Rank-size Distribution Research Articles

Global consistency of urban scaling evidenced by remote sensing

Abstract The urban scaling theory (UST) strives for a universal taxonomy that depicts relationships among urban indicators (e.g., energy consumption, economic output) with city size. However, the lack of international agreement on city definitions and statistics complicates cross-country comparisons of urban scaling performance. Remote sensing provides a uniform standard for measuring cities around the world. To scrutinize the consistency of UST, we quantified changes in remotely sensed urban built-up areas (UBA) and nighttime lights (NTL) distributions from 11,581 cities in 61 countries spanning 2000-2020, representing urban physical elements and socioeconomic activities respectively. We find that UBA is well-described by UST in all analyzed countries, while NTL aligns with 98% of them. UST quantified by remote sensing shows greater robustness than country-dependent aggregate statistics. We also observed disparities of scaling exponents (β) among countries, with UBA all being sub-linear (β < 1), and NTL ranging from 0.46 to 1.22 with a median of 0.94. Both UST and rank-size distributions of urban area and population show stronger scaling relationships for countries with larger networks of built environments, (Brazil, China, Germany, Japan, Russia, USA) suggesting that both size and evolution of urban systems impact the underlying scaling processes. Comparison of scaling properties of remotely sensed UBA and NTL captures complementary physical characteristics of built environments while minimizing the Modifiable Area Unit Problem introduced using spatially aggregated metrics within administrative units. Our findings highlight the consistency of urban growth patterns while confirming the systematic socioeconomic disparities among urban systems of varying size and growth trajectory.

Exploring spatiotemporal heterogeneity of urban green freight delivery parking based on new energy vehicle GPS data

To enhance the efficiency and sustainability of urban freight operations, China has initiated the Urban Green Freight Delivery (UGFD) project, which involves optimizing access control policies and introducing new energy vehicles. Identifying the parking trips of new energy vehicles and exploring the spatiotemporal patterns is crucial to actively promoting the optimal layout of temporary stops and the formulation of parking policies in the UGFD project. In this study, we aim to comprehend the spatiotemporal heterogeneity of parking for new energy vehicles both on roads (on-street) and within urban communities (off-street) for promoting the UGFD project. Its specific content includes: (1) proposing a method for identifying valid parking trips for the loading and unloading of goods based on trajectory data of UGFD new energy vehicles; and (2) mapping the identification results of valid parking trips onto communities and roads to analyze the spatiotemporal heterogeneity. Taking Suzhou, Jiangsu Province, China as an example, the identification results show that the established valid parking trips identification method can outperform state-of-the-art methods. The accuracy, precision, recall, and F1 value were found to be 0.957, 0.908, 0.937, and 0.922, respectively. Further examination of parking patterns indicates a bimodal temporal distribution of delivery demand, with peak activity occurring between 08:00–09:00 and between 14:00–17:00, with a higher delivery demand in the morning. Spatially, delivery demand was aggregated, while the parking time distribution of most delivery activities was normal. Additionally, the parking characteristics of communities and roads conformed to the ‘Rank–size rule’, suggesting that most delivery parking activities were concentrated in a few communities and roads. These findings can also be used in UGFD stop station utilization, travel time, arrival time prediction, and other related fields, all of which can further support relevant management departments in discovering abnormal delivery behaviors and reduce their negative impacts.

Determination of Service Centers Along the East Coast Corridor of Ogan Komering Ilir Regency

This study aims to determine service centers along the East Coast Corridor of Ogan Komering Ilir Regency using two analytical approaches: facility service index analysis and rank-size rule analysis. Perroux's growth center theory (1950) is applied to understand how districts with higher facilities and populations can stimulate surrounding economic growth. The facility service index analysis identifies the availability and distribution of public facilities in each district, while the rank-size rule maps population distribution to pinpoint potential service centers. The results show that Air Sugihan District holds potential as the primary service center due to its facilities, while Tulung Selapan District is designated as a settlement center given its largest population. With a balanced development strategy, it is expected that this can create equitable service access, foster economic growth, and strengthen the socioeconomic resilience of communities in Ogan Komering Ilir Regency.

Rank Size Rule Analysis on the Urban Population Distribution of Indonesia for the Year 2020

The rank-size rule which is known as Zipf``s law is an interesting research topic in the field of urban and regional studies. Conclusions are drawn based on the parameters that examine people and their economic activities in a particular city. this study aims to assess the unique urban development and population distribution of Indonesia by taking into account the complex socioeconomic system and geographically different archipelago. This study uses demographical data from various cities in Indonesia to understand the hierarchal pattern of population distribution in urban areas. Furthermore, Indonesia is an interesting case study regarding urban economics since it has a vast geographical area and contains different cultures and economic activities. The main objective of this research is to understand the population distribution and to evaluate the accuracy of the rank-size rule model in this country. This research also aimed to identify the main factors that deviate the population distribution from the rank-size rule distribution. It is particularly important to develop practical policies on the urban development issues. The minor variations of population distribution in Indonesia from the rank-size rule model are a good tool for decision-makers to allocate resources, develop infrastructure, and contribute to sustainable urban expansion. In addition, this study is significant for understanding how the urban centers developed in Indonesia and how they interact with broader socio-economic perspectives which is essential for further urban development and wise decision-making.

ASSESSMENT OF PORT CITY SERVICE CENTER ALIGNMENT IN THE TERRAIN

Medan, in North Sumatra, is growing rapidly, especially in the sub-districts of Medan Marelan and Medan Belawan, where urban activities and population movement are increasing. As a result, North Medan has been designated as the City Service Centre (CSC) of Medan City, with CSC Labuhan in Medan Labuhan sub-district playing a major role. This highlights the importance of Medan Labuhan in providing essential services and infrastructure to support the growth of the area. In order to effectively serve the community, it is important to ensure that CSC Labuhan meets the area's needs and standards for a city service center. Aims: The objective of this research is twofold: first, to determine the status of the service center system in North Medan; and second, to evaluate the suitability of CSC Labuhan's functions in serving the North Medan region. Methodology and results: The research method is descriptive analysis using both quantitative data and qualitative information. This is achieved through the use of scalogram analysis, centrality index analysis and rank-size rule analysis. Conclusion, significance and implication: The results indicate that the optimal locations for urban service centers are in Sub-District Besar (Medan Labuhan subdistrict), Subdistrict Rengas Pulau (Medan Marelan subdistrict), and Subdistrict Belawan I (Medan Belawan subdistrict). Conversely, Medan City RTRW 2022-2042 designates Labuhan CSC as Subdistrict Pekan Labuhan.

Beyond Zipf’s law: Exploring the discrete generalized beta distribution in open-source repositories

Rank-size distributions, such as Zipf’s Law, have been instrumental in providing insights into the emergence of hierarchies across diverse systems, from linguistic corpuses to urban structures. However, the application of Zipf’s Law reveals limitations, particularly in its focus on distribution tails, sometimes overlooking a large proportion of the data which might play a pivotal role in system dynamics. Yet, fitting rank-size distributions other than a straight line on the log–log scale requires caution. In this study, we re-evaluate the utility of rank-size distributions by contrasting the traditional Zipf’s Law with the Discrete Generalized Beta Distribution (DGBD). We show the need of cautious fitting techniques for rank distributions, including the use of binning to prevent overfitting to data tails. Through both analytical derivation and empirical validation on commit data of open-source repositories, we show that DGBD consistently improves over Zipf distribution for concave rank distributions of large datasets (N≥100). This approach contributes to the advancement of methodologies for analyzing hierarchical systems.

Linguistic Properties of Emojis: A Quantitative Exploration of Emoji Frequency, Category, and Position on Twitter

ABSTRACT Emojis in digital communication have drawn increasing academic attention. Qualitative studies mainly rely on a presumption that emojis share similar properties with units of natural language. It remains to be explored with quantitative methods whether emojis exhibit the same or similar behaviour from linguistic units (like words, morphemes). This study investigates emoji features in relation to language properties based on Zipf’s law and linear regression models. Results show that, firstly, the rank frequency distribution of emojis can be well fitted by Zipf’s law, and the parameters of emoji distribution are closer to those of written language. Secondly, most emoji categories tend to occur in the latter half of the tweet; while in some cases, they can also be at the beginning or in the middle of a tweet. Thirdly, the relative position of the more frequently-used emojis will be further back in the tweet. When emojis’ frequencies are relatively greater, their categories vary more in terms of their positions. In general, our quantitative findings suggest that emojis display linguistic properties to some extent. Our exploratory study demonstrates the value of applying linguistic laws and quantitative methods to investigate emoji features, extending the application of quantitative linguistic methods into emoji studies.

Rapid taxonomic categorization of short, abundant virus sequences for ecological analyses.

Public health concerns about recent viral epidemics have motivated researchers to seek novel ways to understand pathogen infection in native, wildlife hosts. With its deep history of tools and perspectives for understanding the abundance and distribution of organisms, ecology can shed new light on viral infection dynamics. However, datasets allowing deep explorations of viral communities from an ecological perspective are lacking. We sampled 1086 bats from two, adjacent Puerto Rican caves and tested them for infection by herpesviruses, resulting in 3131 short, viral sequences. Using percent identity of nucleotides and a machine learning algorithm (affinity propagation), we categorized herpesviruses into 43 operational taxonomic units (OTUs) to be used in place of species in subsequent ecological analyses. Herpesvirus metacommunities demonstrated long-tailed rank frequency distributions at all analyzed levels of host organization (i.e., individual, population, and community). Although 13 herpesvirus OTUs were detected in more than one host species, OTUs generally exhibited host specificity by infecting a single core host species at a significantly higher prevalence than in all satellite species combined. We describe the natural history of herpesvirus metacommunities in Puerto Rican bats and suggest that viruses follow the general law that communities comprise few common and many rare species. To guide future efforts in the field of viral ecology, hypotheses are presented regarding mechanisms that contribute to these patterns.

China’s urbanisation evolution and metropolitan area expansion, based on the Prolonged Artificial Nighttime-light Dataset (PANDA, 1984–2020)

ABSTRACT China’s massive urbanisation development will undoubtedly serve as a global reference. Due to the uncertainty of statistical data, nighttime light (NTL) data have emerged as alternative data for urbanisation evaluation. Here, the application of the Prolonged Artificial NTL Dataset of China (1984–2020) has been newly expanded. Total-value statistical, complex NTL index (CNLI), NTL concentration degree, rank-size rule, and Markov transfer matrix were used to systematically mine information about urban change from multiple perspectives. China’s urbanisation exhibited rapid growth and outwards expansion. The total percentage increase in NTL brightness and NTL area was 409.38% and 302.58%, respectively. The Yangtze River Delta urban agglomeration of East China had the fastest urbanisation (CNLI Trend = 0.0057/a, p < 0.01). Low NTL areas transitioned to medium NTL areas, and high and extremely-high NTL areas diffused to medium NTL areas. The cities in Eastern China and Southern China typically exhibited extremely-high type NTL areas, whereas other cities primarily exhibited medium NTL areas. The gaps between city sizes decreased over time (q Trend = – 0.0200/a, p < 0.01). Lowest-ranked cities exhibited the highest stability (>95%) in city size type transition. The spatiotemporal changes in NTL obtained were of great significance for monitoring urban expansion patterns, making government decisions, and quantifying China’s sustainable development.

Unraveling intra-urban freight parking patterns: A data-driven geospatial study of shared logistics sector in Hong Kong

Insufficient urban parking in densely populated cities has led to challenges like traffic congestion and unauthorized parking. While existing literature extensively covers parking behaviors among private vehicles and taxis for commuters, limited research has been proposed centering on urban shared freight activities. It can be problematic to neglect shared freight activities because these logistics activities often require more flexible parking space for goods loading/unloading near urban destinations. This study therefore introduces a geospatial data-driven approach to investigate urban parking behaviors from shared freight activities using GPS trajectories collected in Hong Kong. The analysis involves three main folds: using rank-size distribution and log-odds ratio to comparatively examine spatial heterogeneity of parking, evaluating illegal on-street parking events via illegal parking index, and quantifying relationships between parking behaviors and urban functions using random forest (RF) and SHapley Additive exPlanations (SHAP). The findings reveal significant concentrations of parking events on weekdays/weekends and with different parking durations. Particularly, high illegal on-street parking concentration is reported in Tsuen Wan and Kwai Chung, where the port container terminals and industrial activities are located. We further discover that industrial and dining densities both indicate significantly positive impacts on the relationship to the increases of overall parking frequency and illegal parking index. This study can be of great interest to current parking management and enforcement strategies in cities and relevant practical implications are discussed.

Envisaging the Intrinsic Departure from Zipf’s Law as an Indicator of Economic Concentration along Urban–Rural Gradients

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.

Deciphering the evolving trajectories of China's megaregions from 1992 to 2020: A novel morphological approach based on global land cover products

The monitoring of megaregion boundaries (MBs) is of vital importance for optimizing land use decision-making. However, the morphological structure of megaregions is currently insufficiently identified due to the nontransparent proprietary or inherent deficiency data sources. Thus, this study utilized land cover products integrated with morphological approaches to delineate the evolving trajectories of China's megaregions from 1992 to 2020. Firstly, core areas of megaregions and periphery affected areas were extracted from urban land density maps based on the gravity model and landscape metrics. Secondly, contiguous urban-rural fringes were improved based on the morphological operations and the rank-size distribution. Thirdly, urban land was automatically allocated to megaregions, considering planning policies. Furthermore, three indicators (gross domestic product, population, and land use density) are considered to validate our results, and our extracted MBs show a better understanding than the administrative divisions. The dynamic expansion patterns of twenty identified MBs were captured. As urbanization continues, the boundaries between megaregions become increasingly blurred and eventually merged into large-scale continuous megaregions that extend far beyond administrative divisions. Our study provides valuable insights for megaregion planning, including the effective urban management policies, the role of core cities in each megaregion in terms of radiation and capability.

Exploring the contemporary spatial and temporary dynamics of the settlement hierarchy and system in Serbia's Srem Region

Newer trends of balanced regional development emphasize the development of functional integrational areas and strong spatial and functional relations based on the nodal concept and a functional process approach. Therefore, it is essential to determine the hierarchy of settlements in the network, which defines their demographic and functional capacity. This paper identifies the nature and characteristics of urban primacy in the Srem region in Serbia, using the rank-size rule and urban primacy index, as well as the hierarchy of settlements in the network, by calculating the importance of secondary activities and the centralization of the settlements. The results confirmed that while urban primacy is not expressed, the hierarchy of the settlements remains dominated by urban and suburban centres.

A statistical approach to coronavirus classification based on nucleotide distributions

The objective of this study is to analyze specific genomes, namely the RNA of coronaviruses, based on the parameters obtained from the distributions of nucleotide sequences in their RNA. The viral RNA was subjected to distribution based on nucleotide sequences obtained by changing one nucleotide base (adenine) into a "whitespace", with empty sequences denoted as "x". Statistical spectra were constructed in such cases. They exhibited three distinct peaks that were consistent across the studied species. Parameters based on the rank–frequency distributions of the obtained nucleotide sequences, sequence lengths, and some other statistical parameters were calculated. Based on these parameters, the principal components were built, which were the basis for the grouping of the studied viruses. The most relevant parameters formed the model of a naїve Bayes classifier, which analyzes the probability of the virus belonging to a certain group of viruses in the model.

Analysis of Determining Service Center Systems towards the Development of The Eastern Part of Medan City

The issues in the research area include a high concentration of built-up residential areas with the potential for slums and warehousing activities dominating trade and service areas or residential areas, leading to congestion. This requires attention, considering that the Eastern Part of Medan City has a high built-up area, necessitating the provision of affordable infrastructure and basic services for both newcomers and existing residents in the city. Several development theories and concepts can assist in determining and conceptualizing development in the research area. The study focuses on examining the Sub-Central Service System of the City as a center serving sub-urban areas. Various methods are used in determining service centers, including the scalogram method, Marshall's centrality index, and rank-size rule, which will ultimately provide recommendations for the central service system. After issuing recommendations from these three analyses, the determination of the existing urban internal structure is conducted using a scoring method to identify the central service location. Calculations from multiple service center analysis methods are scored and summed to obtain a total score. The neighborhood with the smallest score has the highest hierarchy value, and vice versa. It is revealed that the Sub-Central Service System is present in all sub-districts in the Eastern Part of Medan City, meaning each sub-district in each neighborhood already has comprehensive facilities. Consequently, the four sub-districts in the Eastern Part of Medan City have the potential for development as they meet service needs and can be directed to become service centers serving their respective sub-districts.

Recovering Zipf's law in intercontinental scientific cooperation.

Scientific cooperation on an international level has been well studied in the literature. However, much less is known about this cooperation on an intercontinental level. In this paper, we address this issue by creating a collection of approximately 13.8×106 publications around the papers by one of the highly cited authors working in complex networks and their applications. The obtained rank-frequency distribution of the probability of sequences describing continents and the number of countries-with which authors of papers are affiliated-follows the power law with an exponent -1.9108(15). Such dependence is known in the literature as Zipf's law, and it has been originally observed in linguistics; later, it turned out that it is very commonly observed in various fields. The number of distinct "continent (number of countries)" sequences in a function of the number of analyzed papers grows according to power law with exponent 0.527(14); i.e., it follows Heap's law.

Spatial Pattern and Influencing Factors of Agricultural Leading Enterprises in Heilongjiang Province, China

As one of the major new agricultural business entities, agricultural leading enterprises (ALEs) are responsible for ensuring national food security, leading agricultural and rural modernization, and increasing farmers’ employment prospects and incomes. From the perspective of headquarters and branches, this study used a point pattern analysis, the local Moran’s index, the rank-size rule, and the geographical detector to depict the spatial pattern of ALEs in Heilongjiang Province, detect influencing factors, and reveal the spatial layout mechanism. The main conclusions are as follows. (1) ALE headquarters and branches in Heilongjiang Province had different location requirements, and their layout orientation, clustering areas, and influencing factors were different. (2) The headquarters displayed a pronounced urban and agglomeration orientation, while branches exhibited a significant farm dependence and raw material orientation. (3) Both the headquarters and the branches showed a significant trend towards spatial clustering. The headquarters were mainly in the Harbin municipal district and surrounding counties, including Wuchang, Zhaodong, and Beilin, which showed a high–high cluster pattern. The branches were mainly in the Sanjiang Plain. Tongjiang, Fujin, Hulin, Mishan, Raohe, Baoqing, and Suibin showed a high–high cluster pattern, while the Harbin municipal district and Bei’an showed a high–low outlier pattern. (4) The ALEs’ regional connection network in Heilongjiang Province was radially distributed with the Harbin municipal district as the centre. The agricultural reclamation system deeply influenced it. (5) The number of supporting enterprises, number of permanent residents, gross domestic product, railway mileage, number of people with Bachelor’s degrees or above, and distance from the provincial capital were the main influencing factors of the headquarters spatial pattern in Heilongjiang Province. The number of state farms in Heilongjiang Province, the per capita grain yield, highway mileage, and distance from the provincial capital were the main influencing factors of the branch spatial patterns in Heilongjiang Province. The interaction results indicated that the explanatory power of two-factor interaction was stronger than that of a single factor regardless of headquarters or branches, and most interaction types were bilinear enhancements. This study aims to provide a decision-making reference for the long-term development of ALEs in Heilongjiang Province at the present stage and accelerate the development of agricultural industrialization in major grain-producing areas.

Auerbach, Lotka, and Zipf: pioneers of power-law city-size distributions

Power-law city-size distributions are a statistical regularity researched in many countries and urban systems. In this history of science treatise we reconsider Felix Auerbach’s paper published in 1913. We reviewed his analysis and found (i) that a constant absolute concentration, as introduced by him, is equivalent to a power-law distribution with exponent approx 1, (ii) that Auerbach describes this equivalence, and (iii) that Auerbach also pioneered the empirical analysis of city-size distributions across countries, regions, and time periods. We further investigate his legacy as reflected in citations and find that important follow-up work, e.g. by Lotka (Elements of physical biology. Williams & Wilkins Company, Baltimore, 1925) and Zipf (Human behavior and the principle of least effort: an introduction to human ecology, Martino Publishing, Manfield Centre, CT (2012), 1949), does give proper reference to his discovery—but others do not. For example, only approximately 20% of city-related works citing Zipf (1949) also cite Auerbach (Petermanns Geogr Mitteilungen 59(74):74–76, 1913). To our best knowledge, Lotka (1925) was the first to describe the power-law rank-size rule as it is analyzed today. Saibante (Metron Rivista Internazionale di Statistica 7(2):53–99, 1928), building on Auerbach and Lotka, investigated the power-law rank-size rule across countries, regions, and time periods. Zipf’s achievement was to embed these findings in his monumental 1949 book. We suggest that the use of “Auerbach–Lotka–Zipf law” (or “ALZ-law”) is more appropriate than “Zipf’s law for cities”, which also avoids confusion with Zipf’s law for word frequency. We end the treatise with biographical notes on Auerbach.

Is housing price distribution across cities, scale invariant? Fractal distribution of settlements' house prices as signature of self-organized complexity

Locations and sizes of settlements within regions often exhibit a self-organized central place pattern, where larger settlements are encircled by smaller ones, with the former more set apart. This hierarchical spatial pattern is displayed by the fractal signature of the power law typical of the rank-size distribution of settlements. Is a similar pattern also revealed in the housing prices across settlements? An empirical analysis on the almost 8000 Italian municipalities, whose residential population varies between 33 and ∼3 million, shows that, for 15 out of 20 regions, the upper tails rank-house price distribution of settlements is plausibly following a power law. This result discloses a universality in the function behaviour: a scale invariance. The average scaling parameters have been estimated via Maximum Likelihood Estimator and Kolmogorov–Smirnov statistics, and the goodness-of-fit assessed via bootstrapping. Ordinary Least Squares regressions have also been used for comparative purposes. In average, Maximum Likelihood Estimator found a scaling parameter of 4.4. Namely, doubling the rank of the settlement (in terms of housing price), its housing price decreases by 15 %: between the settlement with the highest average house prices (ranked 1), and the settlement with the second highest average house prices (ranked 2), there is a 15 % difference in the average house prices. Idem between the ranked 5 and 10, 10 and 20 … it does not matter; it is a universal behavioural pattern whose scale invariance suggests a self-organized complexity in the phenomena behind it.

The Impact of Urbanization on the Flow of Traffic in Faisalabad City Pakistan

It is estimated that 50% of the world's population is living on only 2% of land area. Population scientists also believe that the urban population will increase from 80% to 90% by 2030 which is why urban growth is a significant problem in the developing world. Rapid urbanization has created many problems like haphazard development, disturbance in the ecological system, shortage of shelter and food polluted environment, lack of basic facilities, and traffic congestion. Besides traffic jams, traffic flow is also the major reason for different types of pollution like noise pollution and air pollution, which is a cause of health problems. Pakistan being a developing country experiences the same problems because of rapid urbanization. Faisalabad is the third largest city (according to the rank-size rule) and is famous for the agro cottage and textile industry. It covered the 313.21km2 area. Traffic congestion is a significant problem in Pakistan’s cities because of their unplanned infrastructure and explosive population growth. Faisalabad is the first planned city of the subcontinent. With time, Faisalabad’s infrastructure faced many changes due to uncontrolled urbanization. Road transport increased by 3% yearly due to the usage of personal automobiles. In the usage of personal vehicles, the share of motorcycles is 89 % while the share of cars is 50 %. Rapid urbanization is a severe threat to the city environment. Traffic congestion has become the problem of the day. This menace increased over time due to the slow development of roads and insufficient infrastructure as compared to rapid urbanization.