# The Size Distribution, Scaling Properties and Spatial Organization of Urban Clusters: A Global and Regional Percolation Perspective

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. City Clustering and Land Cover Data

## 3. Global City Size Distribution

^{2}.

^{4}inhabitants, resulting in an exponent ${\zeta}_{S}=1.85$. Accordingly, we observe that Zipf’s law approximately holds for the cities on the global scale, whereas the actual exponent is smaller than 2. We explore also l = 4 km and similar power-law size distributions, however with a different exponent in the case of the areas (Figure 3a).

## 4. Percolation Transition and Size Distribution on the Country Scale

#### 4.1. Percolation Transition

_{c}at which the giant cluster component spans within a country territory. The critical value l

_{c}is analogous to the critical occupation probability P

_{c}, which constitutes the control parameter in most of lattice percolation formulations [22]. Both quantities are approximately related by $P\sim {l}^{\beta}$ with $\beta \approx 2$.

_{c}percolation threshold. We find that the average cluster size excluding the largest cluster, ${\langle A\rangle}^{*}$, constitutes a sensitive indicator of the transition. In infinite systems, ${\langle A\rangle}^{*}$ diverges at P

_{c}[22] whereas for finite systems, a (finite) peak occurs. Similarly, one can in principle detect the presence of a peak in ${\langle A\rangle}^{*}$ around a value l

_{c}when applying CCA to the urban land cover of different countries. Since in the limit of small l the urban clusters identified by the CCA approximate the cells of urban land cover, we conjecture that a small l

_{c}value constitutes a proxy indicator of the percolation threshold of the urban land cover.

_{c}= 15 km. As it can be observed, for l < l

_{c}the average cluster size increases strongly with l, yet gradually, and it drops sharply for l > l

_{c}.

#### 4.2. City Size Distribution

_{c}cannot be identified unambiguously, as in the presence of multiple peaks—often a signature of large clusters being disconnected by vastly extended topographic heterogeneities. Therefore, we focus on a selected set of countries exhibiting (i) a clear percolation threshold and (ii) a large number of urban areas.

#### 4.3. Average Size Scaling

#### 4.4. Taylor’s Law For City Size Distribution

#### 4.5. Spatial Correlations

## 5. Fundamental Urban Allometry—Relating Area and Population

## 6. Summary and Discussion

^{2}and a population of around 3.7 million, is classified as bare area.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Auerbach, F. Das gesetz der bevölkerungskonzentration. Petermanns Geogr. Mitt.
**1913**, 59, 73–76. [Google Scholar] - Rybski, D. Auerbach’s legacy. Environ. Plan. A
**2013**, 45, 1266–1268. [Google Scholar] [CrossRef] - Zipf, G.K. Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology (Reprint of 1949 Edition); Martino Publishing: Mansfield, CT, USA, 2012. [Google Scholar]
- Gibrat, R. Les Inégalités Économiques; Libraire du Recueil Sierey: Paris, France, 1931. [Google Scholar]
- Simon, H.A. On a class of skew distribution functions. Biometrika
**1955**, 42, 425–440. [Google Scholar] [CrossRef] - Makse, H.A.; Andrade, J.S.; Batty, M.; Havlin, S.; Stanley, H.E. Modeling urban growth patterns with correlated percolation. Phys. Rev. E
**1998**, 58, 7054–7062. [Google Scholar] [CrossRef] - Gabaix, X. Zipf’s law for cities: An explanation. Q. J. Econ.
**1999**, 114, 739–767. [Google Scholar] [CrossRef] - Rybski, D.; Ros, A.G.C.; Kropp, J.P. Distance weighted city growth. Phys. Rev. E
**2013**, 87, 042114. [Google Scholar] [CrossRef] [PubMed] - Eeckhout, J. Gibrat’s law for (All) cities. Am. Econ. Rev.
**2004**, 94, 1429–1451. [Google Scholar] [CrossRef] - Kriewald, S.; Fluschnik, T.; Reusser, D.; Rybski, D. OSC: Orthodromic Spatial Clustering, R package version 1.0.0. Available online: https://CRAN.R-project.org/package=osc (accessed on 7 April 2011).
- Rozenfeld, H.D.; Rybski, D.; Andrade, J.S., Jr.; Batty, M.; Stanley, H.E.; Makse, H.A. Laws of population growth. Proc. Nat. Acad. Sci. USA
**2008**, 105, 18702–18707. [Google Scholar] [CrossRef] [PubMed] - Stauffer, D.; Aharony, A. Introduction To Percolation Theory; Taylor & Francis: London, UK, 1994. [Google Scholar]
- ESA (European Space Agency). The Ionia GlobCover Project. GlobCover Land Cover 2009 v2.3. Available online: http://ionia1.esrin.esa.int (accessed on 21 August 2011).
- Center for International Earth Science Information Network (CIESIN), Columbia University; International Food Policy Research Institute (IFPRI), The World Bank; Centro Internacional de Agricultura Tropical (CIAT). Global Rural-Urban Mapping Project, Version 1 (GRUMPv1): Settlement Points. Website. 2011. Available online: http://sedac.ciesin.columbia.edu/data/dataset/grump-v1-settlement-points (accessed on 26 November 2011).
- Rozenfeld, H.D.; Rybski, D.; Gabaix, X.; Makse, H.A. The area and population of cities: New insights from a different perspective on cities. Am. Econ. Rev.
**2011**, 101, 2205–2225. [Google Scholar] [CrossRef] - Zanette, D.H.; Manrubia, S.C. Role of intermittency in urban development: A model of large-scale city formation. Phys. Rev. Lett.
**1997**, 79, 523–526. [Google Scholar] [CrossRef] - Batty, M. The size, scale, and shape of cities. Science
**2008**, 319, 769–771. [Google Scholar] [CrossRef] [PubMed] - Schweitzer, F.; Steinbrink, J. Estimation of megacity growth—Simple rules versus complex phenomena. Appl. Geogr.
**1998**, 18, 69–81. [Google Scholar] [CrossRef] - Kinoshita, T.; Kato, E.; Iwao, K.; Yamagata, Y. Investigating the rank-size relationship of urban areas using land cover maps. Geophys. Res. Lett.
**2008**, 35, L17405. [Google Scholar] [CrossRef] - Arcaute, E.; Hatna, E.; Ferguson, P.; Youn, H.; Johansson, A.; Batty, M. Constructing cities, deconstructing scaling laws. J. R. Soc. Interface
**2014**, 12, 20140745. [Google Scholar] [CrossRef] [PubMed] - Small, C.; Elvidge, C.D.; Balk, D.; Montgomery, M. Spatial scaling of stable night lights. Remote Sens. Environ.
**2011**, 115, 269–280. [Google Scholar] [CrossRef] - Bunde, A.; Havlin, S. (Eds.) Fractals and Disordered Systems; Springer-Verlag: New York, NY, USA, 1991.
- Berry, B.J.L.; Okulicz-Kozaryn, A. The city size distribution debate: Resolution for US urban regions and megalopolitan areas. Cities
**2012**, 29, S17–S23. [Google Scholar] [CrossRef] - Makse, H.A.; Havlin, S.; Stanley, H.E. Modeling urban-growth patterns. Nature
**1995**, 377, 608–612. [Google Scholar] [CrossRef] - Bitner, A.; Holyst, R.; Fialkowski, M. From complex structures to complex processes: Percolation theory applied to the formation of a city. Phys. Rev. E
**2009**, 80, 037102. [Google Scholar] [CrossRef] [PubMed] - Murcio, R.; Sosa-Herrera, A.; Rodriguez-Romo, S. Second-order metropolitan urban phase transitions. Chaos Soliton Fract.
**2013**, 48, 22–31. [Google Scholar] [CrossRef] - Arcaute, E.; Molinero, C.; Hatna, E.; Murcio, R.; Vargas-Ruiz, C.; Masucci, P.; Batty, M. Regions and Cities in Britain through Hierarchical Percolation. Available online: http://arxiv.org/abs/1504.08318v2 (accessed on 10 July 2016).
- Clauset, A.; Shalizi, C.R.; Newman, M.E.J. Power-Law distributions in empirical data. SIAM Rev.
**2009**, 51, 661–703. [Google Scholar] [CrossRef] - Vuong, Q.H. Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica
**1989**, 57, 307–333. [Google Scholar] [CrossRef] - Stanley, H.E. Scaling, universality, and renormalization: Three pillars of modern critical phenomena. Rev. Mod. Phys.
**1999**, 71, S358–S366. [Google Scholar] [CrossRef] - Taylor, L.R. Aggregation, variance and mean. Nature
**1961**, 189, 732–735. [Google Scholar] [CrossRef] - Smith, H.F. An empirical law describing heterogeneity in the yields of agricultural crops, Part: 1. J. Agric. Sci.
**1938**, 28, 1–23. [Google Scholar] [CrossRef] - de Menezes, M.A.; Barabasi, A.L. Fluctuations in network dynamics. Phys. Rev. Lett.
**2004**, 92, 028701. [Google Scholar] [CrossRef] [PubMed] - Eisler, Z.; Bartos, I.; Kertész, J. Fluctuation scaling in complex systems: Taylor’s law and beyond. Adv. Phys.
**2008**, 57, 89–142. [Google Scholar] [CrossRef] - Bates, D.M.; DebRoy, S. R-Documentation: Nonlinear Least Squares. Available online: http://stat.ethz.ch/R-manual/R-patched/library/stats/html/nls.html (accessed on 7 April 2011).
- Weinrib, A. Long-range correlated percolation. Phys. Rev. B
**1984**, 29, 387–395. [Google Scholar] [CrossRef] - Prakash, S.; Havlin, S.; Schwartz, M.; Stanley, H.E. Structural and dynamic properties of long-range correlated percolation. Phys. Rev. A
**1992**, 46, R1724–R1727. [Google Scholar] [CrossRef] [PubMed] - Bettencourt, L.; West, G. A unified theory of urban living. Nature
**2010**, 467, 912–913. [Google Scholar] [CrossRef] [PubMed] - Batty, M. Defining city size. Environ. Plan. B-Plan. Des.
**2011**, 38, 753–756. [Google Scholar] [CrossRef] - Rybski, D.; Reusser, D.E.; Winz, A.L.; Fichtner, C.; Sterzel, T.; Kropp, J.P. Cities as nuclei of sustainability? Environ. Plan. B
**2016**. [Google Scholar] [CrossRef] - Sutton, P.; Roberts, D.; Elvidge, C.; Baugh, K. Census from Heaven: An estimate of the global human population using night-time satellite imagery. Int. J. Remote Sens.
**2001**, 22, 3061–3076. [Google Scholar] [CrossRef] - Potere, D.; Schneider, A.; Angel, S.; Civco, D.L. Mapping urban areas on a global scale: which of the eight maps now available is more accurate? Int. J. Remote Sens.
**2009**, 30, 6531–6558. [Google Scholar] [CrossRef] - Zhou, B.; Rybski, D.; Kropp, J.P. On the statistics of urban heat island intensity. Geophys. Res. Lett.
**2013**, 40, 5486–5491. [Google Scholar] [CrossRef] - Jefferson, M. The law of the primate city. Geogr. Rev.
**1939**, 29, 226–232. [Google Scholar] [CrossRef] - Arcaute, E.; Hatna, E.; Ferguson, P.; Youn, H.; Johansson, A.; Batty, M. Constructing cities, deconstructing scaling laws. J. R. Soc. Interface
**2014**, 12, 20140745. [Google Scholar] [CrossRef] [PubMed] - Pisarenko, V.F.; Sornette, D. Robust statistical tests of Dragon-Kings beyond power law distributions. Eur. Phys. J. Spec. Top.
**2012**, 205, 95–115. [Google Scholar] [CrossRef] - Pumain, D.; Moriconi-Ebrard, F. City size distributions and metropolisation. GeoJournal
**1997**, 43, 307–314. [Google Scholar] [CrossRef]

**Figure 1.**Application of city clustering to urban land cover data. The following panels illustrate different aspects of the city of Paris and its surroundings: (

**a**) Remote sensing image as extracted from the ArcGIS 10 component ArcMap; (

**b**) Urban land cover data as obtained from the GlobCover 2009 land cover map. The colors indicate urban (red, class 190), water bodies (blue, class 210), forests and grasslands (green, classes 20-110), and rainfed croplands (yellow, class 14); (

**c**) From the urban land cover and by taking l = 4 km, the identified clusters are color coded according to the logarithm of their size: from small (light blue) via medium (green) to large (red). The cutout has the approximate area of (215 km)

^{2}.

**Figure 2.**Global map of cluster area in km

^{2}for l = 4 km. For better visibility all clusters (15,640) are plotted as single dots, instead of spatial extent. Notice that only clusters with associated population information are displayed. Exemplary the spatial extent is shown for (

**a**) Austria (460 clusters) and (

**b**) France (1907 clusters). Due to the fact that small clusters are much more often than big ones, they are more present in the global dotted map compared to the spatial explicit country examples. Furthermore noticeable is the highly sprawled small urban clusters in the US compared with the concentrated dens urban hot spots in India.

**Figure 3.**Probability density of city size in terms of area and population. (

**a**) Cluster area distribution p(A), as obtained by applying the City Clustering Algorithm (CCA) to global land cover data and extracting all urban clusters on the globe. For A > 0.1 km

^{2}we estimate ${\zeta}_{A}\approx 1.93$ for l = 0.4 km (249,512 clusters) and ${\zeta}_{A}\approx 1.75$ for l = 4 km (46,754 clusters); (

**b**) Cluster population distribution p(S), as obtained from associating population settlement points with the urban clusters identified by means of CCA. For $S>{10}^{4}$ we estimate ${\zeta}_{S}\approx 1.85$ for l = 0.4 km and ${\zeta}_{S}\approx 1.75$ for l = 4 km. In both panels: l = 0.4 km (circles), l = 4 km (squares). The solid grey lines have slope −2.

**Figure 4.**Percolation and Zipf’s law. (

**a**) Average cluster size excluding the largest component ${\langle A\rangle}^{*}$ as a function of the clustering parameter l for Austria. The maximum is located at the percolation transition, which in this example is ${l}_{\mathrm{c}}\simeq 15$ km; (

**b**) Probability density of cluster areas $p\left(A\right)$ for Austria and for $l\simeq \frac{1}{3}{l}_{\mathrm{c}}$ (370 clusters, green triangles) as well as $l\simeq \frac{2}{3}{l}_{\mathrm{c}}$ (87 clusters, brown squares). The dotted grey lines have the slopes −1.71 and −1.27; (

**c**) Estimated power-law distribution exponent ${\zeta}_{A}$ as a function of the rescaled clustering parameter $l/{l}_{\mathrm{c}}$ for various countries as indicated by colored dots. Since we found out that the method proposed in [28] has a significant deviation from the real value for input with less than 100 entries, we estimated the power-law distribution exponents for each country just for those l with at least 100 clusters remaining. The open circles represent averages in logarithmic bins and their error bars the corresponding standard deviations. The exponent decreases with increasing $l/{l}_{\mathrm{c}}$ and takes values between ${\zeta}_{A}\approx 2$ for $l/{l}_{\mathrm{c}}\ll 1$ and ${\zeta}_{A}\approx 3/2$ for $l/{l}_{\mathrm{c}}\to 1$.

**Figure 5.**Fitting of the average size scaling for Austria (

**a**) and Denmark (

**b**). In both cases we have a unique clear peak in the curve of ${\langle A\rangle}^{*}$ against l. The red lines represent the exponents of the function $f\left(x\right)=c\xb7{x}^{a}$ fitted to the green highlighted parts. Fittings yields the parameter $a\approx -1.97$ for Austria and $a=-1.25$ for Denmark.

**Figure 6.**Taylor’s law for city sizes. The standard deviation ${\sigma}_{A}^{*}$ of cluster sizes disregarding the largest cluster is plotted as a function of ${\langle A\rangle}^{*}$ for various values of l. The panels show the results for (

**a**) Austria and (

**b**) Spain. The insets depict the corresponding curve of ${\langle A\rangle}^{*}$ against l (see also Figure 4a). In the panels and insets corresponding coloring is used in order to enable comparison. For Austria, the standard deviation ${\sigma}_{A}^{*}$ reaches its maximum at $l=15$ km (light-blue) and for Spain it is located at $l=13.6$ km (yellow). In the former case this maximum corresponds to the percolation point. In the latter case the maximum corresponds to the first peak one can identify in the curve of ${\langle A\rangle}^{*}$ against l.

**Figure 7.**Spatial correlation computations for Austria (

**a**) and Netherlands (

**b**). The fitted curve (red) on the green-highlighted points which follows the function $f\left(x\right)=c\xb7{x}^{a}\xb7exp(b\xb7x)$ (power-law with exponential cut-off) has the values for Austria $(a,b,c)\approx (-0.18,-1.87\times {10}^{-4},0.01)$ and for the Netherlands $(a,b,c)\approx (-0.65,-1.05\times {10}^{-5},2.02)$. The grey lines (fitted on the corresponding points below the displayed part) have slopes $\approx -0.46$ for Austria and $\approx -0.73$ for the Netherlands. Beyond the shown range, $C\left(d\right)$ fluctuates around zero.

**Figure 8.**Correlations between area and population. (

**a**) $log\left(S\right)$ vs. $log\left(A\right)$ for l = 0.4 km and of all land cover clusters that could be matched with population figures (grey dots). The blue vertical and horizontal lines truncate the fraction q = 0.2 along both axis in order to avoid the discreteness at small A. The green solid line corresponds to the main axis around which the momentum of inertia of the truncated cloud is minimal. In this case, 8786 out of 12321 clusters remain. It’s slope δ is smaller than the diagonal (black dashed line, background); (

**b**) Slope δ (circles) and Pearson correlation coefficients C (squares) vs. clustering parameter l for the cut-off q = 0.2 (green, red), q = 0.3 (magenta), q = 0.4 (orange). The exponent δ is found roughly between 0.82 and 0.87 except for small l where δ up to 0.87 and 0.93 are achieved but always below 1. For q = 0.2 the correlations exhibit a maximum around 5 km.

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**MDPI and ACS Style**

Fluschnik, T.; Kriewald, S.; García Cantú Ros, A.; Zhou, B.; Reusser, D.E.; Kropp, J.P.; Rybski, D. The Size Distribution, Scaling Properties and Spatial Organization of Urban Clusters: A Global and Regional Percolation Perspective. *ISPRS Int. J. Geo-Inf.* **2016**, *5*, 110.
https://doi.org/10.3390/ijgi5070110

**AMA Style**

Fluschnik T, Kriewald S, García Cantú Ros A, Zhou B, Reusser DE, Kropp JP, Rybski D. The Size Distribution, Scaling Properties and Spatial Organization of Urban Clusters: A Global and Regional Percolation Perspective. *ISPRS International Journal of Geo-Information*. 2016; 5(7):110.
https://doi.org/10.3390/ijgi5070110

**Chicago/Turabian Style**

Fluschnik, Till, Steffen Kriewald, Anselmo García Cantú Ros, Bin Zhou, Dominik E. Reusser, Jürgen P. Kropp, and Diego Rybski. 2016. "The Size Distribution, Scaling Properties and Spatial Organization of Urban Clusters: A Global and Regional Percolation Perspective" *ISPRS International Journal of Geo-Information* 5, no. 7: 110.
https://doi.org/10.3390/ijgi5070110