Zipf's Law Research Articles

The role of knowledge intensive business services in economic development: a bibliometric analysis from Bradford, Lotka and Zipf laws

Abstract The international scientific community considers Knowledge Intensive Business Services (KIBS) as one of the main themes related to innovation and economic development. This article presents a review based on Scopus and ISI Web of Knowledge databases, on the KIBS topic in the world, considering the period 2000-2017. The study aimed to understand the role of KIBS in regional development, and they were considered in their roles as innovations’ attributes and resources, methodologies and tools for innovation management, transfer, knowledge diffusion and networks. They concentrate in the areas of business, management and economics, developing approaches to seek innovation, competitiveness and economic development.

Entropy and Temperature of Electromagnetic Radiation

Electromagnetic (EM) radiation is both wave and heat. Waves are characterized by spectral distribution, spatial distribution, time coherence, spatial coherence, energy flux, and polarization. Heat, namely energy transferred from a hot body to a cold one, is characterized by its energy and entropy, and the ratio between them is the temperature. Here we calculate the entropy and temperature of a single radiation mode from the wave properties of the radiation. Using the Heisenberg uncertainty principle and Planck law, we calculate, from the optical properties of the radiation, the number of modes and their occupation number. Then we calculate the entropy and temperature of a single-mode EM radiation. It is shown that the entropy of a single-mode varies from zero for low occupation number, namely in the quantum limit to one Boltzmann constant for high occupation number, namely in the classical limit. The temperature varies from zero Kelvin in the quantum limit to infinity at high energies in the classical limit. This analysis is consistent with Fourier optics and statistical mechanics namely: Stephan-Boltzmann Law, Wein Law, Zipf law, IT File’s entropy, and the canonical distribution.

Higher Order Character Frequency Distribution in Modern Chinese Texts: Application of Zipf's Law

Higher Order Character Frequency Distribution in Modern Chinese Texts: Application of Zipf's Law

Heaps' law, statistics of shared components, and temporal patterns from a sample-space-reducing process

Zipf's law is a hallmark of several complex systems with a modular structure, such as books composed by words or genomes composed by genes. In these component systems, Zipf's law describes the empirical power law distribution of component frequencies. Stochastic processes based on a sample-space-reducing (SSR) mechanism, in which the number of accessible states reduces as the system evolves, have been recently proposed as a simple explanation for the ubiquitous emergence of this law. However, many complex component systems are characterized by other statistical patterns beyond Zipf's law, such as a sublinear growth of the component vocabulary with the system size, known as Heap's law, and a specific statistics of shared components. This work shows, with analytical calculations and simulations, that these statistical properties can emerge jointly from a SSR mechanism, thus making it an appropriate parameter-poor representation for component systems. Several alternative (and equally simple) models, for example based on the preferential attachment mechanism, can also reproduce Heaps' and Zipf's laws, suggesting that additional statistical properties should be taken into account to select the most-likely generative process for a specific system. Along this line, we will show that the temporal component distribution predicted by the SSR model is markedly different from the one emerging from the popular rich-gets-richer mechanism. A comparison with empirical data from natural language indicates that the SSR process can be chosen as a better candidate model for text generation based on this statistical property. Finally, a limitation of the SSR model in reproducing the empirical "burstiness" of word appearances in texts will be pointed out, thus indicating a possible direction for extensions of the basic SSR process.

CIDADES INVENTIVAS NO BRASIL: HIERARQUIA E DETERMINANTES DA INVENÇÃO

O artigo analisa a relação entre a escala urbana e as invenções, medidas por patentes, em uma análise comparada de dois quadriênios (2001-2004 e 2006-2009) no Brasil. Com dados do Instituto Nacional de Propriedade Industrial (Inpi), da Relação Anual de Informações Sociais (Rais), do Instituto Brasileiro de Geografia e Estatística (IBGE) e da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes), o artigo explora a distribuição hierárquica e a aglomeração espacial das invenções a partir de regras de potência (lei de Zipf), além de investigar os principais determinantes do patenteamento municipal com modelos econométricos espaciais. Os resultados apontam que a escala urbana possui efeito significativo no patenteamento, assim como capacidade de P&D universitário, escolaridade dos trabalhadores e presença de setores de média-alta tecnologia.

Comparación multinivel de grandes sistemas urbanos

For the first time the systems of cities in seven countries or regions among the largest in the world (China, India, Brazil, Europe, the Former Soviet Union (FSU), the United States and South Africa) are made comparable through the building of spatio-temporal standardised statistical databases. We first explain the concept of a generic evolutionary urban unit (“city”) and its necessary adaptations to the information provided by each national statistical system. Second, the hierarchical structure and the urban growth process are compared at macro-scale for the seven countries with reference to Zipf’s and Gibrat’s model: in agreement with an evolutionary theory of urban systems, large similarities shape the hierarchical structure and growth processes in BRICS countries as well as in Europe and United States, despite their positions at different stages in the urban transition that explain some structural peculiarities. Third, the individual trajectories of some 10,000 cities are mapped at micro-scale following a cluster analysis of their evolution over the last fifty years. A few common principles extracted from the evolutionary theory of urban systems can explain the diversity of these trajectories, including a specific pattern in their geographical repartition in the Chinese case. We conclude that the observations at macro-level when summarized as stylised facts can help in designing simulation models of urban systems whereas the urban trajectories identified at micro-level are consistent enough for constituting the basis of plausible future population projections. Keywords: urban system, Zipf's law, Gibrat, cities trajectories, BRICS

Service Popularity-Based Smart Resources Partitioning for Fog Computing-Enabled Industrial Internet of Things

Recently, fog computing has gained increasing attention in processing the computing tasks of the industrial Internet of things (IIoT) with different service popularity. In task-diversified fog computing-enabled IIoT (F-IIoT), the mismatch between expected computing efficiency and partitioned resources on fog nodes (FNs) may pose serious traffic congestion even large-scale industrial service interruptions. The existing works mainly studied offloading which type of computing tasks into FNs, but few studies enabled smart resource partitioning of FNs. In this paper, a service popularity-based smart resources partitioning (SPSRP) scheme is proposed for fog computing-enabled IIoT. We first exploit Zipf's law to model the relationship between popularity ranks and computing costs of IIoT services. Moreover, we propose an implementation architecture of the SPSRP scheme for F-IIoT, which decouples the computing control layer from data processing layer of IIoT through a specified SPSRP controller. Besides, a mobility and heterogeneity-aware partitioning algorithm is presented for extending SPSRP scheme to seamlessly support cross-domain resources partitioning. The simulations demonstrate that the SPSRP scheme can bring notable performance improvements on delay time, successful response rate and fault tolerance for fog computing to deal with the large-scale IIoT services.

The size ranking of cities in Germany: caught by a MAUP?

A long standing question in urbanisation studies queries whether an empirical regularity known as Zipf's law applies to the size ranking of cities within countries or regions. Part of the debate addresses the statistical attributes of the size distribution. More recently, the definition of the territories assigned to cities has become a matter of concern in this context. Since the shape of administrative boundaries may not represent economic entities, analysis of the size ranking among cities defined by municipal territories may be affected by a considerable modifiable areal unit problem. Following a methodical approach that relates to current research on natural cities, we study the extent to which variation in the territory assigned to urban areas in Germany affects basic features of the size ranking. We define urban areas according to variable thresholds of population density across 1 km2 grids by a clustering algorithm. We find a systematic but moderate deviation from Zipf's law suggesting scale economies among urban agglomerations if peripheral zones are included.

The Zipf’s Law and Other Ways of Identifying Culture-Specific Linguistics Units

Every language contains a number of linguistic units that serve as a key to understanding the culture of its language speakers. Therefore, they bear specific information that helps to reveal cultural similarities and dissimilarities. Taking into account contemporary studies and focus of the science, this information can be very critical and relevant in intercultural communication. In this paper, we name such linguistic units as cultural keywords. So, we claim that cultural keywords can serve as a tool for intercultural communication. They are indicators for describing and understanding cultures. This can help to study, learn, discuss and describe the cultures of different nations all over the world. In the paper we suggest methodologies to identify cultural keywords. We claim that a methodological basis for the study may consist of three methods: semantic analysis, argumentative analysis and the Zipf's law. However, these are not comprehensive methods for discovering the keywords. Findings of the research can be applied to intercultural and cross-cultural studies as well as other interdisciplinary cultural studies.

Scientific production on workplace bullying and nursing: a bibliometric study.

To verify bibliometric indicators of the scientific production available in online journals that approach workplace bullying and nursing. A bibliometric study making use of Bradford's law, Zipf's law, and textual statistics was carried out with publications in Portuguese, English, and Spanish, made available in national and international databases, from 2000 to 2016. The sample was made up of 111 publications. The main authors had connections with 91 institutions distributed in 24 countries. The United States, Brazil, and Australia were the countries with the most publications. The populations of the studies were made up of nursing professionals and students, and the hospital environment was the most studied setting. The journals with a higher number of publications have international scientific influence. The terms that presented greater semantic power and high frequency in the abstracts were: bullying; assédio moral; and acoso laboral. Indicators showed that workplace bullying occurs in the nursing work environments of several countries, and the number of publications on this theme has tended to increase. Diversifying methods and study settings is important to contribute to the advancement of knowledge and fight against this violence.

Zipf's and Taylor's laws

Zipf's law states that the frequency of an observation with a given value is inversely proportional to the square of that value; Taylor's law, instead, describes the scaling between fluctuations in the size of a population and its mean. Empirical evidence of the validity of these laws has been found in many and diverse domains. Despite the numerous models proposed to explain the presence of Zipf's law, there is no consensus on how it originates from a microscopic process of individuals dynamics without fine tuning. Here we show that Zipf's law and Taylor's law can emerge from a general class of stochastic processes at the individual level, which incorporate one of two features: environmental variability, i.e. fluctuations of parameters, or correlations, i.e. dependence between individuals. Under these assumptions, we show numerically and with theoretical arguments that the conditional variance of the population increments scales as the square of the population and that the corresponding stationary distribution of the processes follows Zipf's law.

Universal rank-size statistics in network traffic: Modeling collective access patterns by Zipf's law with long-term correlations

We analyze network traffic rank-size statistics at different levels and organization. Our results support the emergence of Zipf's law in the rank-size traffic distributions by time, source and destination. The corresponding empirical laws considering typical discreteness and finite-size effects can be well approximated by q-exponential distributions for external IPs as well as by β- and Γ-distributions for internal LAN IPs and time fragments, respectively. Once appropriately normalized, the observed rank-size statistics exhibit rather universal shapes that are well reproduced by nonextensive entropy maximization algorithm for finite system sizes and can be used to model typical network activity patterns for a given community with a given number of active nodes as a sole free parameter.

Manifestations of the onset of chaos in condensed matter and complex systems

We review the occurrence of the patterns of the onset of chaos in low-dimensional nonlinear dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics associated with the attractors at period-doubling accumulation points and at tangent bifurcations to describe features of glassy dynamics, critical fluctuations and localization transitions. We recall that trajectories pertaining to the routes to chaos form families of time series that are readily transformed into networks via the Horizontal Visibility algorithm, and this in turn facilitates establish connections between entropy and Renormalization Group properties. We discretize the replicator equation of game theory to observe the onset of chaos in familiar social dilemmas, and also to mimic the evolution of high-dimensional ecological models. We describe an analytical framework of nonlinear mappings that reproduce rank distributions of large classes of data (including Zipf's law). We extend the discussion to point out a common circumstance of drastic contraction of configuration space driven by the attractors of these mappings. We mention the relation of generalized entropy expressions with the dynamics along and at the period doubling, intermittency and quasi-periodic routes to chaos. Finally, we refer to additional natural phenomena in complex systems where these conditions may manifest.

Majorizatiuon and Zipf-Mandelbrot law

In this paper we show how the Zipf-Mandelbrot law is connected to the theory of majorization. Firstly we consider the Csiszár $f$-divergence for the Zipf-Mandelbrot law and then develop important majorization inequalities for these divergences. We also discuss some special cases for our generalized results by using the Zipf-Mandelbrot law. As applications, we present the majorization inequalities for various distances obtaining by some special convex functions in the Csiszár $f$-divergence for Z-M law like the Rényi $\alpha$-order entropy for Z-M law, variational distance for Z-M law, the Hellinger distance for Z-M law, $\chi^{2}$-distance for Z-M law and triangular discrimination for Z-M law. At the end, we give important applications of the Zipf's law in linguistics and obtain the bounds for the Kullback-Leibler divergence of the distributions associated to the English and the Russian languages.

Sequences of purchases in credit card data reveal lifestyles in urban populations

Zipf-like distributions characterize a wide set of phenomena in physics, biology, economics, and social sciences. In human activities, Zipf's law describes, for example, the frequency of appearance of words in a text or the purchase types in shopping patterns. In the latter, the uneven distribution of transaction types is bound with the temporal sequences of purchases of individual choices. In this work, we define a framework using a text compression technique on the sequences of credit card purchases to detect ubiquitous patterns of collective behavior. Clustering the consumers by their similarity in purchase sequences, we detect five consumer groups. Remarkably, post checking, individuals in each group are also similar in their age, total expenditure, gender, and the diversity of their social and mobility networks extracted from their mobile phone records. By properly deconstructing transaction data with Zipf-like distributions, this method uncovers sets of significant sequences that reveal insights on collective human behavior.

Applying network theory to fables: complexity in Slovene belles-lettres for different age groups

Words are the building blocks of human communication. They are arranged in sentences in a non-trivial and universal way, which implies the existence of fundamental organizational principles that have shaped language development. One of the fundamental examples is the Zipf's law, which says that the frequency of word occurrence is roughly an inverse power-law function of its rank. In our article, we study the structure and complexity of texts in Slovene belles-lettres, with an emphasis on evaluating the differences in the texts for different age groups. We show that the co-occurrence connectivity of words forms a complex and heterogeneous network that is characterized by an efficient transfer of information. Moreover, we show that with the increasing age of readers, the length of texts, the average length of words, different combinations of phrases and the complexity of social interactions between literary characters, also all increase. Conversely, the fraction of unique words decreases. We also show that the word co-occurrence networks for older age groups are less modular and exhibit a higher level of small-worldness, and that the Zipf's exponent as a measure for the linguistic complexity is more negative, than for younger age groups. Taken together, we demonstrate that network theory enables an in-depth theoretical exploration of Slovene belles-lettres, with clear distinctions in statistical properties between different age groups, thus bridging art and exact sciences in a mutually rewarding way.

Iso-height lines of multifractal etched silicon rough surfaces

Morphology of etched silicon wafers during anisotropic etching process is investigated utilizing the geometrical exponents of iso-height contour lines. The images obtained from field emission scanning electron microscopy show pyramidal microstructures which disappear in large time scales and porous silicon structures are formed. The morphology of these microstructures is investigated utilizing the data obtained from atomic force microscopy and contour loop analysis. The results show that the anisotropic etching breaks the up-down symmetry in the height fluctuations of etched silicon surfaces. Area of the contour loops increases by enhancement of pyramids sizes. Fractal dimensions of the loops decrease at the beginning of etching process. Distributions of loops and geometrical exponents of iso-height lines in different etching times are determined. The results indicate that in spite of asymmetry in the height fluctuations, the hyperscaling relation and Zipf's law are verified for the multi-affine silicon surfaces during etching process.

The origins of Zipf's meaning‐frequency law

In his pioneering research, G.K. Zipf observed that more frequent words tend to have more meanings, and showed that the number of meanings of a word grows as the square root of its frequency. He derived this relationship from two assumptions: that words follow Zipf's law for word frequencies (a power law dependency between frequency and rank) and Zipf's law of meaning distribution (a power law dependency between number of meanings and rank). Here we show that a single assumption on the joint probability of a word and a meaning suffices to infer Zipf's meaning‐frequency law or relaxed versions. Interestingly, this assumption can be justified as the outcome of a biased random walk in the process of mental exploration.

Zipf's Law and Heaps' Law in Social Tagging

Zipf's Law and Heaps' Law in Social Tagging

The Zipf's law: A cross-country analysis

The Zipf's law: A cross-country analysis