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A prominent characteristic of. COVID-19 is the long incubation period, the lack of symptoms during that period, and significant geographic variation in prevalence and testing. This article investigates the relationship between testing and infection based on an Israeli sample of 193 municipalities and a report from 11 May 2020. Results support an exponential rise in the fraction of positive. COVID-19 outcomes with the fraction of tests. Finally, under conditions of equal socioeconomic ranking, population density and income inequality, the projected efficiency of tests rises with population density and the Gini index and drops with socioeconomic ranking. [ABSTRACT FROM AUTHOR]

A rank-size rule following Zipf's law was tested along a complete urban–rural hierarchy in Greece using 2021 census data released at different administrative levels. Testing five econometric specifications (linear, quadratic, and cubic forms, together with refined logistic and Gompertz forms) on log-transformed population numbers, deviations from the rank-size rule were assumed as an indicator of economic concentration (considering settlements, population, and activities jointly) along the density gradient in Greece. This hypothesis was verified using progressively disaggregated population numbers at (i) regional units (n = 75), (ii) 'Kallikratis' municipalities (n = 333), (iii) 'Kapodistrian' municipalities (n = 1037), and (iv) local communities (n = 6126). Econometric results were stable across geographical levels and indicate a relatively poor fit of linear specifications, the classical formulation of Zipf's law. Quadratic specifications displayed a good fit for all territorial levels outperforming cubic specifications. Gompertz specifications outperformed logistic specifications under aggregate partitions (e.g., regional units and 'Kallikratis' municipalities). Quadratic specifications outperformed both logistic and Gompertz specifications under disaggregated levels of investigation ('Kapodistrian' municipalities and local communities). Altogether, these findings indicate the persistence of non-linear rank-size relationships estimated over a cross-section of population data at progressively detailed observational units. Such evidence enriches the recent literature on Zipf's law, demonstrating the inherent complexity of rank-size rules tested on real data along the whole density gradient in a given country. [ABSTRACT FROM AUTHOR]

An attempt is made to critically study the measuring urbanisation, recent urban growth patterns, Urban Growth Sustainability (UGS) and addresses the question of how the primate city impacts urban growth. The methods of Urban-Rural Growth Differential (URGD), Composite Annual Growth Rate (CAGR) and Index of Primacy (PI) have been used to understand recent urban growth, impact of primate cities in India based on Census. The top-heavy nature of India's metropolitan structure has a negative impact on the country's balanced regional growth. The study recommends that more infrastructure capacity should be granted for the primate cities due to increases in the city's population. [ABSTRACT FROM AUTHOR]

The strategy of geographically diversifying a portfolio of commercial real estate assets is an intuitive approach for risk management. However, due to high concentrations of these assets in major metropolitan areas, investors may face additional constraints in the portfolio optimization process. The rank-size rule, a log-linear relationship between city rank and size, provides one of the greatest empirical regularities in regional science. As such, it serves as a possible theoretical guide to the weights given to properties by location in a commercial real estate portfolio. This paper sets forth some ideas relating to the concentration side of portfolio variance and the limiting effect that large concentrations may have on the ability to diversify risk. Two variants of the rank-size relationship – the Zipf distribution and the parabolic fractal distribution – are fitted to a variety of datasets to provide a sense of the degree of concentration in the commercial real estate industry. These empirical findings suggest the presence of limitations to geographical diversification that have varying degrees of severity across different property types or sectors of the commercial real estate market. [ABSTRACT FROM AUTHOR]

A rank-size rule following Zipf’s law was tested along a complete urban–rural hierarchy in Greece using 2021 census data released at different administrative levels. Testing five econometric specifications (linear, quadratic, and cubic forms, together with refined logistic and Gompertz forms) on log-transformed population numbers, deviations from the rank-size rule were assumed as an indicator of economic concentration (considering settlements, population, and activities jointly) along the density gradient in Greece. This hypothesis was verified using progressively disaggregated population numbers at (i) regional units (n = 75), (ii) ‘Kallikratis’ municipalities (n = 333), (iii) ‘Kapodistrian’ municipalities (n = 1037), and (iv) local communities (n = 6126). Econometric results were stable across geographical levels and indicate a relatively poor fit of linear specifications, the classical formulation of Zipf’s law. Quadratic specifications displayed a good fit for all territorial levels outperforming cubic specifications. Gompertz specifications outperformed logistic specifications under aggregate partitions (e.g., regional units and ‘Kallikratis’ municipalities). Quadratic specifications outperformed both logistic and Gompertz specifications under disaggregated levels of investigation (‘Kapodistrian’ municipalities and local communities). Altogether, these findings indicate the persistence of non-linear rank-size relationships estimated over a cross-section of population data at progressively detailed observational units. Such evidence enriches the recent literature on Zipf’s law, demonstrating the inherent complexity of rank-size rules tested on real data along the whole density gradient in a given country.

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

In the context of geographical quantitative revolution around the mid-20th century, George K Zipf’s model emerged, which was designed to measure sizes of cities by their population in what is known as the rank-size rule. It is important to apply that scientific base to measure sizes of cities at state level as well as level of administrative regions within the state. This is the scope of this study, which sought to apply the rule to the cities of Qassim region as a geographical unit within Kingdom of Saudi Arabia. The aim of the research is to study the urban system in Qassim and discover the urban hierarchy that runs the governorates within their administrative boundaries. The main question of the study was: What is the balanced geographical rank of the cities of Qassim region? The tools of the geographical statistical method were used to display and analyze data, where most important results were: The urban system in Qassim region is dominated by the city of Buraidah, which contains most of the geographical factors that formed the urban sites and their environmental conditions. The study concluded that the urban system in Qassim enjoys a natural formative background that depends on the fertile lands around its cities, whose characteristics are similar and their main features converge. In spite of this, the consequence of applying Zipf's rank-size rule was that: cities are far from the straight equilibrium line of the vertices of their sizes.

Perturbation analysis of the rank-size rule is made for municipalities that are squeezed in a prefecture. Specifically, attention is focused on Japanese prefectures, for which effects of the archipelagoes, municipal consolidations, and coastal inundations due to the future sea-level rise are examined. In each prefecture, typically tens of municipalities are closely packed with a unique configuration that has been arranged according to a self-organized process. With a mind to sustainable development goals, analysis is made from three points of view. First, comparisons between the results of the mainland and those of the entire prefecture with territorial islands suggest that historical interactions over a long period become a key factor to enhance the statistical rule. Next, to cope with rapidly increasing rural depopulation, simulations are made on the basis of three consolidatory schemes. Of prefectures with extremely depopulated areas, particular emphasis is laid on Fukushima Prefecture, the east coast of which was damaged by the nuclear disaster on 11 March 2011. Lastly, analysis is made implying perturbations due to global warming. Among three parameters that characterize the rank-size relation, stabilities are confirmed for the scaling exponent.

Effectively monitoring the size of a city in real time enables the scientific planning of urban development. Models that utilize the distribution and variations in city size generally use population data as inputs, which cannot be obtained in a timely and rapid manner. However, night-time light (NTL) remote sensing may be an alternative method. A case study was carried out on the Yangtze River Delta (YRD) in China, and the rank–size rule, the law of primate cities, and the Gini coefficient were employed to monitor the variation in city size in the study area. The urban areas extracted based on NTL remote sensing were utilized instead of the traditionally used population data to evaluate the variations in city size from 2012 to 2017. Considering the empiricism and subjectivity of the thresholding method, urban areas were extracted from NTL data combined with the normalized differential vegetation index and land-surface temperature data based on the artificial neural network algorithm. Based on the results, the YRD did not fit the distribution of the primate cities from 2012 to 2017. However, this region satisfied the rank–size rule well, which indicated that the development of medium–small cities was more prominent than that of larger cities, and the dispersed force was larger than the concentrated force. Notably, the city size reached a relatively balanced level in the study area. Further, sensitivity analysis revealed that the relatively low extraction accuracy of urban areas of few small cities had little effect on the results of city size variations. Moreover, the validation of city size computed from statistical population data and its comparison with results calculated based on the statistical data of urban areas aligned with the results of this study, which indicates the rationality and applicability of monitoring the variations in city size using the urban areas extracted from NTL remote sensing instead of population data. [ABSTRACT FROM AUTHOR]

This paper offers a new numerical approach to model the effects of archaeological site placement and population density on environmental suitability using two ecological models, the Ideal Free Distribution (IFD) and Ideal Despotic Distribution (IDD), treating the Oued Loukkos in northern Morocco as an example. This method incorporates local resource depletion with increasing population density consistent with the predictions of the IFD. It also incorporates the potential for the exclusion of part of the population from certain productive areas consistent with the IDD. In this study, we propose a dynamic approach to the impact of site placement on habitat suitability and therefore broader application for understanding changes in settlement distribution with population density through time. Furthermore, resulting settlements can be parameterized by a Zipf-Mandelbrot distribution, entailing a direct relationship between the ecological, demographic, and political factors and the resulting rank-size distribution of sites. This approach allows for a broader range of applications among foraging and agricultural economies. [ABSTRACT FROM AUTHOR]

The sole well-recognized United States (US) urban hierarchy articulation essentially is outdated, even though selected recent work seeks to upgrade it. The primary goal of this paper is to update it in a definitive and comprehensive fashion. This paper describes the conceptual framework underlying such observed orderings, itemizes certain strengths and weaknesses of the existing articulation, and then posits a justifiable renovated US urban hierarchy. Next, recapped analyses expose both contiguity and urban hierarchy spatial autocorrelation components of the upper tiers of the 2020 US metropolitan area population rank size distribution. Noteworthy is that these descriptions entail positive-negative spatial autocorrelation mixtures. Inventoried output from the research efforts leading to this paper includes: a contemporary US urban hierarchy articulation that should prove serviceable for at least the next few decades; and, an apparatus providing a practical contribution for improving cultural, environmental, and social aspects of systems of cities through, for example, better cost containment and more efficient/effective delivery of urban public health services and utilization/consumption. The Earth's scientists need this category of tool to incorporate into methodology combating negative effects of globalization that materialize via spatial diffusion.

Objectives: As the acceleration of global urbanization, traffic congestion and environmental degradation have emerged in the urban area. Based on the light and statistical data, this paper studied the spatial evolution between urban people and land, and revealed the significance of its coupling and coordination characteristics for the sustainable development of cities. Methods: For primary data, a series of preprocessing are needed to eliminate the problems of supersaturation and discontinuity of inter‑annual data. Based on the lighting threshold, this paper excerpted the built‑up areas of urban agglomeration in Zhejiang Province year by year, compared them with the statistical data, and analyzed the corrected data source relatively. Firstly, based on the order‑scale distribution model of urban order and urban population in Zhejiang Province from 1992 to 2020, we made a dynamic analysis on population aggregation and distribution in its development process. Then, we constructed the population‑land urbanization allometric growth model by using population and light area, as well as judged the spatialization of man‑land relationship in urban development. Finally, according to the lag phenomenon of population growth relative to land development, we proposed an allometric growth model with lag factor, and used the lag term model to judge the characteristics of unequal growth of people and land in different cities. Results: The structural system of urban agglomeration is relatively mature in Zhejiang Province, and the distribution of cities is reasonable. However, the primacy of city is gradually improved, and the population is gradually gathering. The results of allometric growth model show that the man‑land relationship of urban agglomeration in Zhejiang Province is generally low in the eastern coastal cities and high in the western inland cities, and the planning of the four metropolitan areas is reasonable. The improvement of fitting accuracy of the proposed model with lag factor obviously shows that there is lag effect between population urbanization and land urbanization, with an average increase of 6.47% and the highest increase of 12.61%. Among the influencing factors on land urbanization, the dominant factor in Hangzhou is the existing population in urban areas, while the dominant factor in Ningbo is the new population in urban areas. Conclusions: The proposed allometric growth model with lag factor can correct the model errors and improve the fitting accuracy of urban man‑land relationship. It can also effectively evaluate the temporal and spatial evolution characteristics of man‑land relationship in regional system, explore its development and put forward countermeasures. [ABSTRACT FROM AUTHOR]

Objectives: It is important to know the pattern of Pakistan̓s regional development for the construction of China‑Pakistan Economic Corridor. Methods: We used visible infrared imager radiometer suite (VIIRS) nighttime light remote sensing image as the main data source, and implemented Savitzky ‑ Golay (SG) filtering algorithm to repair the missing values of the images. Night time light change ratio (NLCR), Mann ‑ Kendall(MK) trend test method and rank ‑ size rule were carried out to study the nighttime light change patterns of province level, city level, geographic grid level, and east‑west zone in Pakistan. Results: The analysis result of NLCR shows that from 2012 to 2019, NLCR of the entire Pakistan is 47.96% and all provincal regions has developed to a certain degree with NLCR larger than 15%. NLCR of Gwadar Port is 126.94%, which is the highest in 15 node cities on east and west lines of China‑Pakistan Economic Corridor. Additionally, the development of 15 node cities on east and west lines of China ‑ Pakistan Economic Corridor was unbalanced. The results of MK trend test show that at the geographic grid scale of 5 km, there are 1 905 grids with the total nighttime light upward trend in Punjab and 338 grids with the total nighttime light downward trend in Sindh. It is found that most grids with the total nighttime light downward trend are located in rural areas of Sindh, which is consistent to the fact that there is a serious power shortage problem in rural areas of Sindh. The analysis results of rank‑size rule show that the Pareto indexes of both eastern and western zones are gradually decreased, indicating that the distribution of urban size in Pakistan becomes more balanced. And the lack of stable power supply may be the reasons for the low brightness of nighttime light in some areas. Conclusions: The regional development patterns of Pakistan under long‑term time series and multiple spatial scales can provide a certain scientific basis for the planning of China‑Pakistan Economic Corridor. [ABSTRACT FROM AUTHOR]

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.

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.

Effectively monitoring the size of a city in real time enables the scientific planning of urban development. Models that utilize the distribution and variations in city size generally use population data as inputs, which cannot be obtained in a timely and rapid manner. However, night-time light (NTL) remote sensing may be an alternative method. A case study was carried out on the Yangtze River Delta (YRD) in China, and the rank–size rule, the law of primate cities, and the Gini coefficient were employed to monitor the variation in city size in the study area. The urban areas extracted based on NTL remote sensing were utilized instead of the traditionally used population data to evaluate the variations in city size from 2012 to 2017. Considering the empiricism and subjectivity of the thresholding method, urban areas were extracted from NTL data combined with the normalized differential vegetation index and land-surface temperature data based on the artificial neural network algorithm. Based on the results, the YRD did not fit the distribution of the primate cities from 2012 to 2017. However, this region satisfied the rank–size rule well, which indicated that the development of medium–small cities was more prominent than that of larger cities, and the dispersed force was larger than the concentrated force. Notably, the city size reached a relatively balanced level in the study area. Further, sensitivity analysis revealed that the relatively low extraction accuracy of urban areas of few small cities had little effect on the results of city size variations. Moreover, the validation of city size computed from statistical population data and its comparison with results calculated based on the statistical data of urban areas aligned with the results of this study, which indicates the rationality and applicability of monitoring the variations in city size using the urban areas extracted from NTL remote sensing instead of population data.

This study examines the nexus between high-speed rail (HSR) stations and city evolution from an urban science perspective. We analyzed 1627 HSR stations built from 1976 to 2021 in 533 Chinese and European cities. Node or activity clusters in cities are identified through the street network. Then, the urban forms of Chinese and European cities are compared and the Zipf's law is tested. We found that activity clusters in Europe tend to be more contiguous and clustered, suggesting a clear downtown. In Chinese cities, these clusters are more scattered, suggesting polycentric urban development. When the Zipf's law is applied to the size distribution of activity clusters, the rank-size rule is supported both at the aggregate and country levels. Generally, it takes roughly two decades for an HSR station to be able to mature in attracting related development, such as residence and businesses, to complete the exploration process of city evolution. Finally, contributory factors for the integration of HSR stations with the urban fabrics are examined. Apart from the size of the city, economic power, station features and the spatial pattern of activity clusters, the relative location of the HSR station in relation to the urban form matters. Our study contributes both to the HSR and urban science literature, and has practical implications for locational decisions about HSR stations. • Analyzed 1627 HSR stations built from 1976 to 2021 in 533 cities • Measured activity clusters from an urban science perspective • Urban forms of Chinese and European cities are compared and Zipf's law is tested. • Contributory factors for integration HSR stations with urban fabrics are identified. • City size, GDP, station features, and size of nearest node clusters matter [ABSTRACT FROM AUTHOR]