Using multifractal analysis to quantify and visualize a social phenomenon of racial segregation

Spatial social systems are complex structures that have frequently been described using multifractal analysis (MFA). One phenomenon conspicuously absent from the list of social topics analyzed using MFA is residential racial segregation. Customarily, segregation data is distributed in an aggregated format as a list of areal units, each quantified by its racial composition. Such a format is not amenable to MFA. However, recently, 2020 US spatial racial data has become available in the form of a high-resolution grid of monoracial cells, called the racial landscape. This paper leverages the racial landscape for the detailed quantification and mapping of binary segregation patterns using MFA. The residency pattern is characterized by a multifractal spectrum function f(α), where α(x,y) represents the local singularity exponent characterizing the scaling behavior or density variation of the pattern at location (x,y), and f(α) indicates the fractal dimension of the subset of the pattern where the singularity exponent takes a given value α. Different values of α(x,y) are visually perceived as different levels of pattern’s gappiness. In the context of biracial populations (a focus sub-population and the remaining population), the gappiness of the focus population’s pattern is intrinsically linked to its segregation from the remaining population. The paper provides a comprehensive description of the methodology (discrete multifractal spectrum) , illustrated by the residency pattern of the Black sub-population in the central region of Washington, DC. Further, the methodology is demonstrated using a sample of patterns of the Black sub-population in fourteen large U.S. cities. By numerically describing each pattern through a multifractal spectrum, the fourteen patterns are clustered into three distinct categories, each having unique characteristics. Maps of local gappiness and segregation for each city are provided to show the connection between the nature of the multifractal spectrum and the corresponding residency and segregation patterns. This method offers an excellent quantification and visualization of biracial segregation patterns within U.S. cities.