# The three-dimensional structure of population density in world cities

Gaëtan Laziou, Rémi Lemoy

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Gaëtan Laziou, Rémi Lemoy. The three-dimensional structure of population density in world cities. npj Urban Sustainability, 2025, 5 (1), pp.75. 10.1038/s42949-025-00262-4. hal-05346981

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# The three-dimensional structure of population density in world cities

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Gaëtan Laziou & Rémi Lemoy

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A good understanding of cities is crucial to implement urban planning policies leading to social and economic sustainability and an efficient use of resources. Urban concentration has been associated with both positive and negative effects, echoing debates on compact cities, but we still lack robust empirical evidence on how intra-urban density systematically evolves with city size. We fill this gap by determining the functional form of the population density radial ("center-periphery") profiles for a global sample of cities, and observing how it changes across the urban hierarchy. Our results uncover strong regularities in the decrease of these profiles with distance from the main center, and in its link with city size, as large cities are denser and more spread out than smaller ones. We actually find that cities can be seen, in terms of population density, as exponential cones that evolve homothetically with city population. This rather simple but universal geometric structure of cities provides a new spatial scaling law, which is an important step forward in understanding how cities work and grow. Some deviations can be observed, which mainly oppose dense cities in the developing world and sprawled cities in high-income countries, associated with high energy use per capita. This suggests that the urban lifestyle in the wealthiest countries has come at the price of negative impacts on environmental outcome. This research has a broad range of applications as it provides a powerful tool to compare cities of different sizes.

Although they cover a small share of the land area, cities have high population densities and concentrate more than one half of the world's population¹,². Therefore, urban density is key in understanding modern lifestyles. Further, it has many implications on various facets of life, as housing, transport, and air quality. Studies by physicists, geographers and economists have associated density with positive effects, for example, increased productivity and more efficient public transport, but also negative effects, as high housing costs and traffic congestion³⁻⁷. While this strand of the literature is widely developed, the precise description of the urban structure linked to city size has received less attention. This is a shortcoming, because it is crucial to understand if similar mechanisms drive the internal structure of cities, in order to make them more sustainable and implement appropriate planning policies.

A milestone in the study of population density structure is the work of Clark (1951)⁸, which confirms earlier observation⁹ that population density tends to decrease exponentially with the distance from the center in many cities around the world. Building on strong assumptions on households' location choice, the Alonso-Muth-Mills model¹⁰⁻¹² proposes a theoretical foundation for Clark's empirical law, by deriving the decrease of density from the center as an outcome of a trade-off between housing and transport costs. Since then, many authors have used a negative exponential function to

model population density within cities¹³⁻¹⁵, but other functional forms have also been used¹⁶⁻¹⁸. Some works provide comprehensive reviews of these population density functions¹⁹,²⁰. In recent decades, however, with growing job decentralization and urban sprawl, an increasing number of studies adopt a polycentric viewpoint. While theoretical studies have shown how multiple centers influence the internal structure of cities²¹, empirical research has examined the decline of population density as a function of distance to several centers²². Today, the radial and polycentric formalisms are recognized as complementary approaches for understanding urban structure²³,²⁴. Largely driven by the growing availability of data, empirical analyses of population density have recently benefited from a renewed interest. Most of this literature focuses on change over time in the distribution of population²⁵⁻²⁷.

This modeling effort has also been followed by several attempts to compare cities with each other, and explain variations between them. From the beginning, it has been suggested that transport costs, income, city size and the development of public transportation have an impact on urban density, although no formal analysis supported this claim⁸. Later work found that the density profile is flatter when income is higher and transport costs lower¹⁵. Besides, there is a consensus on the fact that the urban density radial structure is influenced by city size. Numerous studies in the literature

<figure><img src="image_5.png" /></figure>
highlight that both density decay and central density are dependent on city population<sup>13,28-30</sup>. Building on this observation, some authors have attempted to remove the city size effect in order to compare population concentration between cities<sup>27</sup>.

The scaling framework has developed independently from this intra-
urban research. It views cities as complex systems and suggests that prop-
erties of cities can evolve non-linearly with city size<sup>31-33</sup>. Indeed, cities have
some emergent properties, arising from interactions between individuals.
Regarding population density, studies have discussed in particular the
power-law relationship between city population *N* and area *A* (the two
determinants of density) *A* ~ *N*<sup>γ</sup>, which is called “fundamental allometry”
by some researchers<sup>34</sup>. Although there is a consensus that γ < 1 (large cities
have a higher population density), empirical investigations show very
diverging results, with γ generally ranging between 2/3 and 1<sup>9,35-39</sup>. This lack
of consensus is sometimes related to the diversity of urban systems,
including their hierarchical differentiation, as well as the spatial config-
uration and historical trajectories (path dependence) of cities<sup>40</sup>. Besides, this
urban scaling literature does not consider the considerable heterogeneities
observed within urban areas. By reducing the city to a dimensionless point,
these studies overlook the crucial definition of its boundaries, as peripheral
areas (where population density is low and which can be included or not
depending on the definition) have an important effect on the overall
density<sup>41</sup>.

The primary objective of this work is to bridge intra-urban and inter-
urban research by investigating how the population density center-
periphery structure (“radial profile”) evolves in the urban hierarchy.
Using the urban scaling framework, we first determine empirically the
scaling relationship governing the intra-urban structure of population
density. We obtain unprecedented results at the global scale and for such a
large sample of cities. Using a unique methodology combining radial ana-
lysis and parsimonious non-linear regressions, we then characterize the
functional form of the density radial profiles and uncover how their para-
meters evolve with city size. Despite a great variety of sizes, development,
location on the planet, we observe striking regularities in the urban radial
structure across city sizes and provide a generic formula to describe it, valid
at the global scale. The main fluctuations around the expected behavior can
be explained by the level of development, which questions the sustainability
of today’s cities.

Results

Rescaled population density radial structure

We start by observing how the population density radial (“center-periph-
ery”) structure of cities can be successfully seen through the lens of urban
scaling laws. To do so, we use the GHS-POP gridded dataset⁴² (see Meth-
ods), and first observe the uneven distribution of population within cities of
different sizes through maps. At a similar zoom level, it is very difficult to
find consistencies between cities of different sizes (Fig. 1a, thin edge).
Indeed, cities have very different population sizes, which makes it challe-
ging to compare their urban structure¹³. The main reason is that large cities
like Tokyo (N ≃ 37.4 million inhabitants) are more spread out and tend to
be denser than smaller cities like Sarajevo (N ≃ 0.3 million inhabitants), as
suggested by the above literature. Based on empirical analyses of the average
population density ρ(r) as a function of the distance to the center r, we find
that a rescaling with the power 1/3 of city population N¹/³ in each of the
three dimensions is appropriate to obtain a homogenous population density
across city sizes (see Supplementary Information). Hence, we dilate the
Euclidean space through a zoom on the map r' = rk, simultaneously with
the vertical dimension of density ρ' = ρk. The dilation factor k, which
allows us to rescale any city of population N to the size of Tokyo (the largest
city of our sample, taken as a reference without loss of generality), can be
expressed as k = (NT<sub>Tokyo</sub>/N)¹/³. This approach takes off the size effect,
leading to more comparable population density patterns across cities (Fig.
1a, thick edge). This implies that as cities grow, they experience two
simultaneous phenomena: a densification and an expansion toward the
periphery. Even for a city with shape distortions like Wellington, which is

surrounded by hills and its harbor, the homothetic transformation remains
very effective.

To better capture the radial (“center-periphery”) structure of cities, we
derive population density radial profiles from the maps (Fig. 2a), using the
average density ρ(r) as a function of distance r from the center. Whatever the
size, the population density profile is well-described by a negative expo-
nential function, as straight lines appear on a semi-logarithmic graph. We
sometimes observe a “density crater”, which is already well-documented in
the literature<sup>8,16</sup>, as well as some cities with dense peripheral areas (nearby
cities), such as Mexico or Beijing. Despite these peculiarities, curves are well
clustered by city size at every distance from the center, indicating that large
cities tend to be denser and cover a larger area to accommodate the extra
population.

Following the same approach we used for the maps, we rescale the two
Euclidean dimensions of space and the vertical dimension of population
density. It allows us to get a view of the intra-urban density structure of cities,
for a typical city of the size of Tokyo (Fig. 2b). We notice that our sample of
capital cities exhibits similar variations of rescaled density ρ(r') despite their
different history, size, and location on the planet. These regularities indicate
that the rescaling with the parameter N¹/³ is successful in taking off the size
effect and point to a universal structure of cities arising from similar
mechanisms. These findings are supported by robustness checks, which
suggest that this 1/3 scaling exponent yields the most homogeneous rescaled
radial profiles (Supplementary Fig. 2).

To further document the generic radial structure of all cities and the
deviations, we compute the median rescaled radial profile and the inter-
quartile range (IQR) with a spacing of 1 km around the center. They show a
remarkable concentration of radial profiles around a mean global urban
profile (Fig. 2c). To account for the unequal distribution of city sizes<sup>43,44</sup>, we
also compute those statistics using two additional weighting schemes, the
Person Model (PM) and the P<sup>2</sup> model (P<sup>2</sup>M), which weight cities with city
population and the square of city population, respectively (see Methods).
Setting aside the central crater, the density at the center is ρ'(r' = 0) ≃
40,000 inhab./km<sup>2</sup> for a theoretical city of the size of Tokyo (Fig. 2c), and
drops sharply within the first kilometers from the center. By contrast, our
scaling law of population density predicts a density at the center ρ'(0) ≃
27,000 inhab./km<sup>2</sup> for a city of 10 million inhabitants, and ρ'(0) ≃ 12,500
inhab./km<sup>2</sup> for a city of one million inhabitants. These figures represent the
first estimate of central density in relation to city population, since this size
effect was suggested<sup>9</sup>. We also note that both the median profile and the IQR
remain nearly unchanged under the different weighting schemes, denoting
that our approach is very robust across the urban hierarchy. This is con-
firmed by a look at the mean rescaled profiles of different city size classes,
which show very small remaining fluctuations (Supplementary Fig. 5b).

These regularities in the radial structure of cities also suggest to look at
them through circular disks, instead of rings. This corresponds to inte-
grating (summing) the density from the city center to a given distance,
instead of studying the density in a thin ring. For simplicity, we will speak
here of rescaled distance *r*′ to the center, actually meaning a non-rescaled
radius *r* = *r*′/*k* which differs for each city. These disks we define are illu-
strated for *r*′ = 60 km by circles on Fig. 1a. Given a value of the rescaled
radius *r*′, we define *N*<sub>*r*</sub> as the population count, *A*<sub>*r*</sub> = π*rr*²/*k*² the area, and
*D*<sub>*r*</sub> = *N*<sub>*r*</sub>/*A*<sub>*r*</sub> the density within a disc of rescaled radius *r*′ around the city
center. The rescaled density is given by *D*<sub>*r*</sub> = *D*<sub>*r*</sub> × *k*, where *k* is the
rescaling factor defined above. We test different values of *r*′, in order to look
at the scaling effect from the urban core (*r*′ = 20 km) to the whole city
(*r*′ = 100 km). Urban density decreases as (rescaled) distance *r*′ from the
center increases, because more peripheral areas are encompassed (Fig. 2d).
While we find a clear positive correlation between density *D*<sub>*r*</sub> and city
population *N* (*p*-value < 10<sup>-16</sup>), this correlation vanishes when using the
rescaled density *D*<sub>*r*</sub> (see Supplementary Information), indicating that the
transformation holds throughout the urban system and along the entire
radial profile. This is an important finding for the understanding of urban
structures, as it reveals a rather universal shape of cities, which can only be
seen if we account for the size effect.
<figure><img src="image_6.png" /><figcaption>**Fig. 1 |** View of population density in capital cities of different sizes. **a** Population density in Tokyo (reference city, $N = 3.7 \times 10^7$ and $k = 1$), Mexico and Beijing ($N = 2.2 \times 10^7$ and $2.0 \times 10^7$; $k = 1.2$), Santiago and Khartoum ($N = 6.8 \times 10^6$ and $5.8 \times 10^6$; $k = 1.8$ and 1.9), Lomé and Stockholm ($N = 1.8 \times 10^6$ and $1.6 \times 10^6$; $k = 2.7$ and 2.8), Wellington and Sarajevo ($N = 4.1 \times 10^5$ and $3.4 \times 10^5$; $k = 4.5$ and</figcaption></figure>

4.8). The circles correspond to a rescaled radius of $r' = 60$ km. Top maps of each city are at the same scale (square of side 120 km), while a rescaling has been performed on the bottom maps (thick edge), both on distances $r' = r \cdot k$ and density $\rho' = \rho \cdot k$. The city center is located by the black dot. Scale bars indicate $r = 20$ km. **b** Location of these capital cities on the planet.

Instead of examining density, we now focus on the fundamental relationship between area $A_N$ and population $N$, which is a common question in the urban scaling literature$^{41,45}$. This fundamental allometry$^{34}$ is expressed by the power-law relation

$$A_N \sim A_1 N^\gamma \quad (1)$$

where $A_1$ is a constant corresponding to the theoretical area of a city with one inhabitant, and $\gamma$ a scaling exponent. However, no city delineation is provided by our UN WUP dataset$^{46}$. Nevertheless, we draw concentric circles of rescaled radius $r'$ around the city centers, and look at the relationship between the population $N_r$, and city size $N$. The scaling relationship for $r' = 50$ km returns an exponent $0.99 \pm 0.01$ ($R^2 = 0.83$), and the curve is very close to the straight line $x = y$ (Fig. 2e). This finding suggests a proportionality and even an identity (on average) between the resident population located in a buffer of radius $r' = 50$ km around the center and the one inside the city boundaries used by the UN. In other words, the city definition of the UN (using many criteria) is roughly equivalent, on average, to a circle of rescaled radius $r' = 50$ km, thus following the homothetic behavior of urban density. Then, if the largest city, Tokyo, can be delineated with a circle buffer of radius $r = r' = 50$ km, this distance $r' = 50$ km actually decreases with population, corresponding to $r \approx 34$ km and $r \approx 10$

km for a ten million and a one million inhabitants city, respectively. Since $A_r = \pi r^2/k^2 \sim N^{2/3}$ as $k \sim N^{-1/3}$, these results regarding population density determine a relationship between area and population of the form $A \sim N^{2/3}$. They also confirm the roughly circular shape of cities, featured in classic models of urban geography$^{10,47}$.

## Scaling of the parameters

We now aim to describe these radial profiles of population density mathematically with simple statistical tools and a parsimonious approach. To achieve this, we use a limited number of parameters that can be directly interpreted in terms of urban concentration. As suggested by the previous results, we carry out exponential fits for each city. We use a linear fit $L$ of the logarithm (Eq. (2), see Methods), as well as a non-linear fit of the raw value (NL model, Eq. (3)), with two parameters, $a_N$ measuring the density at the center and $b_N$ the characteristic distance (sometimes also referred to as the density gradient). Besides, we try another non-linear (exponential) fit, where we force the relationship $a_N b_N^2 2\pi = N$ to hold (see Methods), ensuring that the total population in the model corresponds to our UN data. Using the relationship between the two parameters $a_N$ and $b_N$ (Eq. (5), see Methods), this fit is a one-parameter model (1NL model). For comparison purposes, we also fit the profiles using a (negative) power law.
<figure><img src="image_7.png" /><figcaption>**Fig. 2 | Population density radial structure.** Radial profiles for a sample of nine capital cities displayed both before (a) and after (b) rescaling, on a semi-logarithmic plot. Supplementary Fig. 3 shows the radial profiles for all cities. The population *N* provided by the United Nations (UN) is given in the legend, as well as the rescaling factor *k* (s.t. *r*' = *rk* and *ρ*' = *ρk*). c Distribution of population density rescaled radial profiles *ρ*(*r*′) for all cities of the database and under different weighting schemes. d Average rescaled density *D*/*r* in disks of radius *r*′ as a function of city size *N*. To</figcaption></figure>

ensure readability, results are displayed for European cities only (Supplementary Fig. 7 displays the same information for all cities). The correlation coefficient between city size and population density (before and after rescaling, written respectively R and R') is also given. Note that both variables are presented on logarithmic scales. The horizontal lines represent the median rescaled density *D*<sub>*r*</sub>/*r* of cities. e Population within a disc of radius *r*′ = 50 km as a function of the UN population. The solid line is a power-law fit (Eq. (1)) with an exponent 0.99 ± 0.01.

We notice that the exponential function describes empirical data satisfactorily in most world cities, with median R² values of 0.52, 0.95 and 0.91 for the linear, non-linear, and one-parameter non-linear fits, respectively (density plot, Supplementary Fig. 9). These values are high, which suggests that very parsimonious models are able to describe population density profiles. By contrast, the power-law function is clearly outperformed by the exponential (Supplementary Figs. 9–11).

Let us call *α* and *β* the scaling exponents of the fitting parameters *a*<sub>*N*</sub> and *b*<sub>*N*</sub> (Eq. (7)). From the 3-dimensional homothetic scaling with the exponent 1/3 observed before, we expect *α* = *β* = 1/3. This is rather precisely what we find in Fig. 3, where the expected exponent 1/3 is represented by a horizontal dashed line. For nonlinear models, we note that a weighting scheme that distributes statistical weights more evenly along the urban hierarchy (see Methods), such as the Person and P<sup>2</sup> models (PM and P<sup>2</sup>M), brings us closer to these expectations. Conversely, if small cities are weighted as much as megacities (City model CM), we find a higher value for *β* (horizontal scaling) than for *α* (vertical scaling).

We also observe that this scaling behavior is consistent with another
theoretical expectation. If cities are circular and population density
decreases exponentially with the distance to the center, then the total
population can be seen as the volume under the radial profile of density, and
a scaling relationship a_N b_N^2 ~ N holds (see Methods). Hence, the two
scaling parameters are connected by the equality α + 2β = 1 (Eq. (5)),

which aligns with the exponents obtained from the power-law relationships
(Fig. 3), in particular, the homothetic scaling α = β = 1/3.

In light of our findings regarding population density, cities of the world are roughly of one shape, exponential cones, which evolve vertically and horizontally with city size. Therefore, the characteristic distance and central density for a city of any size can be expressed as $b_N = b_1 N^{1/3}$ and $a_N = a_1 N^{1/3}$, respectively, where we estimate $b_1 \simeq 35$ m and $a_1 \simeq 125$ inhab./km² using the results of the 1NL model. Figure 4a provides a theoretical display of the average change in the radial structure of population density, as we move from a small to a large city. Of course, individual cities do not follow precisely these geometric properties, but this lays the foundations for more effective inter-urban comparisons.

Indeed, cities have local characteristics likely to infringe on the
expected radial structure. The scatter plot presented in Fig. 4b illustrates the
relationship between the fitted parameter $b_N$ (from the 1NL model, Fig. 3c)
and city size, along with the fluctuations around the scaling law. Although
many cities exhibit a center-periphery structure close to the expected
homothetic transformation, a closer inspection reveals significant devia-
tions. Some capital cities are rather sprawled, as most of their population is
located in peripheral areas (Brasilia, Canberra, Washington D.C), while
some other cities are much denser (Dhaka, Cairo). We observe that these
different urban forms are linked to the nationwide energy use per capita.
Compact cities (with a low value of $b_N$ considering their population) are
located in energy-efficient countries, while sprawled cities show the opposite
<figure><img src="image_8.png" /><figcaption>**Fig. 3** | Scaling analysis of the fitted parameters. Results of linear fits of the fitted (log) parameters $a_N$ (exponent $\alpha$) and $b_N$ (exponent $\beta$) against the (log) total population $N$. Those parameters $a_N$ and $b_N$ come themselves from the linear fits (a), as well as two-parameters (b) and one-parameter (c) non-linear fits of the radial</figcaption></figure>

profiles. We derive these results for three weighting possibilities: the City Model (CM), the Person Model (PM) and the P²M model. The horizontal dashed line represents the expected value $\alpha = \beta = 1/3$. The error bars indicate the 95% confidence interval (CI).

<figure><img src="image_9.png" /><figcaption>**Fig. 4** | Urban scaling of population density radial profiles. a Illustration of the homothetic behavior of population density radial profiles. The tangent at $x = 0$ (grey dashed lines) is also shown. b Power-law relationship connecting $b_N$ (from the one-parameter non-linear fit 1NL) and city size $N$. The black line gives our</figcaption></figure>

expectations, a scaling with the cube root of total population $N^{1/3}$. The color represents the annual energy use per capita, in kilograms of oil equivalent (in a decimal logarithmic scale).

trend. To further investigate regional and national peculiarities, one option is to account for the city-size effect beforehand. Our approach differs in this respect from many studies, which recognize that urban forms vary with city population, but do not account for this effect systematically.

## Urban density index

Although our urban scaling framework manages for the city size effect surprisingly well, we still observe significant fluctuations across different locations on the planet (Supplementary Figs. 4–6). Our results suggest a size-independent measure of urban concentration, the urban density index UDI = $b_N/N^{1/3}$ = $(b_N/a_N)^{1/3}$ (see Methods). It corresponds to the residual of the scaling relationship between the parameter $b_N$ and population size $N$ (Fig. 4b) and is linked to average density $\rho$ as $\rho \sim UDI^{-2}$ (see Methods), so that a dense city will have a low UDI (Fig. 5a). Each population size $N$ associates indeed with a range of values for the two parameters $a_N$ and $b_N$, and for the UDI. This index provides a more in-depth comparison between cities by examining how the center-periphery structure of cities deviates from

the observed scaling law. It has the dimension of a distance, but could also be considered dimensionless (see Discussion).

The map and boxplots illustrate the geographical variation in urban concentration measured by the UDI (Fig. 5b, c). It reveals a specific distribution, statistically close to log-normal (Fig. 5d), which mainly seems to oppose developed to developing countries. The latter countries (such as Bangladesh, Egypt, India) have very dense cities, while sprawled cities are mainly located in high-income countries (United States, Australia, Canada). However, there are some exceptions. We note that South African cities have a very flat center-periphery structure, which can be explained by the stringent urban regulations during Apartheid¹⁵. We further observe that there are rather small variations within countries, suggesting similar mechanisms at the national level. Among the global driving forces, wealth measured by income per capita seems to be key, as low-income and lower-middle income countries concentrate compact cities (low value of the UDI), and conversely for high-income countries (Fig. 5b).

We deepen this analysis by exploring how the UDI evolves with a set of indicators obtained at national level (Supplementary Table 2 and Fig. 16).
<figure><img src="image_10.png" /><figcaption>**Fig. 5 | Urban density index UDI.** a Relationship between the regression parameters $a_N$ and $b_N$ of the 1NL model and the UDI, for different population sizes. The dots represent the estimated parameters for a selection of capital cities. b Boxplots of the UDI. Results are presented by country and sorted by median value (increasing order, from left to right). The number of cities per country *n* is also given (countries having</figcaption></figure>

fewer than 10 cities are not displayed). The color indicates the income group (World Bank). The horizontal line indicates the global median, UDI = 34.6, and the colored area the 25th-75th percentile range. c Map of the UDI. d Histogram of the logarithm of the UDI. A normal distribution of the same mean 1.5 and standard deviation $\sigma$ = 0.19 (red curve) is used as a guide to the eye.

Our results show that two variables are even more closely linked to the UDI than the GDP per capita, namely energy use and number of cars per inhabitant – these are all indicators of the level of development. Indeed, we have $UDI \sim \varepsilon^{0.4}$ and $\rho \sim UDI^{-2}$, which gives us $\varepsilon \sim \rho^{-1.3}$ (Supplementary Information). The interplay between UDI, population density, and energy use per capita can be related to previous work¹⁸, which finds a negative relationship between average population density $\rho$ and energy use per capita $\varepsilon$, and pleads for more compact cities in order to meet the sustainability challenge. Although the question of the most appropriate urban form is still debated in the literature, along with the causal relationship between density and energy use, our results contribute to highlighting that the environmental issues of sprawling, space-consuming cities are clearly linked to the level of development.

## Discussion

Describing urban population density $\rho_N(r)$ as a function of the distance $r$ to the city center and total population $N$, $\rho_N(r) \sim \frac{N^{1/3}}{2\pi b_N^2} \cdot \exp(-\frac{r}{b_N^{1/3}})$, with only one parameter $b_1 \approx 35$ m, we uncover here what can be considered as the first global result regarding the spatial internal structure of cities. In that

sense, we could relate it to Zipf's law, which has been widely studied and observed, but does not consider the internal structure of cities. This result both gives another benchmark on which to build urban science (for instance, to tackle the still open problem of city definition), and raises many questions and research perspectives. First, as for Zipf's law, to what extent does it hold at the lower end of the size distribution, where effects of polycentricity, and issues of city definition, are more critical?

When it comes to population density, we observe an isometric scaling or homothety in three dimensions. We find an allometry only when we look at the base area of this volume (exponential cone). Why do cities present such a simple geometrical scaling? It is true that there is huge cultural, climatic, political, historical, landscape, and social heterogeneity in the world. But still, there should be more common ground than just the 3-dimensional geometry of Euclidean space. And why is the exponential shape so robust? The pioneering work of Nordbeck³⁵ discusses this universal shape of cities (a “volcano” in his words) and suggests that a homothetic scaling would appear visually appealing and geometrically logical, but this is not a satisfactory explanation. We see here clearly that population density is a variable that behaves like a volume. The two horizontal dimensions *x* and *y*
are geographic and mixed in the radial distance $r = \sqrt{x^2 + y^2}$; therefore, it is rather trivial that they have similar scaling exponents (and hence the $a_N b_N^2 2\pi = N$ relationship of Eq. (5)), which identifies total population N to a “volume” of population). On the other hand, the vertical dimension of population density stands out since it is not really physical, as it involves human beings. This puzzle can be addressed with a simple hypothesis that all inhabitants of a city occupy a roughly similar building volume, which converts the vertical aspect of population density into a physical measure of vertical distance (and makes the UDI dimensionless). While it may seem unusual to combine human residential choice behavior with such purely geometric considerations, our findings bring us to this point.

In order to test this hypothesis, a research perspective is the study of buildings and their volumes, with similar methods as those used here. Additionally, in order to know more precisely how physical the vertical dimension of population density is, it would be interesting to have access to 3-dimensional data on people's locations at various times to understand how high above (or below) ground level city residents live, alongside their horizontal geographic location. No such data is available yet to our knowledge, but it might appear in the future. Likewise, the accuracy of global population density layers and world cities population databases such as those used here will presumably increase, making this research more precise (data precision being probably the major limitation of our study). It can be anticipated that the average 3-dimensional distribution of urban populations will follow a homothetic volume-scaling law similar to the one identified here. Additionally, comparable studies focusing on employment density (rather than residential population density) and commuting patterns would also be interesting steps for improving our understanding of cities. A focus on polycentric urban (or regional) structures and on coastal cities could also lift some limitations of the present work.

Naturally, these empirical results also raise questions on the urban modeling side. By linking intra-urban analysis (inspired in this by theoretical models from urban economics) and inter-urban analysis through the urban scaling framework, we enlighten the still open problem of the most basic allometry of city population N versus area A, $A \sim N^{\gamma}$. With a scaling of population density profiles in the cube root of total population $N^{1/3}$ in the two horizontal geographical dimensions of space, we find that this fundamental allometry has an exponent $\gamma = 2/3$ at the global scale, $A \sim N^{2/3}$. This result, rejoining much earlier ones³⁵, is a fundamental input in urban modeling approaches³⁴,⁴⁵, but as such, there is no attempt to explain it. Urban economic approaches, which offer explanations for the internal spatial structure of cities, lack a realistic description of the urban space, particularly in terms of the radially varying intensity of land use, revealing a modeling gap. The Alonso-LU model⁴⁹ is a good candidate for modelling the radial scaling laws of both land use and population density in an urban economic framework, although it struggles with the scaling of housing price⁵⁰. A gravitational mechanism⁵¹ could also account simultaneously for land use and population density patterns.

Regarding the sustainability of our urban world, these results highlight the positive relationship between urban extent —measured by our size-independent urban density index UDI— and per capita wealth, income, and especially energy use and car ownership. These findings are consistent with the link between income and sprawl in the standard urban economic model¹⁵,²⁹. They are also coherent with the interplay between wealth and environmental footprint at the micro level⁵². We rejoin here, with different methods and data, much discussed earlier work⁴⁸, which observes a negative relationship between average population density $\rho$ and energy use $\varepsilon$. In a world where fossil fuel reserves, formed over millions of years, are rapidly depleting within a few decades, along with the metals required to replace current internal combustion engine vehicles with electric ones, urban planning towards global sustainability should clearly aim at low UDI. This involves promoting denser settlements through smaller homes and compact residential development, aiming to reduce car dependency and per capita energy consumption. In developing countries (as in high-income countries), it is essential to avoid uncontrolled urban sprawl, while of course ensuring that the essential needs of the population are met. Obviously, such policies

could also lead to negative effects, such as higher housing prices. The link between our results on the distribution of urban population and housing prices could actually be an interesting perspective of research, introducing more human behavior, through the housing market, to understand the very geometric scaling observed here.

Our work opens up new avenues for future investigations, since it provides a powerful tool to compare cities with each other. Future research could explore national and continental differences more systematically, in order to gain deeper insights into the factors shaping urban forms. More broadly, we highlight the importance of the intra-urban structure of cities in understanding patterns of human settlement and designing policies that promote sustainable development. Our study contributes to documenting the multi-dimensional impacts of density and suggests that we should aim for compact urban development—however, we must remember that true change mostly relies on a collective will.

# Methods

## Data collection

Various methodologies have been proposed to provide spatially disaggregated population estimates, but this task remains challenging, especially if census data is lacking or inaccurate⁵³. Here, we use the latest version of the GHS-POP layer, which provides residential population estimates in 2020 across the globe, at a resolution of 100 × 100 m⁴². It is produced through a dasymetric top-down approach, where census population from the Gridded Population of the World, version 4.11 (GPWv4.11) is disaggregated to grid cells, proportionally to built-up volume⁵⁴. We note that this approach is rather simple and consistent globally. On the other hand, however, estimates at the pixel level are subject to some uncertainties due to various factors, including the quality of census data and the stable relationship between population and built-up volume⁵⁵. Recently, some authors have proposed a review of large-scale population grids, discussing their methodological approach, resolution, spatial coverage and fitness for use⁵⁶.

We use additional datasets to observe the relationship between socioeconomic variables and the population density radial structure. In detail, we gather a set of indicators at the country level from various institutions. In the Supplementary Information, we provide a detailed description of these datasets (source, description, reference year).

## Sample of cities

Our sample includes 1860 cities in the world with 300,000 inhabitants or more in 2018 from the World Urbanization Prospects (WUP), and gathering some 2.7 billion inhabitants in 2020⁴⁶. Population estimates for each city at 5-year intervals between 1950 and 2035 are also provided. For our reference year 2020, Tokyo was the most populated city (37.393 million inhabitants).

From this initial sample, we remove some cities that can be considered as subcenters of a larger adjacent urbanized area. For full details, one may refer to the Supplementary Information. We end up with 1593 cities worldwide. The most represented countries are China (338 cities), India (155), the US (133), Russia (63), and Brazil (56).

## Population density radial structure

We compute population density as a function of the distance to the city center, whose location is chosen to be the city hall. Indeed, this location is generally very central, associated with a long cultural heritage and the presence of amenities. If not available, we choose instead a central building, such as the central post office or the key train station in India, or the central mosque in the Middle East and North Africa. Thereafter, we reproject the population grid in the local UTM projection and draw concentric rings with a spacing of 1 km around the city center. Each GHS-POP cell is allocated to a ring, taking as a reference the center of the cell. We finally compute the mean population density within each ring. Because large cities have a further-reaching influence than small cities, the population density is computed up to a distance $r_f = r_f \times (N_{Tokyo}/N)^{1/3} = 100$ km, where $r_f$ and N are the maximal radius and the population of a given city, and $N_{Tokyo}$ the
population of Tokyo. Hence, the maximal distance $r_f$ scales with population as $r_f \sim N^{1/3}$.

Thereafter, we carry out exponential fits. We perform both a linear fit of the logarithm (L) and a non-linear fit of the raw value (NL), respectively, following

$$ \log(\rho_N(r)) \sim \log(a_N) - \frac{r}{b_N}, \qquad (2) $$

and

$$ \rho_N(r) \sim a_N \cdot \exp\left(-\frac{r}{b_N}\right), \qquad (3) $$

where $a_N$ is the (theoretical) population density at the center, $r$ the distance to the center (in kilometers), and $b_N$ the distance at which population density has been divided by $e^1 \simeq 2.7$ (it is also the distance at which the highest share of total population can be found, and half of the mean distance to the center of all individuals, see Supplementary Information). Note that both L (Eq. (2)) and NL (Eq. (3)) regressions yield different values of the parameters $a_N$ and $b_N$. Indeed, while the linear fit minimizes the relative error, the non-linear fit minimizes the absolute error, which has an impact on the estimated parameters$^{57}$. Further, we perform a one-parameter non-linear fit (1NL), making use of the fact that integrating the radial profile of density should yield the total population $N$. Supposing an exponential radial profile of population density, the cumulative population $N_r$ in a disc of radius $r$ around the city center can be written as

$$ \begin{aligned} N_r &= \int_0^r a_N \cdot \exp\left(-\frac{x}{b_N}\right) \cdot 2\pi x dx \\ &= 2\pi a_N b_N^2 \left[1 - \left(1 + \frac{r}{b_N}\right) \cdot \exp\left(-\frac{r}{b_N}\right)\right] \end{aligned} \qquad (4) $$

When $r$ is large compared to the characteristic distance $b_N$, we obtain

$$ \lim_{r \to +\infty} N_r = 2\pi a_N b_N^2 = N, \qquad (5) $$

which has already been stressed some decades ago in the early studies of urban population densities$^{8,28}$. We note that the maximal extent $r_f$ respects this condition well since $r_f/b_N \simeq 9$ in our sample, on average, meaning that the theoretical value $N_r$ given by Eq. (4) is only 0.1% smaller than $N$. By using Eq. (5), we can rewrite Eq. (3) as

$$ \rho_N(r) \sim \frac{N \cdot \exp\left(-\frac{r}{b_N}\right)}{2\pi b_N^2} \qquad (6) $$

where $b_N$ is the only parameter to be estimated. Central density $a_N$ is directly given by $a_N = N/b_N^2 2\pi$, thanks to Eq. (5).

## Scaling of urban profiles

We expect the population density at the center $a_N$ and the characteristic distance $b_N$ to scale with city size $N$. Once radial profiles are fitted, we relate the estimated parameters $a_N$, $b_N$ (from both linear and non-linear fits) to population size $N$ via power-law functions

$$ a_N \sim a_1 N^\alpha \text{ and } b_N \sim b_1 N^\beta, \qquad (7) $$

where $\alpha$ and $\beta$ are two scaling exponents, $a_1$ the theoretical density at the center and $b_1$ the characteristic distance for a city of $N = 1$ inhabitant. We estimate the scaling parameters $\alpha$ and $\beta$, with linear fits of the logarithms of $a_N$, $b_N$ and $N$.

To check if the scaling relationship holds throughout the system of cities, we derive the analysis with different weighting possibilities. The usual

City Model (CM) weights all cities equally. In addition, as prescribed by earlier studies$^{44}$, we try the Person Model (PM), which weights cities proportionally to their population $N$. We also add a third possibility, namely the P² Model (P²M), which weights cities with the square of city population, $N^2$. This last weighting scheme is able to cancel the distribution of cities' sizes, as the cumulative probability density associated with Zipf's law$^{43}$ $P(x > N) \sim 1/x$ gives a probability density function $P(x = N) \sim 1/x^2$, thus ensuring an homogeneous distribution of weights at all scales using the P²M model.

## Urban density index

The scaling of urban profiles suggests using a size-independent measure of urban concentration, defined using the parameters of the 1NL model. We introduce the urban density index $UDI = b_N/N^{1/3}$, which corresponds to the residual of the regression displayed in Fig. 4b (see Supplementary Information). Hence, two cities $i$ and $j$ with different population sizes are equal in UDI if $b_i/N_i^{1/3} = b_j/N_j^{1/3}$ hold, or equivalently $b_i/a_i = b_j/a_j$ (Fig. 5a). This index measures how spread a city is, since $b_N \sim UDI$ and how dense it is in the center, since $a_N \sim UDI^{-2}$ (using Eq. (5)), both independently of its size $N$. Thus, every 1% increase in the index associated with a 1% increase in the characteristic distance $b_N$ and a 2% decrease in density at the center $a_N$. Since Eq. (5) holds, we also note that the UDI can be computed as $(b_N/a_N)^{1/3}$, which is linked to the inverse slope of the tangent in Fig. 4a, whose equation is

$$ y = -\frac{a_N}{b_N}x + a_N. \qquad (8) $$

The index synthetically summarizes whether a city is compact and dense or spread out, regardless of its size (Supplementary Fig. 15). If a city has a low UDI value, its population is more densely concentrated near the center. Conversely, a high value of the UDI indicates that city population is spread out.

## Data availability

The population grid at 100 m resolution that is used in this study is available in the European Commission's Joint Research Centre Data Catalog, and can be accessed at https://human-settlement.emergency.copernicus.eu/. Population of largest urban agglomerations (300,000 inhabitants or more) can be downloaded on the United Nations' website, at https://population.un.org/wup/downloads. National socio-economic indicators used in this study are described in the Supplementary Material.

## Code availability

The geospatial data analysis was programmed in R 4.2.2 and using standard packages. The code used for analyzing data is available from the corresponding authors on request.

Received: 28 February 2025; Accepted: 12 August 2025; Published online: 26 September 2025

## References

1. Moreno-Monroy, A. I., Schiavina, M. & Veneri, P. Metropolitan areas in the world. Delineation and population trends. *J. Urban Econ.* **125**, 103242 (2021).

2. United Nations, Department of Economic and Social Affairs, Population Division. World Urbanization Prospects: The 2018 Revision, Online Edition. (2018).

3. Burton, E. The compact city: Just or just compact? A preliminary analysis. *Urban Stud.* **37**, 1969–2006 (2000).

4. Forsyth, A., Oakes, J. M., Schmitz, K. H. & Hearst, M. Does residential density increase walking and other physical activity? *Urban Stud.* **44**, 679–697 (2007).

5. Ahlfeldt, G. M. & Pietrostefani, E. The economic effects of density: A synthesis. *J. Urban Econ.* **111**, 93–107 (2019).
6. Ribeiro, H. V., Rybski, D. & Kropp, J. P. Effects of changing population or density on urban carbon dioxide emissions. *Nat. Commun.* **10**, 3204 (2019).

7. Duranton, G. & Puga, D. The economics of urban density. *J. Econ. Perspect.* **34**, 3–26 (2020).

8. Clark, C. Urban population densities. *J. R. Stat. Soc. Ser. A (Gen.)* **114**, 490–496 (1951).

9. Stewart, J. Q. Suggested Principles of" Social Physics. *Science* **106**, 179–180 (1947).

10. Alonso, W. *Location and Land Use: Toward a General Theory of Land Rent*. (Harvard university press, 1964).

11. Mills, E. S. An aggregative model of resource allocation in a metropolitan area. *Am. Econ. Rev.* **57**, 197–210 (1967).

12. Muth, R. F. Cities and Housing, University Of Chicago Press. Chicago, IL (1969).

13. Berry, B. J., Simmons, J. W. & Tennant, R. J. Urban population densities: structure and change. *Geogr. Rev.* 389–405 (1963).

14. Mills, E. S. Urban density functions. *Urban Stud.* **7**, 5–20 (1970).

15. Bertaud, A. & Malpezzi, S. The spatial distribution of population in 48 world cities: Implications for economies in transition. *Cent. Urban land Econ. Res. Univ. Wis.* **32**, 54–55 (2003).

16. Newling, B. E. The spatial variation of urban population densities. *Geog. Rev.* 242–252 (1969).

17. Ingram, D. R. The concept of accessibility: A search for an operational form. *Reg. Stud.* **5**, 101–107 (1971).

18. Batty, M. & Sik Kim, K. Form follows function: reformulating urban population density functions. *Urban Stud.* **29**, 1043–1069 (1992).

19. Martori, J. C., Suriñach Caralt, J., & others. Urban population density functions: the case of the Barcelona region. (2002).

20. McDonald, J. F. Econometric studies of urban population density: A survey. *J. urban Econ.* **26**, 361–385 (1989).

21. Fujita, M. & Ogawa, H. Multiple equilibria and structural transition of non-monocentric urban configurations. *Reg. Sci. urban Econ.* **12**, 161–196 (1982).

22. Small, K. A. & Song, S. Population and employment densities: structure and change. *J. Urban Econ.* **36**, 292–313 (1994).

23. Clark, W. A. Monocentric to polycentric: New urban forms and old paradigms. *A Companion to the City* 141–154 (2000).

24. Lemoy, R. Monocentric or polycentric city? An empirical perspective. In *Compendium of Urban Complexity* 157–175 (Springer, 2025)..

25. Broitman, D. & Koomen, E. The attraction of urban cores: Densification in Dutch city centres. *Urban Stud.* **57**, 1920–1939 (2020).

26. Qiang, Y., Xu, J. & Zhang, G. The shapes of US cities: Revisiting the classic population density functions using crowdsourced geospatial data. *Urban Stud.* **57**, 2147–2162 (2020).

27. Guterbock, T. M. The beginning and the end of urban population deconcentration in the United States: New insights from application of the Density Distribution Index. *Cities* **118**, 103349 (2021).

28. Weiss, H. K. The distribution of urban population and an application to a servicing problem. *Oper. Res.* **9**, 860–874 (1961).

29. Mills, E. S. & Tan, J. P. A comparison of urban population density functions in developed and developing countries. *Urban Stud.* **17**, 313–321 (1980).

30. Edmonston, B. & Guterbock, T. M. Is suburbanization slowing down? Recent trends in population deconcentration in US metropolitan areas. *Soc. Forces* **62**, 905–925 (1984).

31. Pumain, D. Scaling laws and urban systems. *Santa Fe Institute, Working Paper* 002 (2004).

32. Bettencourt, L. M., Lobo, J., Helbing, D., Kühnert, C. & West, G. B. Growth, innovation, scaling, and the pace of life in cities. *Proc. Natl Acad. Sci.* **104**, 7301–7306 (2007).

33. Bettencourt, L. M. Cities as complex systems. *Modeling complex systems for public policies* 217–236 (2015).

34. Ribeiro, F. L. & Rybski, D. Mathematical models to explain the origin of urban scaling laws. *Phys. Rep.* **1012**, 1–39 (2023).

35. Nordbeck, S. Urban allometric growth. *Geogr. Annaler: Ser. B, Hum. Geogr.* **53**, 54–67 (1971).

36. Fluschnik, T. et al. The size distribution, scaling properties and spatial organization of urban clusters: a global and regional percolation perspective. *ISPRS Int. J. Geo-Inf.* **5**, 110 (2016).

37. Lemoy, R. & Caruso, G. Evidence for the homothetic scaling of urban forms. *Environ. Plan. B: Urban Anal. City Sci.* **47**, 870–888 (2020).

38. Ortman, S. G., Lobo, J. & Smith, M. E. Cities: Complexity, theory and history. *Plos One* **15**, e0243621 (2020).

39. Burger, J. R. et al. Global city densities: re-examining urban scaling theory. *Front. Conserv. Sci.* **3**, 879934 (2022).

40. Bretagnolle, A., Pumain, D. & Vacchiani-Marcuzzo, C. The organization of urban systems. in *Complexity perspectives in innovation and social change* 197–220 (Springer, 2009).

41. Batty, M. & Ferguson, P. Defining city size. *Environ. Plan. B: Plan. Des.* **38**, 753–756 (2011).

42. Schiavina, M., Freire, S., Carioli, A. & MacManus, K. GHS-POP R2023A - GHS population grid multitemporal (1975–2030). https://doi.org/10.2905/2FF68A52-5B5B-4A22-8F40-C41DA8332CFE (2023).

43. Zipf, G. K. *Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology*. (1949).

44. Leitao, J. C., Miotto, J. M., Gerlach, M. & Altmann, E. G. Is this scaling nonlinear? *R. Soc. Open Sci.* **3**, 150649 (2016).

45. Bettencourt, L. M. The origins of scaling in cities. *Science* **340**, 1438–1441 (2013).

46. UNPD. World Urbanization Prospects: The 2018 Revision. (2018).

47. Von Thünen, J. H. *Der Isolierte Staat in Beziehung Auf Landwirtschaft Und Nationalökonomie*. (1826).

48. Newman, P. W. & Kenworthy, J. R. Gasoline consumption and cities: a comparison of US cities with a global survey. *J. Am. Plan. Assoc.* **55**, 24–37 (1989).

49. Delloye, J., Lemoy, R. & Caruso, G. Alonso and the scaling of urban profiles. *Geogr. Anal.* **52**, 127–154 (2020).

50. Laziou, G., Lemoy, R. & Texier, M. L. Radial analysis and scaling of housing prices in French urban areas. *Environ. Plan. B Urban Anal. City Sci.* **52**, 1147–1162 (2025).

51. Li, Y., Rybski, D. & Kropp, J. P. Singularity cities. *Environ. Plan. B: Urban Anal. City Sci.* **48**, 43–59 (2021).

52. Goldstein, B., Gounaridis, D. & Newell, J. P. The carbon footprint of household energy use in the United States. *Proc. Natl Acad. Sci.* **117**, 19122–19130 (2020).

53. Wardrop, N. et al. Spatially disaggregated population estimates in the absence of national population and housing census data. *Proc. Natl Acad. Sci.* **115**, 3529–3537 (2018).

54. Pesaresi, M. & Politis, P. GHS-BUILT-V R2023A - GHS built-up volume grids derived from joint assessment of Sentinel2, Landsat, and global DEM data, multitemporal (1975–2030). https://doi.org/10.2905/AB2F107A-03CD-47A3-85E5-139D8EC63283 (2023).

55. Kuffer, M., Owusu, M., Oliveira, L., Sliuzas, R. & van Rijn, F. The Missing Millions In Maps: Exploring Causes Of Uncertainties In Global Gridded Population Datasets. *ISPRS Int. J. Geo-Inf.* **11**, 403 (2022).

56. Leyk, S. et al. The spatial allocation of population: a review of large-scale gridded population data products and their fitness for use. *Earth Syst. Sci. Data* **11**, 1385–1409 (2019).

57. Kemper, P. & Schmenner, R. The density gradient for manufacturing industry. *J. Urban Econ.* **1**, 410–427 (1974).

## Acknowledgements

The authors acknowledge the help of P. Kilgarriff to locate city centers, and financial support by the French National Research Agency ANR (GreenLand project, ANR-21-CE03-0013). This research makes use of the computing and data-processing resources of the CNRS/TGIR HUMA-NUM Computing Center.
**Author contributions**

G.L. and R.L. designed the research. G.L. performed data collection and processing. G.L and R.L. analyzed the results and wrote the paper.

**Competing interests**

The authors declare no competing interests.

**Additional information**

**Supplementary information** The online version contains supplementary material available at https://doi.org/10.1038/s42949-025-00262-4.

**Correspondence** and requests for materials should be addressed to Gaëtan Laziou.

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