Laws of Population Growth
Hernan D. Rozenfeld, Diego Rybski, Jose S. Andrade Jr., Michael Batty, H. Eugene Stanley, Hernan A. Makse
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Laws of Population Growth
Hern´an D. Rozenfeld , Diego Rybski , Jos´e S. Andrade Jr. ,Michael Batty , H. Eugene Stanley , and Hern´an A. Makse , Levich Institute and Physics Department,City College of New York, New York, NY 10031, USA Departamento de F´ısica, Universidade Federaldo Cear´a, 60451-970 Fortaleza, Cear´a, Brazil Centre for Advanced Spatial Analysis, University College London,1-19 Torrington Place, London WC1E 6BT, UK Center for Polymer Studies and Physics Department,Boston University, Boston, MA 02215, USA
Abstract
An important issue in the study of cities is defining a metropolitan area, as different definitionsaffect conclusions regarding the statistical distribution of urban activity. A commonly employedmethod of defining a metropolitan area is the Metropolitan Statistical Areas (MSAs), based onrules attempting to capture the notion of city as a functional economic region, and is performedusing experience. The construction of MSAs is a time-consuming process and is typically done onlyfor a subset (a few hundreds) of the most highly populated cities. Here, we introduce a new methodto designate metropolitan areas, denoted “City Clustering Algorithm” (CCA). The CCA is basedon spatial distributions of the population at a fine geographic scale, defining a city beyond thescope of its administrative boundaries. We use the CCA to examine Gibrat’s law of proportionalgrowth, which postulates that the mean and standard deviation of the growth rate of cities areconstant, independent of city size. We find that the mean growth rate of a cluster utilizing the CCAexhibits deviations from Gibrat’s law, and that the standard deviation decreases as a power-lawwith respect to the city size. The CCA allows for the study of the underlying process leading tothese deviations, which are shown to arise from the existence of long-range spatial correlations inpopulation growth. These results have socio-political implications, for example for the location ofnew economic development in cities of varied size. . INTRODUCTION In recent years there has been considerable work on how to define cities and how thedifferent definitions affect the statistical distribution of urban activity [1, 2]. This is along standing problem in spatial analysis of aggregated data sources, referred to as the‘modifiable areal unit problem’ or the ‘ecological fallacy’ [3, 4], where different definitions ofspatial units based on administrative or governmental boundaries, give rise to inconsistentconclusions with respect to explanations and interpretations of data at different scales. Theconventional method of defining human agglomerations is through the MSAs [1, 2, 5, 6, 7],which is subject to socio-economical factors. The MSA has been of indubitable importancefor the analysis of population growth, and is constructed manually case-by-case based onsubjective judgment (MSAs are defined starting from a highly populated central area andadding its surrounding counties if they have social or economical ties).In this report, we propose a new way to measure the extent of human agglomerationsbased on clustering techniques using a fine geographical grid, covering both urban andrural areas. In this view, “cities” represent clusters of population, i.e., adjacent populatedgeographical spaces. Our algorithm, the “city clustering algorithm” (CCA), allows for anautomated and systematic way of building population clusters based on the geographicallocation of people. The CCA has one parameter (the cell size) that is useful for the studyof human agglomerations at different length scales, similar to the level of aggregation inthe context of social sciences. We show that the CCA allows for the study of the originof statistical properties of population growth. We use the CCA to analyze the postulatesof Gibrat’s law of proportional growth applied to cities, which assumes that the mean andstandard deviation of the growth rates of cities are constant. We show that populationgrowth at a fine geographical scale for different urban and regional systems at countryand continental levels (Great Britain, the USA, and Africa) deviates from Gibrat’s law.We find that the mean and standard deviation of population growth rates decrease withpopulation size, in some cases following a power-law behavior. We argue that the underlyingdemographic process leading to the deviations from Gibrat’s law can be modeled from theexistence of long-range spatial correlations in the growth of the population, which may arisefrom the concept that “development attracts further development.” These results haveimplications for social policies, such as those pertaining to the location of new economic2evelopment in cities of different sizes. The present results imply that, on average, thegreatest growth rate occurs in the smallest places where there is the greatest risk of failure(larger fluctuations). A corollary is that the safest growth occurs in the largest places havingless likelihood for rapid growth.The analyzed data consist of the number of inhabitants, n i ( t ), in each cell i of a finegeographical grid at a given time, t . The cell size varies for each data set used in this study.We consider three different geographic scales: on the smallest scale, the area of study is GreatBritain (GB: England, Scotland and Wales), a highly urbanized country with populationof 58.7 million in 2007, and an area of 0.23 million km . The grid is composed of 5.75million cells of 200m-by-200m [8]. At the intermediate scale, we study the USA (continentalUSA without Alaska), a single country nearly continental in scale, with a population of303 million in 2007, and an area of 7.44 million km . The grid contains 7.44 million cellsof approximately 1km-by-1km obtained from the US Census Bureau [9]. The datasets ofGB and USA are populated-places datasets, with population counts defined at points in agrid. Since there could be some distortions in the true residential population involved atthe finest grid resolution, we perform our analysis by investigating the statistical propertiesas a function of the grid size by coarse-graining the data as explained in Section IV A. Atthe largest scale, we analyze the continent of Africa, composed of 53 countries with a totalpopulation of 933 million in 2007, and an area of 30.34 million km . These data are griddedwith less resolution by 0.50 million cells of approximately 7.74km-by-7.74km [10]. Moredetailed information about these datasets is found in Section IV A (all the datasets studiedin this paper are available at http://lev.ccny.cuny.edu/ ∼ hmakse/cities/city data.zip). II. RESULTS
Figure 1A illustrates operation of the CCA. In order to identify urban clusters, we requireconnected cells to have nonzero population. We start by selecting an arbitrary populatedcell (final results are independent of the choice of the initial cell). Iteratively, we then grow acluster by adding nearest neighbors of the boundary cells with a population strictly greaterthan zero, until all neighbors of the boundary are unpopulated. We repeat this process untilall populated cells have been assigned to a cluster. This technique was introduced to modelforest fire dynamics [11] and is termed the “burning algorithm,” since one can think of each3opulated cell as a burning tree.The population S i ( t ) of cluster i at time t is the sum of the populations n ( i ) j ( t )of each cell j within it, S i ( t ) = P N i j =1 n ( i ) j ( t ), where N i is the number of cellsin the cluster. Results of the CCA are shown in Fig. 1B, representing the urbancluster surrounding the City of London (red cluster overlaying a satellite image, seehttp://lev.ccny.cuny.edu/ ∼ hmakse/cities/london.gif for an animated image of Fig. 1B). Fig-ure 1C depicts all the clusters of GB, indicating the large variability in their population andsize.A feature of the CCA is that it allows the analysis of the population clusters at differentlength scales by coarse-graining the grid and applying the CCA to the coarse-grained dataset(see Section IV A for details on coarse-graining the data). At larger scales, disconnectedareas around the edge of a cluster could be added into the cluster. This is justified when,for example, a town is divided by a wide highway or a river.Tables I and II in Supporting Information (SI) Section I. show a detailed comparisonbetween the urban clusters obtained with the CCA applied to the USA in 1990, and theresults obtained from the analysis of MSAs from the US Census Bureau used in previousstudies of population growth [5, 6, 7]. We observe that the MSAs considered in Ref. [5] aresimilar to the clusters obtained with the CCA with a cell size of 4km-by-4km or 8km-by-8km. In particular, the population sizes of the clusters have the same order of magnitudeas the MSAs. On the other hand, for large cities the MSAs from the data of Ref. [6] seemto be mostly comparable to our results for cell sizes of 2km-by-2km or 4km-by-4km.Use of the CCA permits a systematic study of cluster dynamics. For instance, clus-ters may expand or contract, merge or split between two considered times as illustratedin Fig. 2. We quantify these processes by measuring the probability distribution of thetemporal changes in the clusters for the data of GB. We find that when the cell size is2.2km-by-2.2km, 84% of the clusters evolve from 1981 to 1991 following the three first casespresented in Fig. 2 (no change, expansion or reduction), 6% of the clusters merge from twoclusters into one in 1991, and 3% of the clusters split into two clusters.Next, we apply the CCA to study the dynamics of population growth by investigatingGibrat’s law, which postulates that the mean and standard deviation of growth rates areconstant [1, 2, 5, 7, 12]. The conventional method [1, 2, 7] is to assume that the populations4f a given city or cluster i , at times t and t > t , are related by S = R ( S ) S , (1)where S ≡ S i ( t ) = P N i j n ( i ) j ( t ) and S ≡ S i ( t ) = P N i j n ( i ) j ( t ) are the initial and finalpopulations of cluster i , respectively, and R ( S ) is the positive growth factor which variesfrom cluster to cluster. Following the literature in population dynamics [1, 2, 5, 7], wedefine the population growth rate of a cluster as r ( S ) ≡ ln R ( S ) = ln( S /S ), and studythe dependence of the mean value of the growth rate, h r ( S ) i , and the standard deviation, σ ( S ) = p h r ( S ) i − h r ( S ) i , on the initial population, S . The averages h r ( S ) i and σ ( S )are calculated applying nonparametric techniques [13, 14] (see Section IV B for details). Toobtain the population growth rate of clusters we take into account that not all clustersoccupy the same area between t and t according to the cases discussed in Fig. 2. Thefigure shows how to calculate the growth rate r ( S ) in each case.We analyze the population growth in the USA from t = 1990 to t = 2000 [9]. We applythe CCA to identify the clusters in the data of 1990 and calculate their growth rates bycomparing them to the population of the same clusters in 2000 when the data are griddedwith a cell size of 2000m by 2000m. We calculate the annual growth rates by dividing r bythe time interval t − t .Figure 3A shows a nonparametric regression with bootstrapped 95% confidence bands [13,14] of the growth rate of the USA, h r ( S ) i (see Section IV B for details). We find that thegrowth rate diminishes from h r ( S ) i ≈ . ± .
004 (error includes the confidence bands)for populations below 10 inhabitants to h r ( S ) i ≈ . ± .
002 for the largest populationsaround S ≈ . We may argue that the mean growth rate deviates from Gibrat’s lawbeyond the confidence bands. While it is difficult to fit the data to a single function for theentire range, the data show a decrease with S approximately following a power-law in thetail for populations larger than 10 . An attempt to fit the data with a power-law yields thefollowing scaling in the tail: h r ( S ) i ∼ S − α , (2)where α is the mean growth exponent, that takes a value α USA = 0 . ± .
08 from OrdinaryLeast Squares (OLS) analysis [15] (see Section IV B for details on OLS and on the estimationof the exponent error). 5igure 3B shows the dependence of the standard deviation σ ( S ) on the initial population S . On average, fluctuations in the growth rate of large cities are smaller than for smallcities in contrast to Gibrat’s law. This result can be approximated over many orders ofmagnitude by the power-law, σ ( S ) ∼ S − β , (3)where β is the standard deviation exponent. We carry out an OLS regression analysis andfind that β USA = 0 . ± .
06. The presence of a power-law implies that fluctuations in thegrowth process are statistically self-similar at different scales, for populations ranging from ∼ ∼
10 million according to Fig. 3B.Figure 4 shows the analysis of the growth rate of the population clusters of GB fromgridded databases [8] with a cell size of 2.2km-by-2.2km at t = 1981 and t = 1991. Theaverage growth rate depicted in Fig. 4A comprises large fluctuations as a function of S ,especially for smaller populations. However, a slight decrease with population seems evidentfrom rates around h r i ≈ . ± .
001 with S ≈ dropping to zero or even negative valuesfor the largest populations, S ≈ . We find that 3556 clusters with population around S = 10 exhibit negative growth rates as well. Thus, the mean rates are plotted on asemi-logarithmic scale in Fig. 4A. When considering intermediate populations ranging from S = 3000 to S = 3 × , the data seem to be following approximately a power-lawwith α GB = 0 . ± .
05 from OLS regression analysis, as shown in the inset of Fig. 4A.Figure 4B shows the standard deviation for GB, σ ( S ), exhibiting deviations from Gibrat’slaw having a tendency to decrease with population according to Eq. (3) and a standarddeviation exponent, β GB = 0 . ± .
04, obtained with OLS technique.The CCA allows for a study of the growth rates as a function of the scale of observation, bychanging the size of the grid. We find (SI Section II.) that the data for GB are approximatelyinvariant under coarse-graining the grid at different levels for both the mean and standarddeviation. When the data of the USA are aggregated spatially from cell size 2000m to8000m, the scaling of the mean rates crosses-over to a flat behavior closer to Gibrat’s law.At the scale of 8000m the mean is approximately constant (with fluctuations). However,we find that, at this scale, all cities in the northeastern the USA spanning from Boston toWashington D.C. form a single cluster. Despite these differences, the scaling of the standarddeviation for the USA holds approximately invariant even up to the large scale of observationof 8000m. 6ext, we analyze the population growth in Africa during the period 1960 to 1990 [10].In this case the population data are based on a larger cell size, so we evaluate the datacell by cell (without the application of the CCA). Despite the differences in the economicand urban development of Africa, Great Britain and the USA, we find that the mean andstandard deviation of the growth rate in Africa display similar scaling as found for the USAand GB. In Fig. 5A we show the results for the growth rate in Africa when the grid iscoarse-grained with a cell size of 77km-by-77km. We find a decrease of the growth rate from h r ( S ) i ≈ . h r ( S ) i ≈ .
01 between populations S ≈ and S ≈ , respectively.All populations have positive growth rates. A log-log plot of the mean rates shown inFig. 5A reveals a power-law scaling h r ( S ) i ∼ S − α Af , with α Af = 0 . ± .
05 from OLSregression analysis. The standard deviation (Fig. 5B) satisfies Eq. (3) with a standarddeviation exponent β Af = 0 . ± .
04. The CCA allows for a study of the origin of theobserved behavior of the growth rates by examining the dynamics and spatial correlationsof the population of cells. To this end, we first generate a surrogate dataset that consists ofshuffling two randomly chosen populated cells, n ( i ) j ( t ) and n ( i ) k ( t ), at time t . This swappingprocess preserves the probability distribution of n ( i ) j , but destroys any spatial correlationsamong the population cells. Figure 4C shows the results of the randomization of the GBdataset, indicating power-law scaling in the tail of σ ( S ) with standard deviation exponent β rand = 1 /
2. This result can be interpreted in terms of the uncorrelated nature of therandomized dataset (SI Section III). We consider that the population of each cell j increasesby a random amount δ j with mean value ¯ δ and variance h ( δ − ¯ δ ) i = ∆ , and that r ≪ n ( i ) j ( t ) = n ( i ) j ( t ) + δ j . Therefore, the population of a cluster at time t can be writtenas S = S + N i X j =1 δ j . (4)It can be shown that (SI Section III.): h S i = h S i + N i X j N i X k h ( δ j − ¯ δ )( δ k − ¯ δ ) i . (5)Randomly shuffling population cells destroys the correlations, leading to h ( δ j − ¯ δ )( δ k − ¯ δ ) i =∆ δ jk (where δ jk is the Kronecker delta function) which implies β rand = 1 / β lies below the random exponent ( β rand = 1 /
2) for all the analyzed data7uggests that the dynamics of the population cells display spatial correlations, which areeliminated in the random surrogate data. The cells are not occupied randomly but spatialcorrelations arise, since when the population in one cell increases, the probability of growthin an adjacent cell also increases. That is, development attracts further development, anidea that has been used to model the spatial distribution of urban patterns [17]. Indeed thisideas are related to the study of the origin of power-laws in complex systems [18, 19].When we analyze the populated cells, we indeed find that spatial correlations in the incre-mental population of the cells, δ j , are asymptotically of a scale-invariant form characterizedby a correlation exponent γ , h ( δ j − ¯ δ )( δ k − ¯ δ ) i ∼ ∆ | ~x j − ~x k | γ , (6)where ~x j is the location of cell j . For GB we find γ = 0 . ± .
08 (see Fig. 4D). In SISection III. we show that power-law correlations in the fluctuations at the cell level, Eq. (6),lead to a standard deviation exponent β = γ/
4. For γ = 2, the dimension of the substrate,we recover β rand = 1 / γ result in the same β since when γ > γ = 0, the standard deviation of the populations growth rates hasno dependence on the population size ( β = 0), as stated by Gibrat’s law, stating that thestandard deviation does not depend in the cluster size. In the case of GB, γ = 0 . ± . β = 0 . ± .
02 approximately consistent with the measured value β GB = 0 . ± . III. DISCUSSION
Our results suggest the existence of scale-invariant growth mechanisms acting at differentgeographical scales. Furthermore, Eq. (3) is similar to what is found for the growth offirms and other macroeconomic indicators [16, 20]. Thus, our results support the existenceof an underlying link between the fluctuation dynamics of population growth and variouseconomic indicators, implying considerable unevenness in economic development in differentpopulation sizes. City growth is driven by many processes of which population growth andmigration is only one. Our study captures only the growth of population but not economic8rowth per se. Many cities grow economically while losing population and thus the processeswe imply are those that influence a changing population. Our assumption is that populationchange is an indicator of city growth or decline and therefore we have based our studies onpopulation clustering techniques. Alternatively, the MSAs provides a set of rules that tryto capture the idea of city as a functional economic region.The results we obtain show scale-invariant properties which we have modelled using long-range spatial correlations between the population of cells. That is, strong development inan area attracts more development in its neighborhood and much beyond. A key finding isthat small places exhibit larger fluctuations than large places. The implications for locatingactivity in different places are that there is a greater probability of larger growth in smallplaces, but also a greater probability of larger decline. Opportunity must be weighed againstthe risk of failure.One may take these ideas to a higher level of abstraction to study cell-to-cell flows (mi-gration, commuting, etc.) gridded at different levels. As a consequence one may definepopulation clusters, or MSAs, in terms of functional linkages between neighboring cells. Inaddition one may relax some conditions imposed in the CCA. Here we consider a cell to bepart of a cluster only if its population is strictly greater than 0. In SI Section V we relaxthis condition and study the robustness of the CCA when cells of a higher population than0 (for instance, 5 and 20) are allowed into clusters and find that even though small clusterspresent a slight deviation, the overall behavior of the growth rate and standard deviation isconserved.
IV. MATERIALS AND METHODSA. Information on the datasets ∼ hmakse/cities/city data.zip.The datasets consist of a list of populations at specific coordinates at two time steps t and t . A graphical representation of the data can be seen in Fig. 1C for GB where each9oint represents a data point directly extracted from the dataset.To perform the CCA at different scales we coarse-grain the datasets. For this purpose,we overlay a grid on the corresponding map (USA, GB, or Africa) with the desired cell size(for example, 2km-by-2km or 4km-by-4km for the USA). Then, the population of each cellis calculated as the sum of the populations of points (obtained from the original dataset)that fall into this cell.Table I shows information on the datasets and results on USA, GB and Africa for thecell size used in the main text as well as some of the exponents obtained in our analysis. TABLE I:
Characteristics of datasets and summary of results
Data Number t t Average Cell Size Number of α β of cells growth rate clustersUSA 1.86 mill 1990 2000 0.9% 2km-by-2km 30,210 0.28 ± ± ± ± ± ± B. Calculation of h r ( S ) i and σ ( S ) and methodology The average growth rate, h r ( S ) i = ln( S /S ), and the standard deviation, σ ( S ) = p h r ( S ) i − h r ( S ) i , are defined as follows. If we call P ( r | S ) the conditional probabilitydistribution of finding a cluster with growth rate r ( S ) with the condition of initial popula-tion S , then we can obtain r ( S ) and σ ( S ) through, h r ( S ) i = Z rP ( r | S )d r, (7)and h r ( S ) i = Z r P ( r | S )d r. (8)Once r ( S ) and σ ( S ) are calculated for each cluster, we perform a nonparametric re-gression analysis [13, 14], a technique broadly used in the literature of population dynamics.The idea is to provide an estimate for the relationship between the growth rate and S andbetween the standard deviation and S . Following the methods explained in Ref. [14] we10pply the Nadaraya-Watson method to calculate an estimate for the growth rate, ˆ r ( S ),with, h ˆ r ( S ) i = P allclusters i =0 K h ( S − S i ( t )) r i ( S ) P allclusters i =0 K h ( S − S i ( t )) , (9)and an estimate for the standard deviation ˆ σ ( S ) with,ˆ σ ( S ) = s P allclusters i =0 K h ( S − S i ( t ))( r i ( S ) − h ˆ r ( S ) i ) P allclusters i =0 K h ( S − S i ( t )) , (10)where S i ( t ) is the population of cluster i at time t (as defined in the main text), r i ( S ) isthe growth rate of cluster i and K h ( S − S i ( t )) is a gaussian kernel of the form, K h ( S − S i ( t )) = e (ln S − ln Si ( t h , h = 0 . α/ α is not related tothe growth rate exponent) quantile we obtain the 95% confidence bands.To obtain the exponents α and β of the power-law scalings for h r ( S ) i and σ ( S ), respec-tively, we perform an OLS regression analysis [15]. More specifically, to obtain the exponent β from Eq. (3), we first linearize the data by considering the logarithm of the independentand dependent variables so that Eq. (3) becomes ln σ ( S ) ∼ β ln S . Then, we apply alinear Ordinary Least Square regression that leads to the exponent β = N c P N c i =1 [ln S i ( t ) ln σ ( S i ( t ))] − P N c i =1 ln S i ( t ) P N c i =1 ln σ ( S i ( t )) N c P N c i =1 (ln S i ( t )) − ( P N c i =1 ln S i ( t )) , (12)where N c is the number of clusters found using the CCA. Analogously, we obtain the expo-nent α by linearizing h| r ( S ) |i and calculating α = N c P N c i =1 (ln S i ( t ) ln h| r ( S i ( t )) |i − P N c i =1 ln S i ( t ) P N c i =1 ln h| r ( S i ( t )) |i N c P N c i =1 (ln S i ( t )) − ( P N c i =1 ln S i ( t )) . (13)Next we compute the 95% confidence interval for the exponents α and β . For this wefollow the book of Montgomery and Peck [15]. The 95% confidence interval for β is givenby, t . ,N c − ∗ se, (14)11here t α ′ / ,N c − is the t-distribution with parameters α ′ / N c − se is the standarderror of the exponent defined as se = s SS E ( N c − S xx , (15)where SS E is the residual and S xx is the variance of S .Finally, we express the value of the exponent in terms of the 95% confidence intervals as, β ± t . ,N c − ∗ se. (16) Acknowledgments
We thank L.H. Dobkins and J. Eeckhout for providing data on MSA and C. Briscoefor help with the manuscript. This work is supported by the National Science Foundationthrough grant NSF-HSD. J.S.A. thanks the Brazilian agencies CNPq, CAPES, FUNCAPand FINEP for financial support. 12
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97: 1639-1666. IG. 1: (A)
Sketch illustrating the CCA applied to a sample of gridded population data. In the topleft panel, cells are colored in blue if they are populated ( n ( i ) j ( t ) > n ( i ) j ( t ) = 0, theyare in white. In the top right panel we initialize the CCA by selecting a populated cell and burningit (red cell). Then, we burn the populated neighbors of the red cell as shown in the lower left panel.We keep growing the cluster by iteratively burning neighbors of the red cells until all neighboringcells are unpopulated, as shown in the lower right panel. Next, we pick another unburned populatedcell and repeat the algorithm until all populated cells are assigned to a cluster. The population S i ( t ) of cluster i at time t is then S i ( t ) = P N i j =1 n ( i ) j ( t ). (B) Cluster identified with the CCA in theLondon area (red) overlaying a corresponding satellite image (extracted from maps.google.com).The greenery corresponds to vegetation, and thus approximately indicates unoccupied areas. Forexample, Richmond Park can be found as a vegetation area in the south-west. The areas in theeast along the Thames River correspond mainly to industrial districts and in the west the LondonHeathrow Airport, also not populated. The yellow line in the center represents the administrativeboundary of the City of London, demonstrating the difference with the urban cluster found with theCCA. The pink clusters surrounding the major red cluster are smaller conglomerates not connectedto London. The figure shows that an analysis based on the City of London captures only a partialarea of the real urban agglomeration. (C)
Result of the CCA applied to all of GB showing thelarge variability in the population distribution. The color bar (in logarithmic scale) indicates thepopulation of each urban cluster.FIG. 2: Illustration of possible changes in cluster shapes. In each case we show how the growthrate is computed. In the first case, there is no areal modification in the cluster between t and t .In the second, the cluster expands. In the third the cluster reduces its area. In the fourth, onecluster divides into two and therefore we consider the population at t to be S = S ′ + S ′′ . In thefifth case two clusters merge to form one at t . In this case we consider the population at t to be S = S ′ + S ′′ . IG. 3: Results for the USA using a cell size of 2000m-by-2000m. (A)
Mean annual growth rate forpopulation clusters in the USA versus initial population of the clusters. The straight dashed lineshows a power-law fit with α USA = 0 . ± .
08 as determined using OLS regression. (B)
Standarddeviation of the growth rate for the USA. The straight dashed line corresponds to a power-law fitusing OLS regression with β USA = 0 . ± . (A) Mean annual growthrate of population clusters in Great Britain versus the initial cluster population. The inset showsa double logarithmic plot of the growth rate in the intermediate range of populations, 3000
05 for this range. (B)
Double logarithmic plot of the standard deviation of the annual growth rates of populationclusters in Great Britain versus the initial cluster population. The straight line corresponds to apower-law fit using OLS with an exponent β GB = 0 . ± .
04, according to Eq. (3). (C)
Scalingof the standard deviation in cluster population obtained from the randomized surrogate datasetof GB by randomly swapping the cells. The data shows an exponent β rand = 1 / S are discussed in the SI Section IV. where we test these results by generatingrandom populations. (D) Long-range spatial correlations in the population growth of cells for GBaccording to Eq. (6). The straight line corresponds to an exponent γ = 0 . ± . (A) Mean growth rate of clusters inAfrica versus the initial size of population S . The straight dashed line shows a power-law fit withexponent α Af = 0 . ± .
05, obtained using OLS regression. (B)
Standard deviation of the growthrate in Africa. The straight line corresponds to power-law fit using OLS providing the exponent β Af = 0 . ± . Fig. B Fig. 1 Fig. 1 t No ChangeExpansionReductionDivisionMerge S S ‘‘ S ‘S S S S S ‘‘S ‘ S r ( S ) = ln S S r ( S ) = ln S S r ( S ) = ln S S r ( S ) = ln S + S S r ( S ) = ln S S + S S = S + S S = S + S S S Fig. B FIG. 3: BC D
FIG. 4: B FIG. 5: UPPORTING INFORMATIONLaws of Population Growth
Hern´an D. Rozenfeld, Diego Rybski, Jos´e S. Andrade Jr.,Michael Batty, H. Eugene Stanley, and Hern´an A. MakseAs supplementary materials we provide the following: In Section V we present tableswith details on our results using the CCA and results presented in previous papers to allowfor comparison between the different approaches. In Section VI we study the stability of thescaling found in the text under a change of scale in the cell size. In Section VII we detail thecalculations to relate spatial correlations between the population growth and σ ( S ) namelythe relation β = γ/
4. In Section VIII we describe the random surrogate dataset used tofurther test our results. In Section IX we further test the robustness of the CCA by proposinga small variation in the algorithm.
V. CLUSTERS AT DIFFERENT SCALES AND COMPARISON WITHMETROPOLITAN STATISTICAL AREAS
In this section, Tables S1 and S2 allow for a detailed comparison of urban clusters obtainedwith the CCA applied to the USA in 1990, and the populations of MSA from US CensusBureau used in previous studies of population growth [5, 6, 7].We can see that the MSA presented by Eeckhout (2004) typically correspond to ourclusters using cell sizes of 4km and 8km. For example, for the New York City regionEeckhout’s data are well approximated by a cell size of 4km, but Los Angeles is betterapproximated when using a cell size of 8km. On the other hand Dobkins-Ioannides (2000)data are better described by cell sizes of 2km or 4km. For instance, Chicago is well describedby a cell size of 4km and Los Angeles is better described by a cell size of 2km.An interesting remark is that the population of Los Angeles when using cell sizes of 2km,4km and 8km does not vary as much as that for New York. This could be caused by thefact that major cities in the northeast of USA are closer to each other than large cities inthe southwest, which may be attributed to land or geographical constraints.23t is important relate the results of Table S2 with an ecological fallacy. As the cell sizeis increased, the population of a cluster also increases, as expected, because the cluster nowcovers a larger area. This is not a direct manifestation of an ecological fallacy which, wouldappear if the statistical results (growth rate vs. S or standard deviation vs. S) gave differentresults as the cell size increases. In Fig. 1 and Fig. 2 in the SI Section VI, we observe thatthe growth rate and standard deviation for the USA and GB follow the same form, exceptfor the case of the growth rate in the USA in which different cell sizes show deviations fromeach other. The later may be an indicative of an ecological fallacy. In this case, it is notobvious what cell size is the “correct” one. We consider this point (the possibility to choosethe cell size) to be a feature of the CCA, since one may appropriately pick the cell sizeaccording to the specific problem one is studying.Table S1:
Top 10 largest MSA of the USA in 1990 from previous analysis ofpopulation growth
Dobkins - Ioannides EeckhoutMSA Population MSA Population1 NYC NY206 9,372,000 NYC-North NJ-Long Is., NY-NJ-CT-PA 19,549,6492 Los Angeles CA172 8,863,000 Los Angeles-Riverside-Orange County, CA 14,531,5293 Chicago IL59 7,333,000 Chicago-Gary-Kenosha, IL-IN-WI 8,239,8204 Philadelphia PA228 4,857,000 Washington-Baltimore, DC-MD-VA-WV 6,727,0505 Detroit MI80 4,382,000 San Francisco-Oakland-San Jose, CA 6,253,3116 Washington DC312 3,924,000 Philadelphia-Wilmington-Atlantic City 5,892,937PA-NJ-DE-MD7 San Francisco CA266 3,687,000 Boston-Worcester-Lawrence, MA-NH-ME-CT 5,455,4038 Houston TX129 3,494,000 Detroit-Ann Arbor-Flint, MI 5,187,1719 Atlanta GA19 2,834,000 Dallas-Fort Worth, TX 4,037,28210 Boston MA39 2,800,000 Houston-Galveston-Brazoria, TX 3,731,131
Top 10 largest clusters of the USA in 1990 from our analysis fordifferent cell sizes.
The city names are the major cities that belong to the clusters andwere picked to show the areal extension of the cluster.
Cell = 1km Cell = 2km Cell = 4km Cell = 8kmCluster Population Cluster Population Cluster Population Cluster Population1 NYC 7,012,989 NYC-Long Is. 12,511,237 NYC-Long Is. 17,064,816 NYC-Long Is. 41,817,858Newark N. NJ-Newark North NJJersey City Jersey City PhiladelphiaD.C.-Boston2 Chicago 2,312,783 Los Angeles 9,582,507 Los Angeles 10,878,034 Los Angeles 13,304,233Long Beach Long Beach San ClementePomona Riverside3 Los Angeles 1,411,791 Chicago 4,836,529 Chicago 7,230,404 Chicago 9,288,345Rockford Gary GaryRockford RockfordMilwaukee4 Philadelphia 1,282,834 Philadelphia 3,151,704 Washington 5,316,890 San Francisco 5,736,479Wilmington Baltimore Santa CruzSpringfield Brentwood5 Boston 759,024 Detroit 2,906,453 Philadelphia 4,935,734 Detroit 4,442,723Trenton Ann ArborWilmington MonroeSarnia6 Newark 581,048 San Francisco 2,601,639 San Francisco 4,766,960 Miami 4,000,432San Jose San Jose Port St. LucieConcord7 San Francisco 507,300 Washington 2,059,421 Detroit 3,722,778 Dallas 3,536,186Alexandria Waterford Fort WorthBethesda Canton8 Washington 504,068 Phoenix 1,556,077 Miami 3,719,773 Houston 3,425,647W. Palm Beach9 Jersey City 438,591 Boston 1,498,208 Dallas 3,134,233 Cleveland 3,233,341Lowell Fort Worth CantonQuincy10 Baltimore 437,413 Miami 1,465,490 Boston 3,064,925 Pittsburgh 3,214,661Brockton YoungstownNashua Morgantown B FIG. 6: Sensitivity of the results under coarse-graining of the data for GB. (A)
Average growthrate and (B) standard deviation for GB using the clustering algorithm for different cell size. Thedashed line represents the OLS regression estimate for the exponents (A) α GB = 0 .
17 and (B) β GB = 0 .
27 obtained in the main text. For clarity we do not show the confidence bands.
VI. SCALING UNDER COARSE-GRAINING
In this section we test the sensitivity of our results to a coarse-graining of the data. Weanalyze the average growth rate h r ( S ) i and the standard deviation σ ( S ) for GB and theUSA by coarse-graining the data sets at different levels.In Fig. 6A we observe that although the results are not identical for all coarse-grainings,they are statistically similar, showing a slight decay in the growth rate. Moreover, we seethat cities of size S ≈ and S ≈ still exhibit a tendency to have negative growthrates for all levels of coarse-graining, as explained in the main text. In the case of the USA(Fig. 7A) there is a crossover to a flat behavior at a cell size of 8000m, although at this scaleall the northeast USA becomes a large cluster of 41 million inhabitants. On the other hand,Figs. 6B, 7B show that the scaling of Eq. (3) in the main text, σ ( S ) ∼ S − β , still holds whenusing the coarse-grained datasets on both GB and the USA. VII. CORRELATIONS
In this section we elaborate on the calculations leading to the relation between Gibrat’slaw and the spatial correlations in the cell population. We first show that when the pop-26 B FIG. 7: Study of results under coarse-graining of the data for the USA. (A)
Average growth rateand (B) standard deviation for the USA using the clustering algorithm for different cell size. Thedashed line represents the OLS regression estimate for the exponents (A) α USA = 0 .
28 and (B) β USA = 0 .
20 obtained in the main text. For clarity we do not show the confidence bands. ulation cells are randomly shuffled (destroying any spatial correlations between the growthrates of the cells), the standard deviation of the growth rate becomes σ ( S ) ∼ S − β rand , where β rand = 1 / β = γ/ r ≪ R = e r ≈ r .Replacing R = 1 + r in Eq. (1) in the main text we obtain S = S + S r. (17)We define the standard deviation of the populations S as σ , which is a function of S : σ ( S ) = q h S i − h S i . (18)This quantity is easier to relate to the spatial correlations of the cells than the standarddeviation σ ( S ) of the growth rates r . Then, since h S i = S + S h r i and h S i = S +2 S h r i + S h r i , we obtain, σ ( S ) ∼ S σ ( S ) , (19)where σ ( S ) = p h r i − h r i as defined in the main text. Therefore, using Eq. (3) in the27ain text, σ ( S ) ∼ S − β . (20)As stated in the main text, the total population of a cluster at time t is the sum of thepopulations of each cell, S = P N i j =1 n ( i ) j , where N i is the number of cells in cluster i . Thepopulation of a cluster at time t can be written as S = S + N i X j =1 δ j , (21)where δ j is the increment in the population of cell j from time t to t (notice that δ j canbe negative). Therefore, the standard deviation σ ( S ) is (cid:16) σ ( S ) (cid:17) = N i X j,k h δ j δ k i − h N i X j δ j i = N i X j,k h ( δ j − ¯ δ )( δ k − ¯ δ ) i . (22)After the process of randomization explained in Section II main text, the correlationsbetween the increment of population in each cell are destroyed. Thus, h ( δ j − ¯ δ )( δ k − ¯ δ ) i = ∆ δ jk , (23)where ∆ = ¯ δ − ¯ δ . Replacing in Eq. (22) and since h n i = (1 /N i ) P N i j n j = S /N i , weobtain (cid:16) σ ( S ) (cid:17) = N i ∆ ∼ S . (24)Comparing with Eq. (20) we obtain β rand = 1 / δ j , decays as a power-lawof the distance between cells indicating long-range scale-free correlations. Thus, asymptoti-cally h ( δ j − ¯ δ )( δ k − ¯ δ ) i ∼ ∆ | ~x j − ~x k | γ , (25)where ~x j denotes the position of the cell j and γ is the correlation exponent (for | ~x j − ~x k | → h ( δ j − ¯ δ )( δ k − ¯ δ ) i tend to a constant). For large clusters, we can approximatethe double sum in Eq. (22) by an integral. Then, assuming that the shape of the clusterscan be approximated by disks of radius r c , for γ < σ ( S )) = N i X j,k ∆ | ~x j − ~x k | γ → ∆ N i a Z r c r d r d θr γ ≈ ∆ (2 − γ ) N i a r − γ +2 c , (26)28here a is the area of each cell and r c the radius of the cluster. Since r c ∼ N i a , we finallyobtain, (cid:16) σ ( S ) (cid:17) ∼ N − γ i . (27)Using S = N i h n i and Eq. (20) we arrive at, β = γ . (28)Equation (28) shows that Gibrat’s Law is recovered when the correlation of the populationincrements is a constant, independent from the positions of the cells; that is when all thepopulations cells are increased equally. In other words, if γ = 0, the standard deviation ofthe populations growth rates has no dependence on the population size ( β = 0), as stated byGibrat’s law. The random case is obtained for γ = d , where d = 2 is the dimensionality of thesubstrate. In this case d = 2 and β rand = 1 /
2. For γ >
2, the correlations become irrelevantand we still find the uncorrelated case β rand = 1 /
2. For intermediate values 0 < γ < < β = γ/ < / VIII. RANDOM SURROGATE DATASET
In this section we elaborate on the randomization procedure used to understand the roleof correlations in population growth.Figure 4C in the main text shows the standard deviation σ ( S ) when the populationof each cluster is randomized, breaking any spatial correlation in population growth. Forclusters with a large population, σ ( S ) follows a power-law with exponent β rand = 1 / S , σ ( S ) presents deviations from the power-law function as seen in Fig. 4Cwith smaller standard deviation than the prediction of the random case. This deviation iscaused by the fact that the population of a cluster is bound to be positive: a cluster with asmall population S cannot decrease its population by a large number, since it would leadto negative values of S . This produces an upper bound in fluctuations of the growth ratefor small S and results in smaller values of σ ( S ) than expected (below the scaling withexponent β rand = 1 / n j ( t ) of each cell j is replaced with random numbers following an exponentialdistribution with probability P ( n j ) ∼ e − n j /n . The decay-constant, n = 150, is extracted29rom the data of GB to mimic the original distribution. This is done through a direct measureof P ( n j ) from the GB dataset and fitting the data using OLS regression analysis. We obtainthe population n j ( t ) = n j ( t ) + δ j of cell j at time t by picking random numbers for thepopulation increments δ j following a uniform distribution in the range − q ∗ < δ i < q ∗ q determines the variance of the increments. Since the population cannot be negativewe impose the additional condition n j ( t ) ≥
0. Figure 8 shows the results of the standarddeviation σ ( S ) for four different q -values for this uncorrelated model. We find that the tailof σ ( S ) reproduces the uncorrelated exponent β rand = 1 /
2. For small S we find that thestandard deviation levels off to an approximately constant value as in the surrogate data ofFig 4C. The crossover from an approximately constant σ ( S ) to a power-law moves to smallervalues of the population S as the standard deviation in the δ j is smaller (smaller value of q ).Such behavior can be understood since the condition n ( i ) j ( t ) ≥ n ( i ) j ( t ) = n ( i ) j ( t ) + δ j . As the initial population gets smaller,the walker “feels” the presence of the wall and the fluctuations decrease accordingly, thusexplaining the deviations from the power-law with exponent β rand = 1 / q decreases, the small population plateau disappears asobserved in Fig. 8. IX. A VARIATION OF THE CCA
In this section we study a variation of the CCA. In the main text we stop growing a clusterwhen the population of all boundary cells have unpopulated, that is, have population exactly0. In other words, clusters are composed by cell with population strictly greater than 0. Itis important to analyze whether this stopping criteria can be relaxed to including cell whichhave a population larger that a given threshold. In Fig. 9A and Fig. 9B we show the resultsfor the population growth rate and standard deviation, respectively, in GB when the cellsize is 2.2km-by-2.2km (as in the main text) but including cells with a population strictlylarger than 5 and 20.Although for small population clusters we observe a slight variation in the growth rateand in the standard deviation, the results show that the thresholds do not influence theglobal statistics when compared to the plots in the main text.30
IG. 8: Standard deviation σ ( S ) for the random data set as explained in the SI Section VIII.The results for σ ( S ) are rescaled to collapse the power-law tails with exponent β rand = 1 / S . The larger the parameter q , thelarger the deviations from the power-law at lower S . In other words, the crossover to power-lawtail appears at larger S as q increases. A B
FIG. 9: Sensitivity of the results under a change in the stopping criteria in the CCA (A)
Averagegrowth rate for GB with a population threshold of 5 (green line) and 20 (black dashed line) and (B) standard deviation for GB with a population threshold of 5 (green line) and 20 (black dashedline). For clarity we do not show the confidence bands.standard deviation for GB with a population threshold of 5 (green line) and 20 (black dashedline). For clarity we do not show the confidence bands.