Scaling laws and applications in urban phenomena

Abstract

Cities are complex systems in which social, economic, and spatial interactions generate robust macroscopic patterns, such as urban scaling laws, which relate variables to population size through power laws; in this work, the underlying geometric and interaction-based mechanisms of these laws are investigated by combining theoretical modeling with empirical analysis applied to a wide set of cities worldwide. A model is proposed based on the probability of interaction between individuals, conditioned by the fractal structure of urban infrastructure and the spatial distribution of the population, allowing the derivation of scaling laws for aggregated socioeconomic variables such as Gross Domestic Product (GDP). Empirically, the fractal dimensions of infrastructure and population are estimated using the Sandbox method, complemented by filters based on Zipf’s law and Pareto fronts to ensure greater statistical robustness; subsequently, Principal Component Analysis (PCA) is employed to identify dominant patterns in the relationships between population, urban area, and built volume. Based on these results, countries are classified into different urban growth regimes, characterized by distinct combinations of horizontal and vertical expansion, revealing relationships with indicators such as GDP per capita and the Human Development Index (HDI). Overall, the findings indicate that urban geometry and social interactions play a central role in determining scaling laws, contributing to a more integrated understanding of the spatial organization and development of cities.