Zipf's Law Research Articles

The Zipf's law: A cross-country analysis

The Zipf's law: A cross-country analysis

Zipf and Heaps laws from dependency structures in component systems.

Complex natural and technological systems can be considered, on a coarse-grained level, as assemblies of elementary components: for example, genomes as sets of genes or texts as sets of words. On one hand, the joint occurrence of components emerges from architectural and specific constraints in such systems. On the other hand, general regularities may unify different systems, such as the broadly studied Zipf and Heaps laws, respectively concerning the distribution of component frequencies and their number as a function of system size. Dependency structures (i.e., directed networks encoding the dependency relations between the components in a system) were proposed recently as a possible organizing principles underlying some of the regularities observed. However, the consequences of this assumption were explored only in binary component systems, where solely the presence or absence of components is considered, and multiple copies of the same component are not allowed. Here we consider a simple model that generates, from a given ensemble of dependency structures, a statistical ensemble of sets of components, allowing for components to appear with any multiplicity. Our model is a minimal extension that is memoryless and therefore accessible to analytical calculations. A mean-field analytical approach (analogous to the "Zipfian ensemble" in the linguistics literature) captures the relevant laws describing the component statistics as we show by comparison with numerical computations. In particular, we recover a power-law Zipf rank plot, with a set of core components, and a Heaps law displaying three consecutive regimes (linear, sublinear, and saturating) that we characterize quantitatively.

Regression model of rank –frequency data of Tamil text

The application of Zipf’s law is universal not only in linguistics but also in various other areas. Mandelbrot modified Zipf law as Zipf Mandelbrot law and it is further we proposed a modification of the ZM law for modeling rank frequency- data of linguistic text. Our model generalized ZM law into a linear regression model involving arbitrary order of Zipfian rank of words in a text .The performance of the proposed model is studied for an English text and it shown to compare favorably with that of Z-M law using Chi-Square goodness of fit test. In this paper we have applied to Tamil text and its performance is also up to the mark and it is been proved by the Chi-Square test and it addresses mainly the lower ranks, we propose to extend the work to higher order ranks using LNRE model in the future.

Asymptotic Analysis of Inhomogeneous Information-Centric Wireless Networks With Infrastructure Support

The information-centric networking (ICN) is a promising and significant approach towards the host-to-content interaction. Motivated by the fact that nodes are inhomogeneously distributed to a certain extent in wireless networks, e.g., the distribution of vehicles exhibits cluster phenomenon in some bustling places of urban areas, we study the impacts of inhomogeneous node distribution and local node cooperation on information-centric wireless networks with infrastructure support. The distribution of nodes is assumed to be inhomogeneous as a multiclustering topology. The base stations are assumed to have access to all the contents, which are requested following a popularity distribution. We propose a scheme where each node retrieves the requested content from other nodes or base stations. Based on the number of clusters and that of base stations, we divide the network topology into two cases, i.e., the cluster-dense case and cluster-sparse case, and evaluate the corresponding throughput and delay performance. Finally, under the Zipf's law for content popularity distribution, we investigate the optimal caching problem in the two cases, and obtain the optimized throughput and delay performance.

The pore length, the pore number and the pore anisotropy distributions in porous materials

The differential pore lengths Li, the differential pore numbers Ni and the differential pore anisotropies Bi in porous materials are estimated in a unified way and presented as a function of pore radius ri. Those parameters can be determined from the differential specific surface area Si and the differential specific pore volume Vi calculated via nitrogen porosimetry, assuming cylindrical pores. The differential pore length is estimated from relation Li = Si2/Vi = Ni·li and corresponds to the total length of Ni pores with similar local length li at each pore group of radius ri estimated at partial pressure (Pi/Po). This parameter bears similarities with the differential pore anisotropy Bi given by Bi = Si3/Vi2 = Ni·bi where bi is the local pore anisotropy. Parameter Bi is suitable for the ranking of pore numbers vs. pore volumes and, for isotropic cavities with bi = 1, leads to the ranking of pores according to the Zipf's law. Parameter Li is suitable for the ranking of pore lengths as a function of pore radii and reveals some morphometric similarities between the pore networks in solids and the branching of trees described by the so-called allometric relations. In addition these relations may be used for the distinction of pore number evolution via either power law or exponential mechanisms, expressed by corresponding distributions. Such effects, observed previously by volcanologists in the vesicles of volvanic magmas in mm scale, are also observed in the present study for random pores of nm scale in lab made materials.

Sensitivity study of experimental measures for the nuclear liquid-gas phase transition in the statistical multifragmentation model

The experimental measures of the multiplicity derivatives, the moment parameters, the bimodal parameter, the fluctuation of maximum fragment charge number (NVZ), the Fisher exponent ($\tau$) and Zipf's law parameter ($\xi$), are examined to search for the liquid-gas phase transition in nuclear multifragmention processes within the framework of the statistical multifragmentation model (SMM). The sensitivities of these measures are studied. All these measures predict a critical signature at or near to the critical point both for the primary and secondary fragments. Among these measures, the total multiplicity derivative and the NVZ provide accurate measures for the critical point from the final cold fragments as well as the primary fragments. The present study will provide a guide for future experiments and analyses in the study of nuclear liquid-gas phase transition.

РЕГИОНЫ С ВЫСОКОЙ КОНЦЕНТРАЦИЕЙ МОНОГОРОДОВ: ПРОБЛЕМЫ ПОВЫШЕНИЯ КАЧЕСТВА ДАННЫХ

With all the diversity of studies of single-industry towns’ development problems, little attention has been given to the quality of statistical data: neither the selective character of official statistics nor the difference between a single-industry town and a municipality are taken into account. The latter makes it impossible to use mathematical methods to simulate the spatial development of towns. The purpose of the current research is to identify the problems of assessment for regions with high concentration of monotowns and to introduce some ways of improving the quality of data by using the case of the Kemerovo region. Research methods include collection and grouping of data on the single-industry towns of the Kemerovo region taking into account the administrative-territorial division of the region, the construction of the logarithmic model of distribution of cities and towns according to the «rank-size» rule and evaluation compliance of the received distribution with the Zipf rule. As a result of research, the author proposes directions of improvement of data on towns and obtained results. In particular, the study specifies the conformity of Kuzbass towns to the Zipf law. The results can be applied in the field of forecasting and management of single-industry towns’ development at the regional level.

Statistics of Shared Components in Complex Component Systems

Many complex systems are modular. Such systems can be represented as "component systems", i.e., sets of elementary components, such as LEGO bricks in LEGO sets. The bricks found in a LEGO set reflect a target architecture, which can be built following a set-specific list of instructions. In other component systems, instead, the underlying functional design and constraints are not obvious a priori, and their detection is often a challenge of both scientific and practical importance, requiring a clear understanding of component statistics. Importantly, some quantitative invariants appear to be common to many component systems, most notably a common broad distribution of component abundances, which often resembles the well-known Zipf's law. Such "laws" affect in a general and non-trivial way the component statistics, potentially hindering the identification of system-specific functional constraints or generative processes. Here, we specifically focus on the statistics of shared components, i.e., the distribution of the number of components shared by different system-realizations, such as the common bricks found in different LEGO sets. To account for the effects of component heterogeneity, we consider a simple null model, which builds system-realizations by random draws from a universe of possible components. Under general assumptions on abundance heterogeneity, we provide analytical estimates of component occurrence, which quantify exhaustively the statistics of shared components. Surprisingly, this simple null model can positively explain important features of empirical component-occurrence distributions obtained from data on bacterial genomes, LEGO sets, and book chapters. Specific architectural features and functional constraints can be detected from occurrence patterns as deviations from these null predictions, as we show for the illustrative case of the "core" genome in bacteria.

Using the parameters of the Zipf–Mandelbrot law to measure diachronic lexical, syntactical and stylistic changes – a large-scale corpus analysis

AbstractUsing the Google Ngram Corpora for six different languages (including two varieties of English), a large-scale time series analysis is conducted. It is demonstrated that diachronic changes of the parameters of the Zipf–Mandelbrot law (and the parameter of the Zipf law, all estimated by maximum likelihood) can be used to quantify and visualize important aspects of linguistic change (as represented in the Google Ngram Corpora). The analysis also reveals that there are important cross-linguistic differences. It is argued that the Zipf–Mandelbrot parameters can be used as a first indicator of diachronic linguistic change, but more thorough analyses should make use of the full spectrum of different lexical, syntactical and stylometric measures to fully understand the factors that actually drive those changes.

Pareto–Zipf law in growing systems with multiplicative interactions

Numerical simulations of multiplicatively interacting stochastic processes with weighted selections were conducted. A feedback mechanism to control the weight w of selections was proposed. It becomes evident that when w is moderately controlled around 0, such systems spontaneously exhibit the Pareto–Zipf distribution. The simulation results are universal in the sense that microscopic details, such as parameter values and the type of control and weight, are irrelevant. The central ingredient of the Pareto–Zipf law is argued to be the mild control of interactions.

Greedy algorithms and Zipf laws

We consider a simple model of firm/city/etc growth based on a multi-item criterion: whenever entity B fares better than entity A on a subset of M items out of K, the agent originally in A moves to B. We solve the model analytically in the cases K = 1 and . The resulting stationary distribution of sizes is generically a Zipf-law provided M > K/2. When , no selection occurs and the size distribution remains thin-tailed. In the special case M = K, one needs to regularize the problem by introducing a small ‘default’ probability ϕ. We find that the stationary distribution has a power-law tail that becomes a Zipf-law when . The approach to the stationary state can also be characterized, with strong similarities with a simple ‘aging’ model considered by Barrat and Mézard.

Optimal growth entails risky localization in population dynamics

Essential to each other, growth and exploration are jointly observed in alive and inanimate entities, such as animals, cells or goods. But how the environment's structural and temporal properties weights in this balance remains elusive. We analyze a model of stochastic growth with time correlations and diffusive dynamics that sheds light on the way populations grow and spread over general networks. This model suggests natural explanations of empirical facts in econo-physics or ecology, such as the risk-return trade-off and the Zipf law. We conclude that optimal growth leads to a localized population distribution, but such risky position can be mitigated through the space geometry. These results have broad applicability and are subsequently illustrated over an empirical study of financial data.

Zipf\u2013Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities

Motivated by the method of interpolating inequalities that makes use of the improved Jensen-type inequalities, in this paper we integrate this approach with the well known Zipf–Mandelbrot law applied to various types of f-divergences and distances, such are Kullback–Leibler divergence, Hellinger distance, Bhattacharyya distance (via coefficient), chi^{2}-divergence, total variation distance and triangular discrimination. Addressing these applications, we firstly deduce general results of the type for the Csiszár divergence functional from which the listed divergences originate. When presenting the analyzed inequalities for the Zipf–Mandelbrot law, we accentuate its special form, the Zipf law with its specific role in linguistics. We introduce this aspect through the Zipfian word distribution associated to the English and Russian languages, using the obtained bounds for the Kullback–Leibler divergence.

Word frequencies: A comparison of Pareto type distributions

Mehri and Jamaati (2017) [18] used Zipf's law to model word frequencies in Holy Bible translations for one hundred live languages. We compare the fit of Zipf's law to a number of Pareto type distributions. The latter distributions are shown to provide the best fit, as judged by a number of comparative plots and error measures. The fit of Zipf's law appears generally poor.

The Evolution of Zipf's Law for U.S. Cities

Exploiting the cascade structure of cities and based on a dataset for U.S. cities between 1840 and 2016, the aim of this short paper is to answer three important questions: First, do we observe that the U.S. city size distribution exhibits a smooth transition to Zipf's law from the beginning or are there periods showing a pronounced departure from Zipf's law? Second, if we observe periods of departure, which alternative laws instead should be used to accurately describe the city size distribution? Third, employing information from the cascade structure of cities, do we always find evidence for primate cities for a specific period of time? Inter alia, we find that the exact Zipf's law has evolved over time from the more general, so-called three-parameter Zipf's law which can be traced back to Mandelbrot (1982).

A scale-free transportation network explains the city-size distribution

Zipf's law is one of the best known empirical regularities in urban economics. There is extensive research on the subject, where each city is treated symmetrically in terms of the cost of transactions with other cities. Recent developments in network theory facilitate the examination of an asymmetric transport network. In a scale‐free network, the chance of observing extremes in network connections becomes higher than the Gaussian distribution predicts and, therefore, it explains the emergence of large clusters. The city‐size distribution shares the same pattern. This paper decodes how accessibility of a city to other cities on the transportation network can boost its local economy and explains the city‐size distribution as a result of its underlying transportation network structure. We confirm our model predictions with US and Belgian data. Finally, we discuss the endogenous evolution of transport networks. Zipf's law city‐size distribution scale‐free network L14 R12 R40

Why Zipf's law: a symmetry-based explanation

Why Zipf's law: a symmetry-based explanation

Inequalities of the Jensen and Edmundson–Lah–Ribarič Type for Positive Linear Functionals with Applications

In this paper we derive some Jensen and Edmundson-Lah-Ribaric type inequalities for positive linear functionals without the assumption about the convexity of the functions that are involved. General results are then applied to generalized f-divergence functional. Examples with Zipf's law and Zipf-Mandelbrot law are given.

Application Of The Criterion Of Links Distribution Between Elements In A Complex System For Evaluation Of Holistic Text Perfection

It is known that texts can be examined as complex social systems and Zipf's law is used when applied for the evaluation of their perfection. It is proposed to add distribution of links between the objects of the system to the list of perfection criteria of texts viewed as complex systems. An empirical law of syntactic and linking words distribution in texts is also proposed. The examples of complex systems are examined in order to define a mathematical representation of this law. The models of such systems are associated with the graph theory, exchange of messages at internet forums, distribution of urban population across city districts, and transportation networks. At the “engineering level of rigor” it is shown that in these systems, the empirical law of links distribution between the objects has a general mathematical representation. Non-linguistic models are provided to prove the system-wide character of the distribution of links ranks feature, but the models are also suitable for the self-consistent use in the corresponding applications. The golden proportion in the number of syntactic and significant words is proposed as an additional criterion of text perfection. Perfection criteria are useful in the analysis of the quality of texts posted in the internet and in the development of recommendations for the improvement of texts used for school purposes.

ФАКТОРЫ ПРОСТРАНСТВЕННОЙ ДИФФЕРЕНЦИАЦИИ ГОРОДСКОГО РАССЕЛЕНИЯ ПРОМЫШЛЕННЫХ РЕГИОНОВ УРАЛА И ЗАУРАЛЬЯ

Introduction. The article deals with current issues of the territorial organization of the social and economic space in the industrial Urals and Trans-Urals regions. The issues have been brought up by growing unevenness in economic development and the living standards in Russian regions, caused by the active process of spatial differentiation of settlement, ultimately leading to the destruction of the established settlement structure under which all social service networks were formed as well as to deterioration of the age and educational structure of the population, and a decrease in the level and quality of life. The aim is to determine trends in the spatial differentiation of urban settlements in the industrialized regions of the Urals and the Trans-Urals and to test the hypothesis that the settlement landscape in Russia resulting from implementation of the regional development policy inevitably leads to segregation of the economic and social space of Russia's regions. Five regions of the Urals and the Trans-Urals, administratively related to the Urals and the Volga federal districts were chosen as the object of the study, namely the Sverdlovsk, Chelyabinsk, Kurgan regions, the Perm Region and the Republic of Bashkortostan. Methods. The analysis of the spatial differentiation of urban resettlement was based on Zipf's law. Results. The analysis revealed the growing disproportion trend in urban resettlement at regional level, especially in industrial areas. It has been identified that the most significant deviations from stable urban settlement system are typical of the largest city in the region, i.e. the regional or republican capital city. Conclusion. The dynamics indicators analysis based on both statistical and spatial methods has revealed negative trends in the current spatial differentiation of urban resettlement in the industrial regions of the Urals and Trans-Urals.