  Cities are complex systems, their complexity manifests itself through
fractality of their spatial structures and by power law distributions (scaling)
of multiple urban attributes. Here we report on the previously unreported
manifestation of urban complexity -- scaling in patterns of residential racial
segregation. A high-resolution racial grid of a city is segmented into racial
enclaves which are patches of stationary racial composition. Empirical PDFs of
patch areas and population counts in 41 US cities were analyzed to reveal that
these variables have distributions which are either power laws or approximate
power laws. Power law holds for a pool of all patches, for patches from
individual cities, and patches restricted to specific racial types. The average
value of the exponent is 1.64/1.68 for area/population in 1990 and 1.70/1.74 in
2010. The values of exponents for type-specific patches vary, but variations
had decreased from 1990 to 2010. We have also performed a multifractal analysis
of patterns formed by racial patches and found that these patterns are
monofractal with average values of fractal dimensions in the 0.94-1.81 range
depending on racial types and the year of analysis. Power law distribution of
racial patch sizes and a fractal character of racial patterns present
observable and quantifiable constraints on models of racial segregation. We
argue that growth by preferential attachment is a plausible mechanism leading
to observed patterns of segregation.
