How the geometry of cities determines urban scaling laws

Abstract

Urban scaling laws relate socio-economic, behavioural and physical variables to the population size of cities. They allow for a new paradigm of city planning and for an understanding of urban resilience and economics. The emergence of these power-law relations is still unclear. Improving our understanding of their origin will help us to better apply them in practical applications and further research their properties. In this work, we derive the basic exponents for spatially distributed variables from fundamental fractal geometric relations in cities. Sub-linear scaling arises as the ratio of the fractal dimension of the road network and of the distribution of the population embedded in three dimensions. Super-linear scaling emerges from human interactions that are constrained by the geometry of a city. We demonstrate the validity of the framework with data from 4750 European cities. We make several testable predictions, including the relation of average height of cities and population size, and the existence of a critical density above which growth changes from horizontal densification to three-dimensional growth.

Keywords: agglomeration effects; fractal geometry; population size; spatial networks; urban scaling.

Figure 1.

(a) Street network in a section of the city of size L. The length of the street network with fractal dimension d_i expands with the linear scale L as $\ell \propto L^{d_i}$. (b) Buildings are located along the street network and are attached to it. Since people live and work mostly in buildings, the fractal dimension of the ‘projected population’ (the actual population fractal projected onto the two-dimensional surface, where streets are embedded) should have a similar fractal dimension $d_{\hspace{0.17em}{p}_{\hspace{0.17em}p}}=d_i$. This is shown in (c), where both dimensions show a strong linear correlation for every city in the UK, with a slope of 1. (d) If all buildings were to have the same height, the fractal dimension of the population, d_p, should be the projected population dimension plus 1, $d_{\hspace{0.17em}p}=d_{\hspace{0.17em}{p}_{\hspace{0.17em}p}}+1$. (e) More realistically, since not all buildings have the same height, the fractal dimension of the populations is $d_{\hspace{0.17em}p}=d_{\hspace{0.17em}{p}_{\hspace{0.17em}p}}+\eta$, where η captures the fractal dimension along the third dimension.

Figure 2.

(a) Fractal dimensions of the street network, d_i, (orange) and for the population, d_p, (blue) for 1000 cities in the UK as a function of their population size p. While the fractal dimensions are strongly size dependent, their ratio, γ_sub = d_i/d_p (black), is not. It is found to be approximately constant, γ_sub ∼ 0.86. (b) The sub-linear relation between street length $\ell$ and $p^{{\gamma }_{\mathrm{sub}}}$ is shown for the empirical data. It follows the theoretical prediction almost perfectly (red line). (c) As an example for a super-linear scaling law the relation between city GDP and $p^{2-{\gamma }_{\mathrm{sub}}}$ is shown. Red lines represent the linear regression and every dot is a city.

Figure 3.

Schematic of how a city grows. (a,b) For low populations, the city expands and densifies mostly horizontally, buildings have one or a few levels. (c,d) From a critical population size upward, buildings begin to grow into the third dimension. In this regime, the urban scaling law in the average building heights, 〈h〉, is expected to hold. (e) The average value of the projected version of the population grows with the same exponent 1 − γ_sub showing an approximated slope of 0.09. Obviously, there is no critical size, as expected. (f ) At the critical population level, the average value of the population in each floor, as measured by 〈p_h〉, saturates, and can no longer grow. Up to this point, the city densifies to absorb the increase of the population. (g) The scaling behaviour of the average building height of UK cities, 〈h〉, is clearly scaling, and follows the theoretical prediction for the exponent 1 − γ_sub. Scaling only appears for populations larger than 100 000, above the saturation level of 〈p_h〉. The red line indicates the scaling region with a slope of 0.10. Below the critical population the growth is marginal. In (e), (f ) and (g) each point is the average for similar sized cities using log-bins. In the case of (f ) and (g), this is a weighted average, using as weights the number of buildings digitized in each city.