Modeling Urban Morphology by Unifying Diffusion-Limited Aggregation and Stochastic Gravitation

Abstract

More than 30 years ago, Diffusion-Limited Aggregation (DLA) has been studied as mechanism to generate structures sharing similarities with real-world cities. Recently, a stochastic gravitation model (SGM) has been proposed for the same purpose but representing a completely different mechanism. Both approaches have advantages and disadvantages, while e.g. the dendrites emerging via DLA are visually very different from real-world cities, the SGM does not preserve undeveloped locations close to the city center. Here we propose a unification of both mechanisms, i.e. a particle moves randomly and turns into an urban site with a probability that depends on the proximity to already developed sites. We study the cluster size distributions of the structures generated by both models and find that SGM generates more balanced distributions. We also propose a way to assess to which extent the largest cluster is a primate city and find that in both models, beyond certain parameter value, the size of the largest cluster becomes inconsistent with being drawn from the same distribution of remaining clusters.Various models have been proposed to generate structures that share similarities with real-world cities. The correlated percolation model combines an exponential radial gradient with correlated random numbers (Makse, Havlin, and Stanley 1995). However, the largest city is too big and consequently omitted when Zipf's law for cities, i.e. a power-law size distribution with specific exponent, is validated. The model by Schweitzer & Steinbrink (1998) is based on two processes, the emergence of new clusters and cluster growth. In this model a similar problem is encountered -due to coagulation the largest cluster dominates the growth process. The authors avoid this by excluding the largest cluster from growth.

Spatiotemporal evolution of urban populations and housing: A dynamic utility-driven market-mediated model

“…Consequently, physical processes that produce fractal structures in other contexts have been used to try to reproduce urban morphologies. These include city formation via diffusion-limited-aggregation (DLA) of immigrants to a city, sometimes with modified probabilities of aggregation, preferential attachment, or directed percolation [12,26]; models of economically driven migration [13]; self-organization that can result in fractal filling of space [14]; agent-based models based on agglomeration principles such preferential attachment [6,9,15]; or network models that take into account the structure of transport networks in conjunction with urban structure [6,9]. Finally, transport models consider the joint spatial structure of economic and social aspects in parallel with the structure of the transport networks that support spatial interactions [6].…”