Quantifying hierarchical patterns in population distribution using local fractal dimensions

Fractal dimension (FD) has been widely used to measure the spatial complexity of human settlements, primarily through 2D analyses of urban form. While 2D FD captures space-filling properties consistent with hierarchical organization such as that in central place theory, it does not take into account the highly uneven and multi-scalar nature of population density. A 3D FD incorporating population counts offers additional structural information. Therefore, this study investigated how 3D FD relates to settlement structure. First, we analyzed theoretical settlement configurations—including regular, clustered, and hierarchical patterns—that can be systematically morphed using structural parameters. Hierarchical systems are constructed following central place theory and population distribution consistent with the rank–size rule, and they are controlled by parameters such as the branching factor (K-value) and degree of metropolitan dominance. We estimated 3D FD for both these configurations to demonstrate how FD varies systematically with changes in hierarchical organization. Results showed that hierarchical patterns consistently yielded FD values greater than 2, while regular or non-hierarchical patterns approached or remained at 2. While FD cannot be defined in the most sparsely populated areas, rural areas exhibited FD greater than 2 as well as the highest FD. Second, building upon these insights, we estimated local 3D FD from gridded population data for the United States using a moving-window approach (400 km × 400 km). The resulting maps revealed substantial regional variation in settlement structure. Large metropolitan regions associated with polycentricity exhibited smaller FD values than smaller cities, while the FD of rural areas varied considerably. Additionally, we measured the fractal depth and the range of scales at which fractal properties are present. Rural areas generally have thinner fractal depth, although those in the arid West were even thinner. These findings provide a structurally grounded interpretation of 3D FD and demonstrate its utility for analyzing settlement systems relevant to the whole urban−rural continuum.