Scale-free Points-of-Interest Distribution in a City Emerging from Homogeneous Poissonian-point Processes

Urban systems often exhibit scale-invariant properties, with power-law distributions observed in various spatial and temporal patterns of human behavior. A prominent example is the distribution of commercial activities and other Points of Interest (POIs) across cities. However, the mechanisms by which such heavy-tailed behaviors emerge from local urban dynamics remain poorly understood. In this work, we demonstrate that global inhomogeneity in the spatial distribution of POIs can arise from the aggregation of locally homogeneous processes. Using Foursquare data from the city of Bologna, we show that POI distributions exhibit clear power-law scaling when analyzed at city scale. We develop a theoretical framework in which this behavior naturally emerges from spatial clusters defined by shared intensity levels across disjoint areas, rather than spatial contiguity. By analytically and empirically linking these local processes to the observed global distribution, we provide a generative explanation for the emergence of scale-free patterns in urban commercial structure. To further relax the assumptions underlying the purely spatial model, and to account for the empirical observation that areas with similar activity intensity can be spatially disjoint, we introduce a hybrid hierarchical approach that combines spatial clustering with statistical heterogeneity across regions of comparable density, modeled via Poisson mixtures. This enables us to capture real-world deviations from local regularity while preserving interpretability. Our findings highlight a key insight: complex global phenomena in cities can arise from the spatial superposition of simple, locally uniform dynamics. This connection between micro-level homogeneity and macro-scale complexity offers new tools for interpreting, modeling, and classifying urban space.