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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

Seen from a satellite, observing land use in the daytime or at night, most cities have circular shapes, organised around a city centre. A radial analysis of artificial land use growth is conducted in order to understand what the recent changes in urbanisation are across Europe and how it relates to city size. We focus on the most fundamental differentiation regarding urban land use: has it been artificialised for human uses (residence or roads for instance) or is it natural, or at least undeveloped? Using spatially detailed data from the EU Copernicus Urban Atlas, profiles of artificial land use (ALU) are calculated and compared between two years, 2006 and 2012. Based on the homothety of urban forms found by Lemoy and Caruso (2018), a simple scaling law is used to compare the internal structure of cities after controlling for population size. We firstly show that when using the FUA definition of cities, a kind of Gibrat's law for land use appears to hold. However, when we examine cities internally, this is no longer clear as there are differences on average between city size categories. We also look at further city groupings using regions and topography to show that artificial land use growth across European cities is not homogeneous. Our findings have important implications relative to the sustainability of cities as this evidence is pointing towards increasing urban sprawl and stagnant growth in urban centres across cities of all sizes. It also has theoretical implications on the nature of sprawl and its scaling with city size., Comment: 20 pages, 7 figures, draft chapter for "Handbook on Entropy, Complexity and Spatial Dynamics: A Rebirth of Theory?" edited by Aura Reggiani, Laurie A. Schintler and Danny Czamanski, forthcoming 20xx, Edward Elgar Publishing Ltd

The scaling of urban characteristics with total population has become an important research field since one needs to better understand the challenges of urban densification. Yet urban scaling research is largely disconnected from intra-urban structure. In contrast, the monocentric model of Alonso provides a residential choice-based theory to urban density profiles. However, it is silent about how these profiles scale with population, thus preventing empirical scaling studies to anchor in a strong micro-economic theory. This paper bridges this gap by introducing power laws for land, income and transport cost in the Alonso model. From this augmented model, we derive the conditions at which the equilibrium urban structure matches recent empirical findings about the scaling of urban land and population density profiles in European cities. We find that the Alonso model is theoretically compatible with the observed scaling of population density profiles and satisfactorily represents European cities. This compatibility however challenges current empirical understanding of wage and transport cost elasticities with population, and requires a scaling of the housing land profile that is different from the observed. Our results call for revisiting theories about land development and housing processes as well as the empirics of agglomeration benefits and transport costs., Comment: 37 pages, 7 figures

In this paper we analyse the profile of land use and population density with respect to the distance to the city centre for the European city. In addition to providing the radial population density and soil-sealing profiles for a large set of cities, we demonstrate a remarkable constancy of the profiles across city size. Our analysis combines the GMES/Copernicus Urban Atlas 2006 land use database at 5m resolution for 300 European cities with more than 100.000 inhabitants and the Geostat population grid at 1km resolution. Population is allocated proportionally to surface and weighted by soil sealing and density classes of the Urban Atlas. We analyse the profile of each artificial land use and population with distance to the town hall. In line with earlier literature, we confirm the strong monocentricity of the European city and the negative exponential curve for population density. Moreover, we find that land use curves, in particular the share of housing and roads, scale along the two horizontal dimensions with the square root of city population, while population curves scale in three dimensions with the cubic root of city population. In short, European cities of different sizes are homothetic in terms of land use and population density. While earlier literature documented the scaling of average densities (total surface and population) with city size, we document the scaling of the whole radial distance profile with city size, thus liaising intra-urban radial analysis and systems of cities. In addition to providing a new empirical view of the European city, our scaling offers a set of practical and coherent definitions of a city, independent of its population, from which we can re-question urban scaling laws and Zipf's law for cities., Comment: 19 pages, 11 figures

Thermal noise in a cellular automaton refers to a random perturbation to its function which eventually leads this automaton to an equilibrium state controlled by a temperature parameter. We study the 1-dimensional majority-3 cellular automaton under this model of noise. Without noise, each cell in this automaton decides its next state by majority voting among itself and its left and right neighbour cells. Transfer matrix analysis shows that the automaton always reaches a state in which every cell is in one of its two states with probability 1/2 and thus cannot remember even one bit of information. Numerical experiments, however, support the possibility of reliable computation for a long but finite time., Comment: 12 pages, 4 figures

In order to use the advanced inference techniques available for Ising models, we transform complex data (real vectors) into binary strings, by local averaging and thresholding. This transformation introduces parameters, which must be varied to characterize the behaviour of the system. The approach is illustrated on financial data, using three inference methods -- equilibrium, synchronous and asynchronous inference -- to construct functional connections between stocks. We show that the traded volume information is enough to obtain well known results about financial markets, which use however the presumably richer price information: collective behaviour ("market mode") and strong interactions within industry sectors. Synchronous and asynchronous Ising inference methods give results which are coherent with equilibrium ones, and more detailed since the obtained interaction networks are directed., Comment: 14 pages, 6 figures

We introduce a new local search algorithm for satisfiability problems. Usual approaches focus uniformly on unsatisfied clauses. The new method works by picking uniformly random variables in unsatisfied clauses. A Variable-based Focused Metropolis Search (V-FMS) is then applied to random 3-SAT. We show that it is quite comparable in performance to the clause-based FMS. Consequences for algorithmic design are discussed., Comment: 7 pages, 3 figures

We consider a simple stochastic model of a urban rental housing market, in which the interaction of tenants and landlords induces rent fluctuations. We simulate the model numerically and measure the equilibrium rent distribution, which is found to be close to a lognormal law. We also study the influence of the density of agents (or equivalently, the vacancy rate) on the rent distribution. A simplified version of the model, amenable to analytical treatment, is studied and leads to a lognormal distribution of rents. The predicted equilibrium value agrees quantitatively with numerical simulations, while a qualitative agreement is obtained for the standard deviation. The connection with non-equilibrium statistical physics models like ratchets is also emphasized., Comment: 12 pages, 5 figures, to appear in J. Stat. Mech

In statistical physics, the conservation of particle number results in the equalization of the chemical potential throughout a system at equilibrium. In contrast, the homogeneity of utility in socio-economic models is usually thought to rely on the competition between individuals, leading to Nash equilibrium. We show that both views can be reconciled by introducing a notion of chemical potential in a wide class of socio-economic models, and by relating it in a direct way to the equilibrium value of the utility. This approach also allows the dependence of utility across the system to be determined when agents take decisions in a probabilistic way. Numerical simulations of a urban economic model also suggest that our result is valid beyond the initially considered class of solvable models., Comment: 6 pages, 3 figures, final version

Linking the microscopic and macroscopic behavior is at the heart of many natural and social sciences. This apparent similarity conceals essential differences across disciplines: while physical particles are assumed to optimize the global energy, economic agents maximize their own utility. Here, we solve exactly a Schelling-like segregation model, which interpolates continuously between cooperative and individual dynamics. We show that increasing the degree of cooperativity induces a qualitative transition from a segregated phase of low utility towards a mixed phase of high utility. By introducing a simple function which links the individual and global levels, we pave the way to a rigorous approach of a wide class of systems, where dynamics is governed by individual strategies., Comment: 5 pages, 5 figures, submitted to PNAS