Rank-size Distribution Research Articles

Combining rank-size and k-means for clustering countries over the COVID-19 new deaths per million

This paper deals with the cluster analysis of selected countries based on COVID-19 new deaths per million data. We implement a statistical procedure that combines a rank-size exploration and a k-means approach for clustering. Specifically, we first carry out a best-fit exercise on a suitable polynomial rank-size law at an individual country level; then, we cluster the considered countries by adopting a k-means clustering procedure based on the calibrated best-fit parameters. The investigated countries are selected considering those with a high value for the Healthcare Access and Quality Index to make a consistent analysis and reduce biases from the data collection phase. Interesting results emerge from the meaningful interpretation of the parameters of the best-fit curves; in particular, we show some relevant properties of the considered countries when dealing with the days with the highest number of new daily deaths per million and waves. Moreover, the exploration of the obtained clusters allows explaining some common countries' features.

Spatial structure and carbon emission of urban agglomerations: Spatiotemporal characteristics and driving forces

This paper explores the impact of UASS on carbon emissions. Population and economic factor are taken into the measurement of UASS. GTWR model is to analyze the spatial and temporal differences. BTH, YRD, CP and MRYR show the characteristics of multi-center structure in 2019. PGDP and EI have positive impacts on carbon emissions in all urban agglomerations. To highlight the impact mechanism of spatial structure on carbon emission, this study calculates the spatial structure index of six urban agglomerations in China from the perspective of population and economy by using the rank-size rule. The Geographically and Temporally Weighted Regression (GTWR) model is used to analyze the spatial and temporal differences of the impact of spatial structure on carbon emissions in different urban agglomerations. The research results reveal that the carbon emissions of Yangtze River Delta (YRD), Beijing-Tianjin-Hebei (BTH) and Central Plains (CP) are much higher than that of the other three urban agglomerations. Chengdu-Chongqing (CY) and Pearl River Delta (PRD) show the characteristics of single-center structure, while the other four urban agglomerations show the characteristics of multi-center structure in 2019. The impact of spatial structures on carbon emissions of Yangtze River Delta and Pearl River Delta have experienced the process from positive impact to negative impact. The findings show that the multi-center structure of some urban agglomerations does not achieve the effect of carbon emission reduction to a certain extent, but promotes carbon emission. Therefore, it is not advisable to set the spatial structure of urban agglomeration as a unified model. The government should optimize the spatial structure of urban agglomerations according to the time evolution and internal characteristics of urban agglomerations.

An Attempt to Appreciate Climate Change Impacts From a Rank-Size Rule Perspective

For representative observational stations on the globe, rank-size analyses are made for vectors arising from sequences of the monthly distributions of temperatures and precipitations. The ranking method has been shown to be useful for revealing a statistical rule inherent in complex systems such as texts of natural languages. Climate change is detectable through the rotation angle between two 12-dimensional vectors. The rankings of the angle data for the entire station are obtained and compared between the former (from 1931 to 1980) and the latter (from 1951 to 2010) period. Independently of the period, the variation of the angles is found to show a long tail decay as a function of their ranks being aligned in descending order. Furthermore, it is shown that for the temperatures, nonlinearities in the angle-rank plane get stronger in the latter period, confirming that the so-called snow/ice-albedo feedback no doubt arises. In contrast to the temperatures, no sign of a feedback is found for the precipitations. Computed results for Japan show that the effect is consistent with the global counterpart.

Evolution of the Structure of the Urban Land-Price System in China Based on the Rank-Size Law

China’s cities have experienced rapid and uneven development in recent decades, which influenced urban land prices crucially. Understanding the distribution of the urban land-price system and its evolution would help to provide an in-depth understanding of the development of China’s urban system. Therefore, we combined rank-size law and fractal theory to analyze the patterns and variations of the rank-size distributions of China’s urban land-price system based on land transfer data for 2007–2019. We concluded the following: (1) the rank-size law explains the distribution of the urban land-price system in China—the system was balanced and showed an equilibrium–disequilibrium development; (2) the actual land prices were higher than the theoretical land prices in southeastern China, especially in urban agglomerations, while they were lower in northwestern China; (3) the number of cities located in China’s urban development system was decreasing, and the unevenness of urban land-price system was increasing; and (4) the imbalance of land-price rank-size distributions was higher in the Beijing–Tianjin–Hebei and Guangdong–Hongkong–Macau Greater Bay Area urban agglomerations, while they were more balanced in the Chengdu–Chongqing and Central Plains urban agglomerations. We suggest that the development of small cities in China has a large upside, which should be noticed.

Birth, Annexation, and Squeezing of Cities in a Prefecture: Can the Ranking of Competitive Areas of Municipalities Obey the Authentic Power Law?

As the first step for revealing potential rules inherent in cities that are closely squeezed in a sectioned domain, municipalities in the entire prefecture in Japan are considered and their distributions of the areas are analyzed in details according to a rank-size procedure. Computed results suggest that among the population, area, and population density, the last becomes the most important factor in finding the rank-size rule. Indeed, of the 47 Japanese prefectures the Metropolis of Tokyo and Fukuoka Prefecture exhibit the most typical rank-size rules, where the former possesses the exceptionally high population density as well as urbanized rate. The underlying mechanism of the rule can be supported by a toy model with a tournament game using a sequence of random numbers, where teams (municipalities) are highly competitive in gaining the final wins (broadest territory). A stability analysis implying perturbations due to global warming allows one to confirm unexpected robustness of the rank-size relation. Finally, the authenticity of the log-log relation in the rank-area data of Tokyo Metropolis is tested statistically.

Factor analysis of the parameters of the ergatic natural-production system of the carting out of timber assortments

ABSTRACT Factor analysis of 35 factors in 162 experiments was conducted for carting out timber from the forest to a forest products mill using the method of identifying stable patterns. The sampling power of the total correlation matrix was 352 = 1225 values of the correlation index for the identified stable patterns. According to the identified patterns of rank-size distributions and binary relations, the number of pre-selected factors decreased from 35 to 24 in summer (68.57%) and to 25 in winter (71.43%) using the proposed method of factor analysis. Basically, factors with a correlation index of less than 0.3 were excluded. Structural indicators become a system parameter of the set of regularities of binary relations, which decrease with an increase in the “permissible” correlation index. In total there are 1190 pairs of binary relations. According to the research results, among the influencing variables, the factor “type of snow and ice cover” was in first place, “number of the month in the year” was in second place, and “time of operation of the logging truck” was in third place. Among the dependent indicators in comparison with the general correlation matrix and the matrix of strong connections, the first and second places were: “the height of the snow cover (off the road)”; the given eastern longitude. The factor “the wheel formula of the logging truck” was in third place.

Spatio-temporal evolution of the container port system along the 21st-century Maritime Silk Road

ABSTRACT Research on the spatio-temporal evolution of the port system along the 21st-century Maritime Silk Road (MSR) can reveal the development law of ports. This study adopts the Herfindahl-Hirschman index, rank-size law, and an improved spatial and temporal correlation to investigate the spatial distribution and spatio-temporal correlation characteristics of container ports along the MSR during 2000–2019. The results show that 1) the distribution trends among port clusters have significant differences, and the gravity centre of the port system gradually shifts from Europe to Asia (market shares of European and Asian ports decline from 28.15% to 19.83% and increase from 63.80% to 66.78%, respectively); 2) the port system undergoes periodic ‘decentralisation–agglomeration–decentralisation’, and its structure gradually develops a ‘spindle’ or ‘olive’ shape; 3) the initial effect of MSR arises on the economic development of ports, and the spatial occupation ability of small- and medium-sized port clusters increases steadily; and 4) the spatial and temporal correlation gradually increases. The evolution of temporal correlation is more complicated with a bifractal structure in 2000–2010. This study can provide decision support for port development and thus clarifies the investment environment and construction of MSR.

Системы центральных мест: континуальное развитие на ранних этапах

The article is devoted to determining the conditions for the evolution of central place systems in the early stages – in the framework of the new and classical economic geography. We come to the conclusion that the new economic geography is not operational in relation to evolutionary processes, since it proceeds from postulates that violate the central place theory and based on incorrect assumptions (for example, validity of Bunge’s problem). We postulate the need to revise three main provisions of the classical approach: 1) on the initial correspondence of systems to the rank-size rule; 2) on increasing the compliance of systems with the rank-size rule as the share of urban population increases to 50%; 3) about the striving of central place systems to the state of equilibrium and – upon reaching it, for mature systems – an insignificant in its intensity ‘fluctuation’ around it. The authors define the conditions for the evolution of central place systems as follows: 1) the systems correspond slightly to the rank-size rule in the early stages and much better to the Christaller distribution; 2) as the share of urban population increases, the very slight compliance of systems with the rank-size rule worsens more and more; an increase in the share of urban population is accompanied by an improvement in the fit to the Christaller distribution; 3) systems strive for a state of equilibrium at any stage of their evolution; ‘oscillations’ around it for young systems are characterized by a significantly larger amplitude compared to that for mature systems

Formation of the settlement hierarchy: Zipf’s law vs Central place theory

This study compares the possibilities and advantages of forming the settlement hierarchy aсcording to Zipf and according to Christaller, starting from the first stages of the settlement systems development. The author found that the minimum number of hierarchy levels required to start the formation of the Christaller distribution, bypassing the Zipf distribution stage, for small values of the urbanization level was two: the first one, represented by one central place, and the level of rural settlements. With virtually the same possibilities of forming a hierarchy according to Zipf and according to Christaller at its initial stages in relation to the share of urban population, the advantages of the latter give large values of the maximum share of each central place in the population of the area it serves and lower costs of the system for the redistribution of the population between the existing settlements and the settlements emerging at a certain stage of evolution. At the same time, with an increase in the share of the urban population, the correspondence of the real rank-size distribution of cities to the ideal one (according to Zipf) decreases. The author established that the distribution according to Zipf and according to Christaller is a kind of two “immiscible liquids”: if the first is based on stochastic processes, then the second is based on non-random, even deterministic ones. The transition of the settlement system from the distribution according to Zipf to the hierarchy according to Christaller is a rather weak construct in the methodological and logical respect.

Economic Freedom: The Top, the Bottom, and the Reality. I. 1997-2007.

We recall the historically admitted prerequisites of Economic Freedom (EF). We have examined 908 data points for the Economic Freedom of the World (EFW) index and 1884 points for the Index of Economic Freedom (IEF); the studied periods are 2000–2006 and 1997–2007, respectively, thereby following the Berlin wall collapse, and including 11 September 2001. After discussing EFW index and IEF, in order to compare the indices, one needs to study their overlap in time and space. That leaves 138 countries to be examined over a period extending from 2000 to 2006, thus 2 sets of 862 data points. The data analysis pertains to the rank-size law technique. It is examined whether the distributions obey an exponential or a power law. A correlation with the country’s Gross Domestic Product (GDP), an admittedly major determinant of EF, follows, distinguishing regional aspects, i.e., defining 6 continents. Semi-log plots show that the EFW-rank relationship is exponential for countries of high rank (≥20); overall the log–log plots point to a behaviour close to a power law. In contrast, for the IEF, the overall ranking has an exponential behaviour; but the log–log plots point to the existence of a transitional point between two different power laws, i.e., near rank 10. Moreover, log–log plots of the EFW index relationship to country GDP are characterised by a power law, with a rather stable exponent () as a function of time. In contrast, log–log plots of the IEF relationship with the country’s gross domestic product point to a downward evolutive power law as a function of time. Markedly the two studied indices provide different aspects of EF.

Demystifying the Geography of Urbanization in Indonesia (Case Study: Southeast Sulawesi Province)

What is a city? How can we define the boundaries of a city? Despite being the popular subject of research in rapid urbanization, the discussion of the spatial dimension of urbanization is still few in Indonesia. This paper aims to make the geography of urbanization in Indonesia clearer and easier to understand by applying the degree of urbanization (DEGURBA/DoU) methodology. The case study location proposed is Southeast Sulawesi province (SSP)which has distinctive urbanization trajectories with rapid urbanization trends and a relatively stable rural population, but a lack of research regarding its urbanization patterns. This research used Global Human Settlement Layer (GHSL) data as the primary source to conduct spatial analysis to produce the degree of urbanization with complementary national statistical and spatial data to compare with the current conventional classification methodology. The findings show that based on DoU, 39.87% of the SSP population live in urban centers, 22.58% in urban cluster/periurban, and 37.56%. SSP has a total of 49 urban units with 6 urban centers/cities, and 43 urban clusters/towns. By testing the rank-size distribution of SSP’s urban units, it shows that SSP’s urban systems pattern satisfy Zipf’s law with R2 = .9885, and the slope of the fitted line is -0.9554. Based on this result, currently, SSP has 4 medium cities, 2 small cities, 21 large towns, and 20 small towns. In addition, the DoU also has an opportunity to understand how should urban areas be governed and managed, and by whom. Despite the Greater Kendari Urban Area has not yet fulfilled the national’s Metropolitan population threshold (1 million pop), the urban management and development should be conducted more collaboratively with neighboring regencies such as Konawe and South Konawe. The advantages are shown by these findings highlight the importance of scaling up this research to a national scale and using finer and nationally available data.

The Distribution Patterns of Valency-changing Verbs: An Approach of Quantitative Linguistics

The present study attempts to explore the distribution patterns of the valency-changing verbs from the perspective of quantitative linguistics. We took authentic spoken language data as the research materials. The corpus used in this paper is a self-built spoken English corpus containing about 21,000 words. We half-manually annotated the corpus with the help of SpaCy, a natural language processing tool. According to the annotation results and statistical data, we obtained a total of 217 valency-changing English verbs and 248 sentence components governed by them. After analysis, the current study came to the following conclusions: First, bivalent verbs are most frequent among the three types of valency-changing verbs; second, after fitting all the language data to different probability distributions, we found that the rank-frequency distributions of all the valency-changing English verbs with different numbers of obligatory arguments obey the power law, and the frequencies of bivalent valency-changing verbs obey other kinds of distributions such as the mixed Poisson distribution.

The rank-size distribution of cities in South Africa

The rank-size distribution of cities in South Africa

On the Evolution of Hierarchical Urban Systems in Soviet Russia, 1897–1989

One piece of evidence of the inefficiency of the spatial economy of modern Russia presented in the seminal work of Hill and Gaddy (2004) is that Russian urban agglomerations are non-viable. This was demonstrated using Zipf’s rank-size distribution, which does not hold for Russian urban systems. Hill and Gaddy explained this through the legacy of the Soviet command-administrative planning. Having constructed an original dataset, which incorporated comprehensive historical data for all the cities in the former Soviet Union republics and tested the rank-size distributions for the respective years, the study yielded more nuanced findings. First, unlike the modern Russian hierarchical urban systems, the Soviet ones followed rank-size distribution fairly well. Second, the Soviet urban systems were evolving. In the late Imperial era and early Soviet period, they followed the Zipf’s law prediction. However, between 1939 and 1959, the rank-size distribution diverged from the predicted one. Yet again, the Soviet hierarchical urban systems revealed a trend of convergence toward the traditional rank-size distribution in the late Soviet era. A corollary to such evidence from data trajectory appears that the evolution of the Soviet hierarchical urban systems was not necessarily the ultimate product of the urban development policies of the command-administrative system. It can be thus presumed that, contrary to the established belief, command administrative urban development might be ineffectual even in centrally planned socialist economies.

Assessing the Balance of the Urban Settlement System in the European North of Russia

Introduction. In modern Economics, one of the most common and simplest methods of analyzing the balance of urban settlement systems is to assess their compliance with Zipf's law or the rank–size rule. The basis of this pattern is the relationship between urban population and its place in the hierarchy of towns ranked in descending order of size. Based on the results of the study conducted, the article assesses the balance of the urban settlement system of the European North Russia, as one of its regions, by analyzing its compliance with Zipf’s law. Materials and Methods. The official data from the Federal State Statistics Service on the population of towns in the European North of Russia for 1959, 1989 and 2019 were used as materials of the study. The method of constructing a linear regression between the logarithm of the actual population and the logarithm of the rank of the town was used to verify Zipf's law for the urban network of the region in a certain period. In order to substantiate the conclusions drawn, an analysis of the dynamics of the number of towns and the share of the population living in them was carried out. The monographic method, as well as the methods of tabular and graphical data visualization, was used to interpret the results of the calculations. Results. Based on the analysis of data on the application of the rank–size rule for the towns in the European North of Russia, it has been found that Zipf’s law was not fully observed in any time period, which indicates the imbalance of the existing urban settlement system. In the period from 1959 to 2019, there was an increase in the concentration of the population in the major cities of the region. The imbalance is also caused by the growing number of small towns with a population that does not correspond to the optimal value according to Zipf's law. Discussion and Conclusion. Based on the calculations, the author has come to the conclusion that the cities of Arkhangelsk and Cherepovets have the potential for growth, as well as some others with a population of up to 100 thousand people. The practical significance of the study lies in the possibility of using the results obtained to prognosticate the population of towns in the European North of Russia when planning the location of production facilities, as well as transport and social infrastructure in the region.

A Fractal Approach to Urban Boundary Delineation Based on Raster Land Use Maps: A Case of Shanghai, China

Given the diverse socioecological consequences of rapid urban sprawl worldwide, the delineation and monitoring of urban boundaries have been widely used by local governments as a planning instrument for promoting sustainable development. This study demonstrates a fractal method to delineate urban boundaries based on raster land use maps. The basic logic is that the number of built-up land clusters and their size at each dilation step follows a power-law function. It is assumed that two spatial subsets with distinct fractal characteristics would be obtained when the deviation between the dilation curve and a straight line reaches the top point. The top point is regarded to be the optimum threshold for classifying the built-up land patches, because the fractality of built-up land would no longer exist beyond the threshold. After that, all the built-up land patches are buffered with the optimum threshold and the rank-size distribution of new clusters can be re-plotted. Instead of artificial judgement, hierarchical agglomerative clustering is utilized to automatically classify the urban and rural clusters. The approach was applied to the case of Shanghai, the most rapidly urbanizing megacity in China, and the dynamic changes of the urban boundaries from 1994 to 2016 were analyzed. On this basis, urban–rural differences were further explored through several fractal or nonfractal indices. The results show that the proposed fractal approach can accurately distinguish the urban boundary without subjective choice of thresholds. Extraordinarily different fractal dimensions, aggregation and density and similar average compactness were further identified between built-up land in urban and rural areas. The dynamic changes in the urban boundary indicated rapid urban sprawl within Shanghai during the study period. In view of the popularization and global availability of raster land use maps, this paper adds fuels to the cutting-edge topic of distinguishing the morphological criteria to universally describe urban boundaries.

Rank–size distributions for banks: A cross-country analysis

Rank–size analysis is a powerful methodological instrument for describing the main features of a unified system by moving from an available sample of its disaggregated elements. We here study the rank–size relationship of banks’ total assets in a sample of 13 advanced economies. We show that a standard power-law function is unable to provide an adequate fit of the data, while a Discrete Generalized Beta Distribution provides a statistically satisfactory fit, being able to capture the behaviour of the distribution at low and high ranks. We then analyse the relationship between the parameters of the Discrete Generalized Beta Distribution estimated in each country and the main features of the country’s banking industry. Our results point to a connection with the degree of development of the banking sector and with some regulatory and institutional characteristics, but no relationship with riskiness.

Mapping hierarchical urban boundaries for global urban settlements

Urban boundary is the spatial basis for urban statistics and urban planning. However, the city definition by which the urban boundary is determined is not comparable among different countries. A globally consistent delimitation of hierarchical urban boundaries (HUBs) is rare but urgently needed. In this research, we proposed a model to consistently identify HUBs according to physical urban entity. The main quantitative basis of our model is the density, size and spatial relationship of the artificial impervious surface. We produced a vector dataset of global HUBs (named GHUB) based on 30-meter resolution remote sensing products. Cross-product comparison shows the effectiveness and advantages of our model. We furtherly, for the first time, investigated and compared the physical form and spatial characteristics of global urban settlements through the same lens supported by GHUB. We have identified 10,242 urban settlements with an area above five square kilometers, 39.18% of them are located within 100 km of the coastline. Urban settlements are not the “reinforced concrete forest” as commonly thought. The global average proportion of open space within urban settlements is 41.51%. The total area of urban open space in the world is 279.47 thousand square kilometers. Notably, we verified the Rank-size rule of urban settlement is universal, meanwhile, the verification results of different urban systems in 47 countries show that the Zipf’s exponent varies from country to country. Our work can help to consistently investigate global urban settlements without being affected by various definitions of cities. The GHUB can not only be applied to various urban studies but also can support global affairs such as the Sustainable Development Goal 11 due to its international comparability.

Research on Layout Optimization of Villages in Poverty-Stricken Counties —A Case Study of Wangmo County, Guizhou, China

Against the background of rural revitalization in China, this study takes Wangmo County as the study area, and uses mathematical models such as entropy method and rank-size rule to quantitatively evaluate the rural development potential of Wangmo County and rural development scale and hierarchical structure in Wangmo County. Based on this, the study puts forward suggestions for village layout optimization. According to the results, (i) evaluation elements of rural subject, industrial development, resource endowment, and habitat environ­ment in Wangmo County are presented in the spatial pattern of being scattered as a whole and be agglomerated locally. (ii) The rank-size distribution of village is that there are more villages in the medium-ranking position, while there are fewer high-and-low-ranking villages with spindle structure of “being small on both ends, and large in the middle”, showing that the agglomeration of village elements has a weak degree of spatial polarization. (iii) By comprehensively evaluating results and field investigation situation, a township hierarchy of central village-general village-merged village is constructed to divide the development types of three rural areas, including agglomeration type for improvement, equilibrium, and stable type, as well as relocation and merger type. The results of the study in the case area can provide a reference for the local scientific response to the decreasing trend of rural population and change the predicament of low efficiency of public resource allocation caused by small-scale and scattered distribution of the rural areas.

The distribution of city sizes in Turkey: A failure of Zipf’s law due to concavity

AbstractThe linearity of the distribution of city sizes is often assumed in the existing literature. Although different functional forms were tried, almost all of them impose a certain functional shape. In this study, we investigate the urban hierarchy and Zipf’s law using data for 973 Turkish subprovincial cities in 2019. We contribute to the literature in several ways. We force no definite functional form to observe the natural shape and employ nonparametric and quadratic regressions. We incorporate formal procedures of spatial dependence in regression models. We demonstrate that the linear model overestimates the Pareto exponent for small cities and underestimates it for bigger cities. We show that city sizes are unevenly distributed in Turkey. The rank–size rule is not valid in Turkey, either above or below a certain city‐size truncation level. Thus, the Pareto exponent estimated from the linear model is not a reliable indicator as quadratic regressions perform much better.