Singularity cities

Abstract

We propose an upgraded gravitational model which provides population counts beyond the binary (urban/non-urban) city simulations. Numerically studying the model output, we find that the radial population density gradients follow power-laws where the exponent is related to the preset gravity exponent γ. Similarly, the urban fraction decays exponentially, again determined by γ. The population density gradient can be related to radial fractality and it turns out that the typical exponents imply that cities are basically zero-dimensional. Increasing the gravity exponent leads to extreme compactness and the loss of radial symmetry. We study the shape of the major central cluster by means of another three fractal dimensions and find that overall its fractality is dominated by the size and the influence of γ is minor. The fundamental allometry, between population and area of the major central cluster, is related to the gravity exponent but restricted to the case of higher densities in large cities. We argue that cities are shaped by power-law proximity. We complement the numerical analysis by economics arguments employing travel costs as well as housing rent determined by supply and demand. Our work contributes to the understanding of gravitational effects, radial gradients, and urban morphology. The model allows to generate and investigate city structures under laboratory conditions.

On the influence of density and morphology on the Urban Heat Island intensity

“…Simulations with mono-centric urban clusters. We repeat the analysis for 50 clusters (generated by a gravitational urban growth model 39,40 , see "Methods" section) that vary by size and compactness. In the Supplementary Fig.…”

“…The investigated urban clusters do exhibit a range of compact or scattered shapes ( Supplementary Fig. 1), but more complex spatial features, e.g., as captured by the fractal dimension 31,46 , can hardly be analysed based on those clusters since their fractal dimension covers a comparably small range 39 . Accordingly, we perform further simulations with more extreme urban forms that are beyond real life cities.…”

“…where γ is the main parameter, d i,j is the Euclidean distance from cell i to j, v j is the value in cell j, M i is a site-specific normalisation constant 39,40 , and G is another parameter determining the overall rate of growth. Starting with a single v = 1 cell in the centre, the model is run iteratively incrementing the counts v i → v i + 1, if z < pi , where z is a random number between 0 and 1.…”

Quantitative evidence for leapfrogging in urban growth

“…Further, we apply the regression q ( d ), Equation (2), to the binned probabilities. Now we simulate structures resembling settlements with a gravitational model (Li et al., 2021; Rybski et al., 2013) and study them the same way as the real-world data, that is, calculate q d and fit q ( d ). The gravitational model itself also involves a probability which is fundamentally different since it is based on the distance to all urban pixels and not only to the closest one.…”

“…Moreover, by using the shortest distance to urban land cover we distinguish land cover conversion which occurs close to existing settlements from that which takes place further away. The results are complemented by simulations with a numerical model that reproduces gravitational urban growth (Li et al., 2021; Rybski et al., 2013). The simulations support the empirical findings.…”