A worldwide model for boundaries of urban settlements

1. Introduction

What are cities? In The Death and Life of the Great American Cities, Jacobs argues that human relations can be seen as a proxy for places within cities [1]. A modern view of cities establishes that they can be defined by the interactions among several types of networks [2,3], from infrastructure networks to social networks. In recent years, an increasing number of studies have been proposed to define cities through consistent mathematical models [4–15] and to investigate urban indicators at inter- and intra-city scales, in order to shed some light on problems faced by decision-makers [16–31]. Despite the efforts of such studies, properly defining the boundaries of urban settlements remains an open problem in the science of cities. A minimum criterion of acceptability for any model of cities seems to be the one that retrieves a conspicuous scaling law found for USA, UK and other countries, known as Zipf’s law [6,7,32–42]. In 1949, Zipf [43] observed that the frequency of words used in the English language obeys a natural and robust power law behaviour, i.e. a few words are used many times, while many words are used just a few times. Zipf’s law can be represented generically by the following relationship between the size S of objects from a given set and its rank R:

where ζ=1 is Zipf’s exponent. The size of objects is, in the original context, the frequency of used words. On the other hand, if such objects are cities, then the sizes stand for the population of each city, taking into account Zipf’s law and reflecting the fact that there are more small towns than metropolises in the world. We emphasize that it is not straightforward that Zipf’s law, despite its robustness, should hold independently of the city definition, as other scaling relations are not, such as the allometric exponents for CO₂ emissions and light pollution [24,31]. Many other man-made and natural phenomena also exhibit the same persistent result, e.g. earthquakes and incomes [44,45].

Here, we propose a worldwide model to define urban settlements beyond their usual administrative boundaries through a bottom-up approach that takes into account cultural, political and geographical biases naturally embedded in the population distribution of continental areas. After all, it is not surprising that two regions, e.g. one in western Europe and another one in eastern Asia, spatially contiguous in population or in commuting level have different cultural, political or geographical characteristics. Thus, it is also not surprising that such issues yield different stages of the same mechanics of growth. The main goal of our model is to be successful in defining cities even in large regions. Our conjecture is straightforward: there are hierarchical mechanisms, similar to those present in previous studies of cities in the UK [14] and brain networks [46], behind the growth and innovation of urban settlements. These mechanisms are ruled by a combination of general measures, such as the population and the area of each city, and intrinsic factors which are specific to each region, e.g. topographical heterogeneity, political and economic issues, and cultural customs and traditions. In other words, if political turmoil or economic recession plagues a metropolis for a long time, all of its satellites are affected too, i.e. the entire region ruled by the metropolis will be negatively impacted.

2. The models

2.1. City clustering algorithm

In 2008, Rozenfeld et al. [6] proposed a model to define cities beyond their usual administrative boundaries using a notion of spatial continuity of urban settlements, called the city clustering algorithm (CCA) [6–8,11,15,30,24,31]. The CCA is defined for discrete or continuous landscapes [7] by two parameters: a population density threshold D* and a distance threshold ℓ. These parameters describe the populated areas and the commuting distance between areas, respectively. Here, we adopt the following strategy to improve the discrete CCA performance. (i) Supposing a regular rectangular lattice L_x×L_y of sites where the population density of the kth site is D_k, we perform an initial agglomeration by D* to identify all clusters. If D_k>D*, then the kth site is populated and we aggregate it with its populated nearest neighbours. Otherwise, the kth site is unpopulated. (ii) For each populated cluster, we define its shell sites, i.e. sites in the interface between populated and unpopulated areas. (iii) Lastly, we perform a final agglomeration by ℓ, taking into account only the shell sites. If d_ij<ℓ, where d_ij is the distance between the ith and jth shell sites, and if they belong to different clusters, then the ith and jth sites belong to the same CCA cluster, even with spatial discontinuity. Otherwise, they indeed belong to different CCA clusters. This simple strategy improves the algorithm’s computational performance because the number of shell sites is proportional to L, where L=L_x≈L_y is a linear measure of the lattice.

2.2. City local clustering algorithm

We propose a worldwide model based on the CCA, called the city local clustering algorithm (CLCA), not only to define cities beyond their usual administrative boundaries, but also to take into account the intrinsic cultural, political and geographical biases associated with most societies and reflected in their particular growing dynamics. The traditional CCA, with fixed ℓ and D*, when applied to a large population density map, can introduce biases defining a lot of clusters in some regions, while in others just a few. We present the CLCA with the aim of defining cities even in large regions in order to overcome such CCA weakness. Hence, it is possible that other models, such as the models based on street networks proposed by Masucci et al. [13] and Arcaute et al. [14], carry the same CCA burden and that local adaptations are necessary for their applications into large regions.

The main idea of our model is to analyse the change of the CCA clusters through the variation of D* under the perspective of different regions. First, we define a regular rectangular lattice L_x×L_y of sites, where the population density of the kth site is D_k. We sort all the sites in a list according to the population density, in descending order. Therefore, the site with the greatest population density is the first entry in this list, which we call the first reference site. The reference site can be considered as the current core of the analysed region. Second, we apply the CCA to the lattice, keeping a fixed value of ℓ, for a range of D* decreasing from a maximum value D^(max) to a minimum value D^(min) with a decrement δ. During the decreasing of D*, clusters are formed and they spread out to all regions of the lattice. Eventually, the cluster that contains the reference site (from now on the reference cluster), together with one or more of the other clusters, will merge from D⁽ⁱ⁾ to D⁽ⁱ⁺¹⁾, where D⁽ⁱ⁺¹⁾=D⁽ⁱ⁾−δ. In order to accept or deny the merging of these clusters, we introduce three conditions:

• (i) If the area A_r(D⁽ⁱ⁾) of the reference cluster r, i.e. the cluster that contains the rth reference site at D⁽ⁱ⁾, obeys

  $$A_r(D^{(i)})<A^{\ast },$$2.1
  then the reference cluster r always merges with other clusters, because it is still considered very small. In this context, the area A* can be understood as the minimal area of a metropolis.

• (ii) If the difference between the areas of the reference cluster r at D⁽ⁱ⁺¹⁾ and D⁽ⁱ⁾ obeys

  $$A_r(D^{(i+1)})-A_r(D^{(i)})>H^{\ast }{A}_r(D^{(i)}),$$2.2
  then the reference cluster r has grown without merging (figure 1a) or there is a merging of at least two large clusters (figure 1b). In the last case, we emphasize that if there are more than two clusters involved in the merging process, the reference cluster r may not be one of the largest. As the first case is not desirable, we can avoid it by reducing the value of δ and keeping the value of H* relatively high. The parameter H* can be understood as the percentage of the area of the reference cluster r at D⁽ⁱ⁾. If the second case happens, we consider the entire region inside of the reference cluster r at D⁽ⁱ⁺¹⁾, but the clusters of this region (which we call the usual clusters) are defined by those at D⁽ⁱ⁾. The usual clusters are the CCA clusters at the imminence of the merging process between D⁽ⁱ⁾ and D⁽ⁱ⁺¹⁾. This includes the reference cluster r itself and one or more of the other clusters before the merging (figure 1b). Furthermore, all of the sites of the reference cluster r at D⁽ⁱ⁺¹⁾ are removed from the initial list of reference sites. This condition is necessary because we should not merge two large metropolises.

• (iii) In condition (ii), when a reference cluster r is merging with another cluster that covers one or more regions already defined by previous reference clusters at different values of D*, there is a strong likelihood of the emergence of a forbidden region within that cluster. In this case, we force the region already defined by the largest value of D* to grow to the limits of the forbidden region (figure 1c). The forbidden regions are the complementary areas of the reference clusters already defined within the usual clusters. As a consequence of this procedure, some CCA clusters that were hidden after the analysis of the previous reference cluster arise in this forbidden region. We justify this condition by the idea that a metropolis rules the growth of its satellites, as it plays a fundamental role in their socioeconomic relations.

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We apply the same procedure to the second reference cluster, to the third reference cluster and so on. Finally, we also define the isolated clusters with the minimum value of D* for all the cases accepted in condition (ii). In order to make our model clearer, we chose the descending order to sort the population density for one reason: to favour the merging process of the high-density clusters that arose from the decreasing of D*. In practice, we run our revised discrete CCA just once for the entire range of input parameters and store all of the outputs in order to improve the performance of the model. The apparent simplicity of this task hides a RAM management problem of storing all of the outputs in a medium-performance computer. We overcome such a barrier through the zram module [47], available in the newest linux kernels. The zram module creates blocks which compress and store information dynamically in the RAM itself, at the cost of processing time.

3. The dataset

We use the GRUMPv1 [48], available from the Socioeconomic Data and Applications Center (SEDAC) at Columbia University, to apply the CLCA to a single global dataset. The GRUMPv1 dataset is composed of georeferenced rectangular population grids for 232 countries around the world in the year 2000 (figure 2). Such a dataset is a compilation of gridded census and satellite data for the populations of urban and rural areas. These data are provided at a high resolution of 30 arc-seconds, equivalent to 30/3600^° or a grid of 0.926×0.926 km at the Equator. We note that despite the heterogeneous population distributions that built the GRUMPv1, its overall resolution is tolerable to the CLCA, since we can identify well-defined clusters around all continents in the raw data.

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We calculate the area of each site by the composition of two spherical triangles [49]. The area of a spherical triangle with edges a, b and c is given by

where s=(a/R_e+b/R_e+c/R_e)/2, s_a=s−a/R_e, s_b=s−b/R_e and s_c=s−c/R_e. In this formalism, R_e=6378.137 km is the Earth’s radius and the edge lengths are calculated by the great circle (geodesic) distance between two points i and j on the Earth’s surface: The values of λ_i (λ_j) and ϕ_i (ϕ_j), measured in radians, are the longitude and latitude, respectively, of the point i (j). Thus, we are able to define the population density for each site of the lattice, since its population and area are known.

We also pre-process the GRUMPv1 dataset, dividing all countries and continents—and even the entire world—into large regions which we call clusters of regions, to apply our model in a feasible computational time using medium-performance computers. These regions are defined by the CCA with lower and upper bound parameters D*=50 people km⁻² and ℓ=10 km, respectively. We believe that such large clusters can hold the socioeconomic and cultural relations among different urban settlements of a territory. Figure 3a shows the largest clusters of regions in the USA; as we can see, all of the eastern USA is considered a single cluster.

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4. Results

To show the relevance of our model, we apply the CLCA to the GRUMPv1 dataset at three different geographical levels: countries, continents and the entire world. For each case, we consider only a single set of CLCA parameters. We justify our choices with the following assumptions: (i) D^(min)=100 people km⁻², a value slightly greater than the lower bound CCA parameter (D*=50 people km⁻²) used to define the regions of clusters; (ii) D^(max)=1000 people km⁻², a loosened value of $D^{(\max )}=\mathrm{\infty }$; (iii) δ=10 people km⁻², a small enough value to avoid the reference clusters growing without merging; (iv) ℓ=3 km, the critical distance threshold, already extensively analysed by previous CCA studies [6,7,24]; (v) A*=50 km², the minimum area of a metropolis, as it is required that A* be reasonably greater than the minimum unit of area from the dataset and smaller than a metropolis’ area; and (vi) H*=0.05, a large enough value to favour the merging of clusters which are similar in size. Figure 3b shows the CLCA cities defined by the single set of CLCA parameters. For other regions, see the electronic supplementary material.

We study the population distribution using the maximum-likelihood estimator (MLE) proposed by Clauset et al. [50]. Their approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on Kolmogorov–Smirnov statistic. Figure 4 shows the log–log behaviour of the cumulative distribution function (CDF) for the population of the CLCA cities, considering only the countries with the highest number of CLCA cities for each continent (for other countries, see the electronic supplementary material). The $Pr(\mathcal{P}\ge P)$ represents the probability that a random population $\mathcal{P}$ takes on a value greater than or equal to the population P. In all CDF plots, we also show the maximum-likelihood power-law fit, as well as the value of the exponent ζ=α−1, where α is the MLE exponent, and the value of $P_{min}$, the lower bound of the MLE.

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In figure 5, we show a normalized histogram, with frequency F, of the ζ exponents for all countries (145 out of 232) with at least 10 CLCA cities in the region covered by the maximum-likelihood power-law fit. The mean value of the ζ exponents is $\overline{\zeta }=0.98$, with variance σ²=0.09. The dashed red line stands for the normal distribution $\mathcal{N}(\overline{\zeta },{\sigma }^2)$. In spite of the ζ exponent heterogeneity illustrated by figure 5, Zipf’s law holds for most countries around the globe. We emphasize that such results corroborate with previous studies performed for one country or a small number of countries [6,7,32–42]. In particular, the figure 5 also endorses an astute meta-analysis performed by Cottineau [51]. Cottineau provided a comparison among Zipf’s law exponents found in 86 studies. Our results strongly corroborate those presented in such study, except that our exponents are ranged between 0 and 2.

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Furthermore, we challenge the robustness of our model at higher geographical levels: continents and the entire world. We performed the same analyses and find that our results persist on both scales, i.e. the CLCA cities follow Zipf’s law for continents and the entire world, as illustrated in figures 6 and 7.

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We summarize our results in a set of seven tables: tables 1–6, for countries from Africa, Asia, Europe, North America, Oceania and South America, respectively. Table 7 contains similar information for all continents and the entire world. In all cases, we show the name of the considered region (country, continent or globe), the ISO 3166-1 alpha-3 code associated (only for countries), the number of cities obtained by the CLCA and those covered by the MLE, the lower bound $P_{min}$ and the Zipf exponent ζ.

It is remarkable that the top CLCA city, with a population of 63 585 039 people, is composed of three large urban settlements (Alexandria, Cairo and Luxor) connected by several small ones. Figure 8a–c shows the largest cluster of regions in Egypt for the GRUMPv1 dataset, CLCA cities and night-time lights from the National Aeronautics and Space Administration (NASA) [52], respectively. We believe the main reason for this finding has been present in the northeast of Africa since before the beginning of ancient civilization—namely, the Nile river. Actually, it is well known that almost the entire Egypt population lives in a strip along the Nile river, in the Nile delta and in the Suez canal on 4% of the total country area (10⁶ km²), where there are arable lands to produce food [53]. The river and delta regions are composed by some large cities and a lot of small villages, making them extremely dense. Therefore, our results raise the hypothesis that the cities and villages across the Nile can be seen as a kind of ‘megacity’, despite spatially non-contiguous, due to the socioeconomic relation, reflected in the high commuting levels, among close subregions.

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Table 8 shows the top 25 CLCA cities in the entire world by population, and their associated areas. After the top CLCA city, Alexandria-Cairo-Luxor, we emphasize that the 13 next-largest CLCA cities are in Asia. Indeed, we can see that the shape of the tail end of the entire world population distribution (in figure 7) is roughly ruled by the greater CLCA city in Africa and several CLCA cities in Asia.

These facts are not in line with what was recently reported by the United Nations (UN) [54], e.g. the largest CLCA city, Alexandria-Cairo-Luxor, is just the 9th largest city according to the UN, and the largest UN city, Tokyo, is just the 4th largest according to our analyses.

5. Conclusion

We propose a model to define urban settlements through a bottom-up approach beyond their usual administrative boundaries, and moreover to account for the intrinsic cultural, political and geographical biases associated with most societies and reflected in their particular growing dynamics. We claim that such a property qualifies our model to be applied worldwide, without any regional restrictions. We also propose an alternative strategy to improve the computational performance of the discrete CCA. We emphasize that the CCA can still be used to define cities; however, it depends upon a different tuning of its parameters for each large region without direct socioeconomic and political relations. Furthermore, we show that the definition of cities proposed by our approach is robust and holds to one of the most famous results in social science, Zipf’s law, not only for previously studied countries, e.g. the USA, the UK or China, but for all countries (145 from 232 provided by GRUMPv1) around the world. We also find that Zipf’s law emerges at different geographical levels, such as continents and the entire world. Another highlight of our study is the fact that our model is applied upon one single dataset to define all cities. Furthermore, we find that the most populated cities are not the major players in the global economy (such as New York City, London or Tokyo). The largest CLCA city, with a population of 63 585 039 people, is an agglomeration of several small cities close to each other which connects three large cities: Alexandria, Cairo and Luxor. Finally, after the top CLCA city of Alexandria-Cairo-Luxor, we find that the next-largest 13 CLCA cities are in Asia. These facts are not in full agreement with a recent UN report [54]. According to our results, the largest CLCA city, Alexandria-Cairo-Luxor, is just the 9th largest city according to the UN, while the largest UN city, Tokyo, is just the 4th largest according to our analyses.

Data accessibility

The data supporting this article are available at http://sedac.ciesin.columbia.edu/data/collection/grump-v1. More specifically, the reader can click on ‘Data sets’ and, after that, on ‘Population Count Grid, v1 (1990,1995,2000)’. We also provide the codes for the proposed model that are available at https://doi.org/10.5061/dryad.968nq8n [55].

Authors' contributions

E.A.O. performed the data analysis, the algorithm of the proposed model, and the statistical analysis. He also participated in the design of the study and drafted the manuscript. V.F. carried out the funding acquisition and helped draft the manuscript. J.S.A. participated in the design of the study, carried out the funding acquisition, and helped draft the manuscript. H.A.M. conceived, designed, and coordinated the research, as well as carried out the funding acquisition and helped draft the manuscript. All authors approved the manuscript.

Competing interests

We declare we have no competing interests.

Funding

We gratefully acknowledge funding by CNPq, CAPES, FUNCAP, NSF, ARL Cooperative Agreement no. W911NF-09-2-0053 (the ARL Network Science CTA), NIH-NIBIB 1R01EB022720, NIH-NCI U54CA137788/U54CA132378 and NSF-IIS 1515022.

Footnotes

Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.4089005.