dependin non-trivial ways on the scale and size of the system under study, or mostlyneglect the importance of large-scale spatial correlations, or depend onparameters which make it hard to compare different systems on equal grounds. Wepropose here two non-parametric measures of spatial variance and local spatialdiversity, based on the statistical properties of the trajectories of randomwalks on graphs. We show that these two quantities provide a consistent andintuitive estimation of segregation of synthetic spatial patterns, and we usethem to analyse and compare the ethnic segregation of large metropolitan areasin the US and the UK. The results confirm that the spatial variance and localdiversity as measured through simple diffusion on graphs provides meaningfulinsights about the spatial organisation of ethnicities across a city, andallows us to efficiently compare the ethnic segregation of urban areas acrossthe world irrespective of their size, shape, or peculiar microscopiccharacteristics.

Quantifying ethnic segregation in cities through random walks