Urban Complexity through the Spatial Structure of Amenities: An Information-Theoretic Approach

Abstract Urban amenities represent the everyday infrastructure through which people organize their lives, from schooling and shopping to leisure, health care, and transport. The way these amenities are distributed across space shapes residential choice and neighborhood satisfaction, a relationship widely discussed in the literature on urban quality of life. Counting amenities nevertheless captures only part of this picture, since the complexity of a living space also depends on which kinds of amenities coexist (composition) and how they are arranged relative to each other (configuration). This study explores the extent to which information-theoretic measures can capture both dimensions and be translated into a comparable measure of urban liveliness. We combine OpenStreetMap points of interest with the official geographies of Statistics Netherlands to characterize 342 municipalities, 3,324 districts, and 13,943 neighborhoods, organizing amenities into 13 main and 53 sub-categories under three filter levels. Shannon entropy is compared with two spatial entropy measures, namely Leibovici and Altieri entropy, on synthetic point patterns and on the empirical data. Our findings reveal that Shannon entropy is insensitive to spatial arrangement, Leibovici entropy responds only when clustering follows amenity categories, and Altieri entropy responds to clustering in both cases. Within each filter level the three measures are strongly correlated, whereas the choice of which amenities to include alters the results considerably, especially for municipalities. Weighting spatial entropy by the degree of urbanity yields a measure of urban liveliness whose distribution becomes bimodal, separating urban from rural districts. Local Moran's I further reveals hot spots of liveliness in the large Dutch cities and cold spots across much of the north, with 36.83\% of districts being statistically significant. The results are condensed into an interactive dashboard that identifies areas with comparable amenity profiles, offering a transparent tool for residential choice and spatial planning.