Abstract

Power law distributions characterise several natural and social phenomena. Zipf'south police for cities is ane of those. The study views the question of whether that global regularity is independent of different spatial distributions of cities. For that purpose, a typical Zipfian rank-size distribution of cities is generated with random numbers. This distribution is then cast into two different settings of spatial coordinates. For the interpretation, the variables rank and size are supplemented past considerations of spatial dependence inside a spatial econometric arroyo. Results propose that distance potentially matters. This finding is further corroborated by iv state analyses even though estimates reveal merely modest effects.

Citation: Bergs R (2021) Spatial dependence in the rank-size distribution of cities – weak merely not negligible. PLoS I 16(2): e0246796. https://doi.org/10.1371/journal.pone.0246796

Editor: Yannis Ioannides, Tufts University, U.s.

Received: June 3, 2020; Accepted: January 26, 2021; Published: February nine, 2021

Copyright: © 2021 Rolf Bergs. This is an open up access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in whatever medium, provided the original author and source are credited.

Information Availability: Data are available from Harvard Dataverse: https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/EK4CNU

Funding: The writer is affiliated as partner with PRAC. PRAC is a private institute organised as a partnership visitor. The enquiry represented by the paper at manus is a secondary upshot of a current research project, being funded by the European Commission „Horizon2020" program (grant No. 727988). The funder has not been involved in the study blueprint, data collection, decision to publish and preparation of the manuscript. The only criterion to be fulfilled by the author is the thematical relevance of this manuscript for the purpose of the a.m. research projection. The funder provided support in the form of salaries for authors [RB], but did non have whatever additional role in the study pattern, data collection and analysis, decision to publish, or grooming of the manuscript. The specific roles of these authors are articulated in the 'writer contributions' section. My commercial amalgamation (PRAC) did not play any role in this context. The paper was exclusively prepared and written by myself.

Competing interests: The author is affiliated with PRAC. This does not alter our adherence to PLOS I policies on sharing information and materials.

1 Introduction

Zipf'south law of the rank-size distribution of cities is regarded equally an enthrallment of rare social physics. Krugman [one] has described this phenomenon even equally an embarrassment for economic theory (p. 42–46). At first glance, the human relationship betwixt size and rank suggests a tautological relationship because size direct determines rank and vice versa. The contained variable is thus possibly no true predictor but could just be function of a elementary universal statistical miracle. The power law exponent typically shut to -one and a determination coefficient (R2) close to one are an indication of that. Therefore some emphatically questioned the relevance for economic analysis, notably Gan et al. [2]. Yet, such a hit ubiquitous regularity has motivated the exploration of hidden explanatory factors behind. Diverse authors, such as Gabaix, Fujita et al., Brakman et al., Reggiani and Nijkamp and Ioannides take washed this [3–7]. In dissimilarity to the Zipf distribution of frequency of words in languages [8], the rank-size distribution of cities appears slightly more varied between countries as e.g. shown past Rosen and Resnick [ix] and less stable over fourth dimension as e.g. plant by Brakman et al. [10] but there is a secular convergence that is explained by Gibrat'southward law and its resulting steady country [3]. The fact that this happens in all countries, regardless of their economic structures and histories, yet lacks a truly sufficient explanation. When comparing such power law distributions for dissimilar types of data one benchmark could perhaps add pocket-sized insight: A spatial versus non-spatial context. Spatial dependence in terms of contiguity or distance between cities of different rank or size may affect Zipf'south police force for cities; in contrast, infinite tin can never predict the rank-distribution of words.

To have a closer look at that context, I first reflect on theoretical considerations of spatial dependence in Zipf'southward law earlier simulating a typical rank-size distribution of cities for a varied distribution of spatial coordinates. The objective is to see how much influence spatial distance could have on the ranks of cities and thus on the shape of the distribution. Especially in countries with a geographical concentration of bigger cities at that place is reason to presume that these cities have evolved due to certain spatial advantages (e.g. raw materials, climate, accessibility or certain random determinants). Those city clusters are frequently characterised by specific industries of national importance. Whether and how dispersion and concentration forces determine the rank-size distribution of cities has been a widely researched object in urban economics.

Surprisingly, there has been fiddling inquiry shedding light on spatial dependence associated with Zipf's police force. Lalanne views the dichotomic urban construction of Canada [11]. She rejects the Zipf police force and its underlying scale invariance and shows that the Canadian urban construction has evolved in a deterministic process based on urban size (inhabitants inside administratively defined boundaries), previous growth and the spatial setting. Coefficients for the years 1971 to 2001 vary between -0.77 and -0.81. The spatial component is not part of the Pareto regression; instead growth of cities is regressed on size and previous growth in standard spatial regression models (SEM and SAR). Le Gallo and Chasco explore Zipf'southward police force for Spain by applying a SUR model which they cast into spatial autoregressive and error specifications. Zipf'southward police does non hold betwixt 1900 and 2001. While the elementary OLS estimate varies betwixt -0.54 and -0.66, the extended spatial models deviate even further from Zipf'south constabulary, thus revealing spatial impacts [12]. Cheng and Zhuang [13], who look at urban evolution in cardinal China under consideration of Zipf's law, bear witness that the estimation of the Pareto coefficient has displayed an undulatory pattern between 1985 and 2009. They cannot ostend Zipf's law at any point of time. The OLS estimates are then augmented by the use of spatial autoregressive or spatial error specifications. Like in the study of Le Gallo and Chasco [12], spatial dependence increases the deviation from Zipf's police. It is, nonetheless, to be stressed that in the three studies on Canada, Kingdom of spain and Prc the size of cities is non defined by functionality simply the number of inhabitants inside administratively defined boundaries of all cities, i.e. including the lower tail of the distribution. This is a reason why Zipf's constabulary often does not hold [5: 301–306].

In the report at hand I intend to testify (i) whether and how spatial dependence of a Zipfian rank-size distribution varies among different geographical settings of cities and (ii) how these settings behave differently along the unabridged distribution of cities and specifically the upper Pareto tail. Evidence suggests that the rank-size distribution is non homogeneously post-obit a Pareto shape, just rather a combination of an upper Pareto and a lower lognormal section. By using a switching model, Ioannides and Skouras [14] bear witness for United states cities with 2000 Census places information that in that location is a narrow transition corridor around slightly more than 60,000 inhabitants where the upper tail Pareto distribution merges with a lower tail lognormal distribution. They reject Eeckhout's standpoint [15], that the entire rank-size distribution of cities is best described by a lognormal distribution. Evidence of the hybrid distribution was corroborated by several further studies, eastward.yard. Malevergne et al. [16], then that this is explicitly considered in my paper.

In this paper I first explore an ideal type random-generated hybrid distribution with ii different spatial settings. This should demonstrate that the called econometric methodology is powerful enough to observe distance effects if the patterns are sufficiently potent. The simulation exercise is and then followed by four country studies covering the USA, the United kingdom of great britain and northern ireland, Federal republic of germany and Slovenia, the latter representing a former smaller province of Yugoslavia. For Slovenia, not just population was used as the size variable but also the detected extent of natural cities to better represent their true functional size. In essence, the paper is a theory-led artificial simulation of Zipf's law enriched with real world studies.

2 Some theoretical considerations

The spatial relevance of the rank-size distribution of cities was already emphasised by Zipf himself in his widely noticed lemma [17]: If effort of interaction among the possible pairs of cities is optimum (with least effort for all individuals), the cities (settlements) of different size are ranked in a way and so that the full population of a country Sc equals the sum of a harmonic series: (1) where South p is the population of the primacy city, r is the rank of an private urban center and α is the power exponent determining the shape; in example of a perfect Zipfian relationship, the cumulative distribution function then follows (ii) or in its reversed Pareto form with R (rank) as the dependent variable: (3) with α = 1 and B = C. This ability police force describes a scale-invariant pattern with very few large and very many small items as it is institute in many natural systems. Co-ordinate to Zipf, the slope of that item distribution necessitates the endeavor of interaction between the communities to be at a minimum when multiplied by the distances d between the communities. Zipf'southward lemma describes a stylized equilibrium model in which in that location is a scattered distribution of settlements close to the raw materials (first economy) and one big urban center where all the raw materials are processed (2nd economy). Since living in either place will create opportunity costs for any dweller, both economies are in disharmonize over unification and diversification. The conflict betwixt those forces plays a key office in the determination of the endeavour-minimizing number, location and sizes of settlements or, with other words: the built environment is created and then that costs of primary product, processing, and the transport of goods and factors between the two economies are minimized. Apparently, an equilibrium is found when the magnitude of the centrifugal and centripetal forces is equal. In this optimum case α = 1, and the equilibrium is and then Pareto efficient. If one imagines a growing network of cities, it becomes obvious that the number of connections represents an economical value (utility), and this again is highest the minimum possible effort is needed. As explained by Kak [18], the value of a potential network with n items (cities) so grows in proportion to n ∙ log(northward). This explains a power law beliefs as a precondition of least effort.

According to Zipf [17], this design only works in social systems that exactly produce what they consume and where all members of the population receive an equal share of the national income. This understanding very obviously assumes constant returns to scale in both economies. Consequently, if the organisation is non in an equilibrium (due east.g. with the occurrence of increasing returns), diversifying (centrifugal) and unifying (centripetal) forces do not offset each other. In this example the slope of the power police changes. It is to be stressed that Zipf's spatial equilibrium substantially depends on the beingness of spatial heterogeneity. Perfect divisibility of space would rule out any equilibrium (Starret's spatial impossibility theorem) [5, xix]. Hence, (i) for cities to evolve anyway, indivisibility is needed and (ii) for cities to evolve efficiently with optimum allotment of resources needed for interaction, their rank-size distribution should converge to Zipf's police force. This deserves some closer examination since indivisibility of infinite unveils an important explanatory limitation of Zipf's considerations of a spatial equilibrium.

In Zipf's model, the difference between the showtime and second economic system is solely explained by their corresponding functional roles. Notwithstanding, the development of the spatial economy, characterised by a dynamic rural-urban differentiation, essentially exhibits spatial cistron and goods price differentials that originate from several interrelated determinants, such as increasing returns in manufacturing production [20], monopolistic competition and higher existent urban income through the supply of a variety of substitutable goods and lower transport costs [21], stronger knowledge spillovers in agglomerated urban settings [22], trickle-down effects of individual specialised skills on the local qualification and productivity levels [23] or agglomeration fuelled past entrepreneurial dubiousness and risk [24] to mention some. In those settings consumers aim to maximise their utility non only past minimising effort of transport just also to maximise their real income through an optimum choice of expenditure on nutrient and on the variety of substitutable manufactured products. The "honey for variety consequence", scale economies and less transportation effort are circularly caused. They are a bonus for larger markets [v] and constitute an urban amenity.

Interestingly, the major thread of the subsequent theoretical literature on Zipf's police since the 1950s centered around statistical and largely non-economic explanations based on random growth of population [25]. Afterwards the size- and variance-independent growth of cities was discussed to explicate the inherent fractal dimension of Zipf's police by Gibrat's law. In these models the potential spatial dimension was largely ignored. Indeed, in Gabaix's [three] model of zippo normalized city growth, infinite and distance between cities practise not propose to be meaningful factors under the strict supposition of Gibrat's police. In this model, the economic foundation of Gibrat's law is explained past scale-independent regional and policy shocks with the same variance for all cities in addition to specific shocks that affect particular industries, thus implying a decreasing variance with city size. Withal, for the upper tail of the city size distribution, industrial shocks may dice out so that, according to Gabaix, variance rather depends on the policy and regional shocks.

While urban economics and regional scientific discipline in the 20th century generated an affluence of theoretical models to explicate agglomeration economies, near of them had pursued a fractional focus. During the last twenty years, economic geography models became more consolidated every bit to nest different hypotheses that may plant forces of agglomeration. An important seed of those efforts was the seminal piece of work of Fujita, Krugman and Venables [4] which reveals the sensitivity of a spatial evolution path faux by the relationship between the share of manufacturing employment and transport costs adjusted with few decisive parameter settings (substitution elasticity, iceberg losses). Those determine tipping points (bifurcations) causing either centrifugal or centripetal spatial evolution paths. This piece of work also contributed to more than insight into the economical determinants of the rank-size distribution of cities. Brakman et al. [v] develop a core economical geography model with monopolistic competition and farther extensions to explore the beliefs of the spatial economy nether dissimilar parameter adjustments. By extending their model with congestion, every bit the major counterforce of bunch, they simulate Zipf'south law historically and bear witness that at that place is an Northward-shaped blueprint of the Zipf coefficient over time from the pre-industrial to the post-industrial era (for log of city size as the dependent variable and log of rank as the predictor). In this model, aimed to explain Zipf'southward law, infinite is explicitly considered, but in terms of agglomeration, rather than inter-city distances.

Only recently, a growing tape of inquiry stressing the relevance of distance and accessibility in the evolution of urban space tin can be observed. Indeed, the functional differentiation between Zipf's first and second (spatial) economic system closely corresponds to the Central Places theory provided the being of agglomeration economies is not ignored. Altitude, or the endeavor to comprehend it, is then the major friction in city interaction. This makes spatial altitude not only important for city interaction but somewhen besides relevant to the rank-size distribution of those cities.

In a highly comprehensive assay to capture the determinants of Zipf's law economically and spatially, Ioannides [7] demonstrates the limits of explanatory ability of otherwise well-founded theories. This relates to independently and identically distributed (i.i.d.) growth rates of cities when using normalised city sizes and to an entirely lognormal city size distribution determined by Gibrat'southward police [3, fifteen]. In a growing and increasingly urbanizing economy the assumption that urban center growth rates are essentially i.i.d. may not farther hold. Particularly the relationship betwixt the variation of stock-still costs and the number of product sites appears to exist an important chemical element to explain a power law distribution in the upper tail of the city sizes. The finding reveals important explanatory ability of the Central Places theory. Firms with lower fixed costs are spatially more dispersed (i.e. in big and minor cities), while those with high fixed costs locate shut to those with lower fixed costs (usually in larger cities). This refers to the work of Hsu [26] who concludes that "… The ability police force for cities and firms and the NAS ["Number-Average-Size"] dominion arise when the distribution of scale economies is regularly varying. In fact, this is the condition for ensuring that a central place hierarchy is a fractal structure…" (p. 923). The Key Identify theory tin can be thus reconciled with the mainstream economic theories and a power law beliefs of the upper tail of the city distribution: Larger cities are not simply more diversified than small ones because many small and big industries are agglomerated there simply considering bigger cities specialize in industries with higher calibration economies.

In a very recent assay of this thread of studies, Mori et al. [27] compare real with random city systems at national level and within the hierarchy of a system with central places. They detect strong evidence of a fractal dimension in the rank-size distribution of cities, simply this is not governed by random growth of cities just rather by local urban center systems surrounding major cities. In another study, Jiang et al. [28] explores the system of cities from the viewpoint of the design of space and finds that cities are non isolated only coherent entities inside a connected whole, whereas cities themselves comprise coherent hotspots. Also arguing with the Central Place theory, Jiang concludes that the club of the built surroundings corresponds to the order in nature and that scaling police and spatial dependence "… are key not merely to geographical phenomena, but too to any other living structure that recurs between the Planck length and the size of the universe itself. …" (p. 311).

In addition to economic geography models in that location are likewise further recent contributions aimed to explain a spatial Zipf law from the viewpoint of social physics. With a probability based arroyo of urban evolution, Rybski et al. simulate formation and growth of cities under the assumption of Tobler's police, namely that urban growth takes place close to other urban settlements [29]. In comparative simulations with dissimilar numbers of iterations and different strengths of distance disuse γ they show, that for sites in a grid with a central site already occupied (w = 1) any other site j in that grid (w = 0) will be occupied with a probability: (4) where d j,k is the Euclidian distance betwixt locationas j and k, and A is a normalized constant and then that the maximum probability is 1. Hence, in this simulation, supported by a real world study on the urban development of Paris, evolution of new sites solely depends on altitude to sites already existing. Despite the fact that the simulated urban evolution is non random just deterministic the authors confirm Zipf's constabulary and calibration invariance of the clusters generated, except the primate one. Thus, Zipf'due south law can exist also reproduced by "spatial explicit preferential attachment".

A well-known example of such peculiar spatial trajectories is the Ruhr area in Federal republic of germany. Hither the theory of Central Places seems to neglect in explaining the fact that bigger cities (more than 100,000 inhabitants) are just medium centers or fifty-fifty cities with small-scale central relevance. Findings of Dobkins and Ioannides [30] for United states of america cities also confirm that big cities tend to take large neighbors. This may suggest a possible inconsistency with the Central Place theory, non for the regional setting of cities (as new neighbors entering are withal relatively minor compared to the older ones) merely perhaps from a national viewpoint with a larger variation of urban center sizes within the different axis classes. Here it may happen that size of cities in urban clusters does not non anymore correspond to axis. Only also an opposite type of spatial settings is imaginable, such as for big urban areas surrounded by but specially modest municipalities with piffling centrality role, e.yard. the Berlin urban zone.

Such peculiar agglomeration phenomena and so substantially imply positive or negative spatial autocorrelation of city ranks or sizes and may accept an influence on the Pareto coefficient at national scale, then that size and growth of such cities are not necessarily random simply spatially autoregressive (or disturbed by spatially autocorrelated mistake) and thus partly depending on rank or size of neighbor settlements and their growth.

In the real globe, Zipf's law for cities is never absolutely perfect. Many empirical studies have shown this, as mentioned earlier. Reasons for that can stem from the regional political economic system, notably a functionally inadequate authoritative depiction of urban space or the politically emphasized weight of the primate metropolis (in many cases the uppercase). A farther reason can be the typical hybrid distribution form mentioned earlier (Pareto and lognormal) and whether the full or a concise range of metropolis sizes is regarded; coefficients estimated tin can differ strongly. In addition to such data bug, increasing returns or congestion can accept an influence on the rank-size distribution of cities. This does not at all mean that Zipf's law fails in such cases. A spatial distance influence can better the fit of the ability law or it tin move α further abroad from 1. But as long every bit development of big cities in urban clusters tends to exhibit spatial dependence in the rank-size distribution, its furnishings would be substantially concealed by a non-spatial regression. In conclusion, this suggests a spatial econometric arroyo when testing Zipf's law.

iii Methodological approach

The key assumption is as follows: For the typical upper Pareto (3,ane) tail the expected exponent α is approximately i (±0.1) equally empirically confirmed ubiquitously. In its log-linear class the above described cumulative distribution role (iii) to be estimated is: (v) where the residual error is assumed i.i.d with ε ∽ North(0,σ 2). OLS or maximum likelihood are possible estimators of α. For the combined Pareto and lognormal tails the exponent normally does not fit Zipf's law only but for the Pareto department. In addition to that, the theoretical considerations put forrard earlier suggest that the rank-size distribution of cities is potentially affected by spatial distance in the sense of Tobler's law. Such spatial forces are however not incorporated into Zipf'southward police, and then that in theory a Pareto exponent α≈1 in one land may remain stable under consideration of spatial dependence of rank while, in another one, the incorporation of such interaction could perhaps atomic number 82 to a minor or major change of α. For spatial dependence to be considered for the rank-size distribution of cities, some methodological considerations are relevant: Compared to gravity estimations that address the interaction of places (i.e. the number of combinations), the estimation of the rank-size distribution cannot include altitude as one regular predictor. Either distance enters the model equally a large matrix of single independent variables (one for every city combination) or ane controls for spatial spillover or fault of the residuals in regression analyses. Information technology is to be stressed that a stand-lone construction of numerous contained distance variables would ignore the possible endogeneity of distance (spillover effects of the dependent variable or residual spatial autocorrelation). The underlying economic rationale is the utility of interaction with respect to spatial distance between cities of either similar or very different ranks. A more than precise approach would exist thus a spatial econometric procedure [31]: (half dozen) or (7) where W is a North 10 North row-standardised weight matrix (changed distance) to capture a potential distance upshot and C is a constant while ρ (spatial spillover) and λ (spatial autocorrelation in the residuals) in add-on to α (direct effects) are the coefficients estimated. The error term ν in the SEM case consists of spatial error and the residuum ε.

The right choice betwixt both models can be determined by different tests, such as the z-score of Moran's I of the residuals and (Robust) Lagrange multiplier (LM) tests. The different estimation types in Tables i and two correspond to the respective choice.

Spatial lags can exist likewise expected for the independent variable. In an extended Spatial Durbin model, both, the dependent also every bit independent variables appear simultaneously as lagged variables. Every bit proposed by Halleck Vega and Elhorst, a simpler approach to consider the spatial lag of the predictor is offered past the SLX model [32]: (eight) The coefficients α and θ can be estimated past OLS. This model is applied in improver to the SEM/SAR estimations to command for spatial dependence of the variable S.

Ane major shortcoming of all such spatial econometric procedures needs to exist stressed: Changed distance never properly represents the effort needed to access a close or afar place. Natural transport infrastructure, topographical characteristics and the energy resources bachelor also decide mutual accessibility and city interaction. Altitude is thus only a proxy for try and time needed, given it is understood in the same manner every bit Zipf had divers the problem in his lemma. But this caveat applies to all such spatial econometric models equally long as there are no differentiated data that can replace inverse distance in the spatial weight matrix. In the end, a comparative view over time (different years) based on a true endeavour-specific weight matrix could much better reveal the important dynamic of spatial dependence in specific geographical settings during phases of major structural change. Merely this would be a discipline of future research.

iv Information

The ii faux "countries" describe a distribution with in each case 109 observations for cities, the upper 50 beingness Pareto (three,i) distributed and the lower 59 with a lognormal shape. Both distributions are consecutively random-generated and so matched into one data set. The get-go pace is the generation of 100 observations for both distributions. The upper 50 observations of the Pareto set are and so matched at the signal with the next smaller observation in the lognormal gear up. The proportion of observations in both tails could be different, e.g. exhibiting a larger lognormal tail, but this would just affect the shape of the full distribution. The merely purpose is to simulate one realistic rank-size distribution and to explore how sensitive it reacts on changing geographic coordinates. In the next footstep this hybrid rank-size distribution is combined with unlike distributions of coordinates 10 and Y, the start 1 being randomly generated to fit a normal distribution (Fig 1). Based on this configuration a spatial weight matrix is derived. With a normal distribution of both Ten and Y coordinates big and small cities are spread evenly.

In the second variation (Fig ii) with the same rank distribution the usually distributed coordinates of 10 and Y are both ranked equally well, and so that all cities are geographically positioned on a diagonal line, ordered forth rank, the biggest city in the outer N-East, the smallest one in the outer South-West (similar a one-dimensional von-Thunen assembly).

It is expected that in this case both, rank too equally size, exhibit spatial autocorrelation even though such a setting is hardly encountered in the real world.

The only purpose of this extreme setting is to show the possible potential of distance impact depending on the distribution of coordinates. With other words, identical coefficients confirming Zipf'south law may have a different meaning for different countries.

The two simulated cases are then compared with respect to the stability of α and the significance of the spatial parameters ρ or λ respectively. I hypothesise that with a normally distributed arrangement of coordinates the spatial weight parameters are insignificant and meaningless, representing a Zipf distribution of the upper tail similar to that of words in a language. However, when modifying the coordinates and the spatial distribution of cities past edifice clusters of cities with a different level of size we may await some stronger and significant altitude impact. The interesting question is then how stable the original Zipf distribution of the upper tail remains.

Finally, and in addition to the simulation analysis, this bogus exercise needs to exist examined in the real world. The two questions are: do we observe countries with significant spatial dependence in the rank-size distribution of cities and, if yep, how stiff could it exist? For that purpose I examine the spatial altitude influence on Zipf'due south constabulary with population information on U.s., German, British and Slovene urban areas respectively. For Slovenia as a detail instance of immature and small-scale country, as well natural cities (extracted from nocturnal satellite imagery) are explored in order to ameliorate capture the truthful functional urban infinite in that state. The respective paradigm segmentation methodology is further explained in Bergs [33]. The source of data is the National Oceanic and Atmospheric Assistants (NOAA) [34].

For the Us, the Britain and Germany the database is truncated below 100,000 inhabitants. This is in line with Gabaix [3], Giesen and Südekum [35] and Brakman et al. [5]. Slovenia, being a former small-scale province of Yugoslavia, represents a lower scaling level with the primate city slightly more than twice every bit large as the in a higher place truncation betoken. Therefore cities larger than 10,000 inhabitants are covered. To demonstrate that all cities regarded are inside the upper Pareto tail, a Shapiro-Wilk test was carried out for the log-transformed observations of population (or natural size).

Equally for the simulations, the SAR and the SEM model were used. In order to control for possible spatial dependence of the predictor in the country studies, SLX as an culling spatial model was as well tested. In all country models, the dependent variable was modified to R-1/2 (Gabaix-Ibragimov estimate) to avoid a potential bias of standard errors [36].

5 Results

First I take a expect at the unlike results of the simulation exercise. Tabular array 1 shows the full rank-size distribution, while Table 2 displays the estimates just for the upper (Pareto) tail. Regarding example I, Zipf's law is only confirmed for the curtailed distribution: α≈1 (Table ii). The lognormal extension with the smaller cities reduces the estimate of α to a large extent. As expected, the spatial coefficients λ of the full distribution and ρ of the Pareto tail are not significant. The normal distribution of coordinates of large and pocket-size cities leads to zero spatial dependence. With or without the spatial extension of the model the α coefficient remains the same. Zipf'due south law is well confirmed for the Pareto tail.

Example Ii shows the same rank-size distribution but ordered geographically along the coordinates. Now, there is a pregnant spatial mistake influence confirmed by λ for the full distribution and a spillover effect ρ for the Pareto tail. The accented value of α decreases essentially when incorporating spatial dependence. Zipf's law cannot exist further confirmed for the upper (Pareto) tail, even though the random-generated distribution had been a Pareto (3,ane) one. At a first glimpse this finding may be puzzling, only it simply confirms the potential result of a spatial arrangement with extremely enhanced spatial autocorrelation.

The false distributions of coordinates prove that, in theory, spatial distance may have a potential impact on the coefficient of the rank-size distribution of cities. The stylised models to a higher place are however artificial and not likely to be encountered in the existent world. Therefore data of four countries (3 big ones, ane minor) are used to run into how the spatial organization of cities may influence the estimate of rank-size distribution in the real globe. As expected, the results generated are less spectacular than for simulation Two merely yet suggesting spatial dependence to play a part in the rank-size distribution of cities for some countries:

For all samples the Shapiro-Wilk test rejects the Null hypothesis of normal distribution of the log-transformed observations in the upper tail, so the distributions regarded correspond the corresponding Pareto tail.

Estimates for the rank-size distribution of US urban zones are very much in line with Zipf's law (α = 1.005). The spatial error issue is small, still highly significant, and slightly improves the approximate. Hence, there is a very minor distance effect. A similar issue is obtained for Germany: In the distribution of urban zones a meaning spatial fault improved the Pareto coefficient from 0.930 to 0.948. I besides compared this estimate with German cities proper. In this sample, a coefficient of α = 1.239 could not ostend Zipf's law; however even here a significant spatial lag upshot moves the estimate slightly closer to Zipf (α = i.227).

In contrast to the United states and Deutschland, distance effects are insignificant in the case of the U.k., both, for spatial error (not displayed) too as spatial lag.

The Slovene case constitutes itself a fleck dissimilar. There is no distance result on the rank-size distribution of municipalities larger than 10,000 inhabitants. However, when viewing natural cities extracted from nighttime satellite images we find that spatial spillovers (ρ) are pregnant at the p<0.05 level; the α coefficient, yet, changes from 0.983 (inside the Zipf tolerance of α = one±0.1) to 0.860. In this case, spatial dependence implies a deterioration of the rank-size distribution.

To complete the econometric findings by viewing a possible spatial lag of the predictor, the SLX model did not reveal meaning spillover effects for whatever of the countries (Table iii). Hence, spatial dependence is merely found for the dependent variable R.

Now, an interesting question is, from where such spatial dependence furnishings of rank may originate. For this purpose the local Moran I coefficients (LISA indicator) may offer insight [37]. The spatial weight matrices are those generated for the SEM/SAR regressions (information sources: see Table 4). In Fig iii the resulting coefficients and the p-values for the United states of america, Frg, the UK and Slovenia are displayed and compared. The p-values are particularly important for the interpretation.

A hit evidence is that about of the 25 biggest urban zones in the Usa exhibit significant spatial autocorrelation. Negative LISA coefficients prevail simply at that place is a remarkable spread especially for the largest observations, e.g. a LISA coefficient of +3.ix for New York, outside the range of the Y centrality. For Deutschland, the nine biggest urban zones exhibit meaning spatial autocorrelation; the strongest being the Ruhr expanse on rank i (for German language cities proper only five out of the biggest). For the UK, a significant LISA coefficient is but found for the first three urban zones. For Slovenia, spatial autocorrelation can but be established for the majuscule city expanse (Ljubljana). Getting back to the spatial econometric estimates, information technology is to be remembered that the almost significant distance impact is found for the United states of america urban areas, followed by German urban areas. This seems to be reflected by the LISA coefficients.

To summarize, the estimations discussed above brandish partly significant though pocket-sized spatial dependence in the metropolis rank-size distribution of few selected countries. The existence of a significant blazon II error thus confirms the existence of spatial dependence. Probably in that location may be stronger or weaker such disturbances in other countries which are not regarded in this modest sample. A large comparative study covering all countries, ideally over fourth dimension and with more realistic weight matrices in the spatial econometric models (encounter earlier), could shed light on the global variation of spatial dependence in Zipf'due south law for cities.

6 Conclusion and further interpretation

Compared to Zipf's law for words in languages the results propose that in case of cities, their spatial organisation matters: Zipf's police for cities will behave similar Zipf'south law for words simply if small and big cities are normally distributed in space. This is shown by the two simulations. In the predominant theory, during time cities may change their size, simply the slope of the rank-size distribution remains rather stable [three]. This has been explained by its scale invariance and city growth independent of city size (Gibrat's law). Hence, the change of city ranks might be well explained by economic forces, but information technology is not directly visible in a changing slope of the rank-size distribution of cities. For this thread of argumentation spatial touch on has no item relevance. But studies combining Zipf's police with the Cardinal Place theory evidence that a spatial relationship between centers of different layers is besides in line with scale invariance [26–28]. Zipf'southward law can be also established in a model where distance exclusively governs the probability of city formation and growth [29]. In that location is thus reason to presume that dispersion and concentration forces determine the geographical distribution and centrality levels of cities, occasionally with more or less spatial dependence in their rank-size distribution. A spatial econometric approach suggests to shed light on such balance spatial dependence. If Gan et al. [2] were right, and Zipf's law represents nada more than a pure statistical human relationship, the extension of the model with spatial distance effects would not modify α. Where such spatial impact is significant, whether potent or modest, Zipf's law for cities is certainly more than a pure statistical phenomenon.

Supporting information

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