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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