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Urban scaling laws summarize how urban attributes evolve with city size. Recent criticism questions notably the aggregate view of this approach, which leads to neglecting the internal structure of cities. This is all the more relevant for housing prices due to their important variations across space. Based on a dataset compiling millions of real estate transactions over the period 2017-2021, we investigate the regularities of the radial (center-periphery) profiles of housing prices across cities, with respect to their size. Results are threefold. First, they corroborate prior findings in the urban scaling literature stating that largest cities agglomerate higher housing prices. Second, we find that housing price radial profiles scale in three dimensions with the power 1/5 of city population. After rescaling, great regularities between radial profiles can be observed, although some locational amenities have a significant impact on prices. Third, it appears that our rescaled profiles approach fails to explain housing price variations in the city center across cities. In fact, prices near the city center rise much faster with city size than those in the periphery. This has strong implications for low-income households seeking homeownership, because prohibitive prices in the center may contribute to pushing them out into peripheral locations.

Do cities have just one or several centers? Studies performing radial or monocentric analyses of cities are usually criticised by researchers stating that cities are actually polycentric, and this has been well known for a long time. Reversely, when cities are studied independently of any center, other researchers will wonder how the variables of interest evolve with the distance to the center, because this distance is known to be a major determinant at the intra-urban scale. Both monocentric and polycentric formalisms have been introduced centuries (respectively, decades) ago for the study of urban areas, and used both on the empirical and the theoretical side in different disciplines (economics, geography, complex systems, physics...). The present work performs a synthesis of both viewpoints on cities, regarding their use in the literature, and explores with data on European urban areas how some cities considered to be the most polycentric in Europe compare to more standard cities when studied through a combination of radial analysis and scaling laws., Comment: Compendium of Urban Complexity. Chapter 7. 16 pages

The difference of a city's artificial land use (ALU) radial profile to the average ALU profile is examined for 585 European cities. Using Urban Atlas 2012 data, a radial (or monocentric) approach is used to calculate a city's land use profile in relation to distance to the city centre. A scaling law is used which controls for city size and population. As a consequence, cities of varying degrees of size can be contrasted in a comparable way. Utilising the mean ALU profile for the entire sample of 585 cities, the difference to the mean profile is calculated for each city. Using these differences allows us to examine heterogeneity of the ALU across European cities but also examine these differences within cities. We utilise city groupings by city size and country to attempt to understand these differences. Combining Urban Atlas and Corine Land Cover data, the impact of water on the ALU profiles is examined. A city classification is also introduced which considers the difference to the average curve. Ordering methods are used to visualise cities within these classifications. Results highlight the level of heterogeneity between cities. Removing water, we can see that the cities with the highest levels of water have a higher level of ALU on average. Spain and France are found to have contrasting levels of ALU, Spanish cities having below average ALU and France above average. Using seriation techniques enables us to group and order cities into a typology which can be used to benchmark cities., Comment: 33 pages, 11 figures