Multiplicative random cascade models of multifractal urban structures

Abstract

Spatial structures of urban systems are best described as multifractals. To provide insight on a hidden mechanism responsible for such scaling we investigate the multiplicative random cascade (MRC) model of urban spatial structure. MRC is a multiplicative process capable of producing multifractal 2D patterns, which is expressed in a closed analytical form. We use street intersection points pattern (SIPP) data as a proxy of urban structure. Renyi’s generalized dimensions (RGD) spectra, probability distributions of intersections densities, and urban layouts generated by the MRC model are compared to the same descriptors extracted from SIPP data for six U.S. metropolitan statistical areas (MSAs). We found that the RGD spectrum of the best-fit MRC model closely matches the observed spectrum in all cases. We also found that CDF of models’ density distributions do not match closely observed density distributions, although, in some cases, differences are not large. Observed density distributions have forms that are close to the log-normal distribution which suggests that a hidden process is multiplicative. Finally, we found that the MRC model is broadly compatible with the observed urban layouts. Given our findings, we suggest that the hidden process of urban scaling is the preferential attachment dynamics — development is attracted to areas in proportion to the degree of their existing development.