Mandelbrot Law Research Articles

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.

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.

Using Zipf–Mandelbrot law and graph theory to evaluate animal welfare

This work deals with the construction and testing of metrics of welfare based on behavioral complexity, using assumptions derived from Zipf–Mandelbrot law and graph theory. To test these metrics we compared yellow-breasted capuchins (Sapajus xanthosternos) (Wied-Neuwied, 1826) (PRIMATES CEBIDAE) found in two institutions, subjected to different captive conditions: a Zoobotanical Garden (hereafter, ZOO; n=14), in good welfare condition, and a Wildlife Rescue Center (hereafter, WRC; n=8), in poor welfare condition. In the Zipf–Mandelbrot-based analysis, the power law exponent was calculated using behavior frequency values versus behavior rank value. These values allow us to evaluate variations in individual behavioral complexity. For each individual we also constructed a graph using the sequence of behavioral units displayed in each recording (average recording time per individual: 4 h 26 min in the ZOO, 4 h 30 min in the WRC). Then, we calculated the values of the main graph attributes, which allowed us to analyze the complexity of the connectivity of the behaviors displayed in the individuals’ behavioral sequences. We found significant differences between the two groups for the slope values in the Zipf–Mandelbrot analysis. The slope values for the ZOO individuals approached −1, with graphs representing a power law, while the values for the WRC individuals diverged from −1, differing from a power law pattern. Likewise, we found significant differences for the graph attributes average degree, weighted average degree, and clustering coefficient when comparing the ZOO and WRC individual graphs. However, no significant difference was found for the attributes modularity and average path length. Both analyses were effective in detecting differences between the patterns of behavioral complexity in the two groups. The slope values for the ZOO individuals indicated a higher behavioral complexity when compared to the WRC individuals. Similarly, graph construction and the calculation of its attributes values allowed us to show that the complexity of the connectivity among the behaviors was higher in the ZOO than in the WRC individual graphs. These results show that the two measuring approaches introduced and tested in this paper were capable of capturing the differences in welfare levels between the two conditions, as shown by differences in behavioral complexity.

The Diary of Boima Kiakpomgbo from Mando Town (Liberia): A Quantitative Study of a Vai Text

We analyse a Liberian text from the first quarter of the twentieth century written in Vai, a Mande language of West Africa, using an indigenous syllabic script. Rank–frequency distributions of syllabic symbols and words are studied. The distribution of symbols is well-fitted by the 1-displaced negative hypergeometric distribution, confirming previous studies on alphabetic scripts, with a similar interpretation of parameters. Two regimes are discovered for the saturation of symbols in the text (growth of the number of types approaching the complete set versus the number of tokens). For the distribution of words, the Zipf–Mandelbrot law is found to be a proper model rather than the simple Zipfian dependence.

Earthquakes economic costs through rank-size laws

This paper is devoted to assessing the presence of some regularities in the magnitudes of the earthquakes in Italy between January , 2016 and January , 2017, and to propose an earthquakes cost indicator. The considered data includes the catastrophic events in Amatrice and in the Marche region. To our purpose, we implement two typologies of rank-size analysis: the classical Zipf–Mandelbrot law and the so-called universal law proposed by Ausloos and Cerqueti (2016 PLoS One 11 e0166011). The proposed generic measure of the economic impact of earthquakes moves from the assumption of the existence of a cause-effect relation between earthquakes magnitudes and economic costs. At this aim, we hypothesize that such a relation can be formalized in a functional way to show how infrastructure resistance affects the cost. Results allow us to clarify the impact of an earthquake on the social context and might serve to strengthen the struggle against the dramatic outcomes of such natural phenomena.

Estimations of f- and Rényi Divergences by Using a Cyclic Refinement of the Jensen’s Inequality

The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this paper, we introduce a new functional, based on the f-divergence functional, and then, we obtain some estimates for the new functional, the f-divergence and the Renyi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Renyi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.

Ley de Zipf/Mandelbrot y teoría de la probabilidad aplicadas a la caracterización de reacciones adversas a medicamentos en adultos mayores

Abstract Introduction. Zipf/Mandelbrot Law establishes a fractal auto-organization of phenomena such as language, the B and T immune repertoire and the fetal cardiac behavior, among other possible applications. Objective. The application of the probability theory has allowed the prediction of the cardiac behavior, epidemics and the union of peptides in molecular biology. Materials and methods. The adverse reactions rate to medications (ARM) and problems related to medications (PRM) in adults above 44 years of age in Bogota, Colombia, from January to December, 2012, were analyzed by means of the Zipf- Mandelbrot law, separating the ages by ratios every 5 years. Additionally, the probability of the number of reports of the 19 pharmacological groups with higher ARM and PRM were assessed. Results. The ARM and PRM rates have a fractal statistic behavior with fractal dimension of 0,983 and a R2 of 0,8618. The probability of the pharmacological groups varied between 0,0383 for anticonvulsants and lipid-lowering medications and 0,2743 for anti-bacterial ones. The probabilities of the pharmacological groups show differences of 10 in magnitude orders in regard to the maximum probability. Conclusion. The incidence rate of RAM and PRM is characterized by Zipf/Mandelbrot law; It was found a not equiprobable behavior of the distribution of reports by pharmacological groups, which could show useful trends in pharmacological surveillance.

Black–Scholes–Schrödinger–Zipf–Mandelbrot model framework for improving a study of the coauthor core score

The data and findings by Miskiewicz (2013) on the relationship between the number (J) of publications ranked according to their decreasing importance, for some scientist with her/his coauthors (CA), i.e. J∝1/rα, as found in Ausloos (2013), when specific types of publications, i.e. proceedings (in a generalized sense) and peer-review journals, are considered, are reexamined along the Zipf–Mandelbrot law Mandelbrot (1977), i.e. J∝1/(ν+r)ζ. The statistics are much improved. The exponent α and ζ are compared. The ma core value, i.e. the core number of CAs [1] is unaffected, of course. A Black–Scholes–Schrödinger model framework is proposed to describe the findings.

Mandelbrot Law of Evolving Networks

We show that the degree distribution of a growing network with linear preferential attachment approximately follows the Mandelbrot law, and propose an analytical method based on a recursive formula that can be used to obtain a more accurate expression of the shifting coefficient. Simulations demonstrate the advantages of our method. This work provides a possible mechanism leading to the Mandelbrot law of evolving networks, and refines the mainstream analytical methods for the shifting coefficient.

Asymptotic Mandelbrot law for some evolving networks

In this letter, we study some evolution networks that grow with linear preferential attachment. Based upon some recent results on the quotient Gamma function, we give a rigorous proof of the asymptotic Mandelbrot law for the degree distribution $p_k \propto (k + c)^{-\gamma}$ in certain conditions. We also analytically derive the best fitting values for the scaling exponent $\gamma$ and the shifting coefficient $c$.

Constructing n-gram rules for natural language models through exploring the limitation of the Zipf–Mandelbrot law

The Zipf–Mandelbrot law is widely used to model a power-law distribution on ranked data. One of the best known applications of the Zipf–Mandelbrot law is in the area of linguistic analysis of the distribution of words ranked by their frequency in a text corpus. By exploring known limitations of the Zipf–Mandelbrot law in modeling the actual linguistic data from different domains in both printed media and online content, a new algorithm is developed to effectively construct n-gram rules for building natural language (NL) models required for a human-to-computer interface. The construction of statistically-oriented n-gram rules is based on a new computing algorithm that identifies the area of divergence between Zipf–Mandelbrot curve and the actual frequency distribution of the ranked n-gram text tokens extracted from a large text corpus derived from the online electronic programming guide (EPG) for television shows and movies. Two empirical experiments were carried out to evaluate the EPG-specific language models created using the new algorithm in the context of NL-based information retrieval systems. The experimental results show the effectiveness of the algorithm for developing low-complexity concept models with high coverage for the user’s language models associated with both typed and spoken queries when interacting with a NL-based EPG search interface.

Discovering power laws in computer programs

The power-law regularities have been discovered behind many complex natural and social phenomenons. We discover that the power-law regularities, especially the Zipf’s and Heaps’ laws, also exist in large-scale software systems. We find that the distribution of lexical tokens in modern Java, C++ and C programs follows Zipf–Mandelbrot law, and the growth of program vocabulary follows Heaps’ law. The results are obtained through empirical analysis of real-world software systems. We believe our discovery reveals the statistical regularities behind computer programming.

Les lois de Pareto et Lévy, leur cas particulier : la loi rang-fréquence de G.K. Zipf

A family of statistical laws, the first of which was discovered in 1896 by Vilfredo Pareto in relation to the distribution of incomes, is as universal and as general as the “ normal law” : these are the laws expressed by “ power functions”. They are relevant in various domains such as urban Geography, the distribution of wealth and incomes (in Economics), physical Geography, Granulometry, Financial Mathematics, and Lexicology, where they are known as the Zipf and Estoup laws, or — in their most general form — as the Mandelbrot law. They are characterized by the fact that very large values of the variable are not unlikely ; hence, in general, the variance of a distribution is infinite, and possibly also its mean : the last case is frequent in Lexicology. Therefore they must be handled with care, notably when interpreting their application to empirical data. Due to the universality of these laws, there exists a great number of “ mathematical models” that can explain them. These range from absolutely deterministic mechanisms to pure probabilistic theories : Paul Lévy stable laws.

Modeling hypermedia-based communication

In this article, we explore two approaches to modeling hypermedia-based communication. It is argued that the classical conveyor-tube framework is not applicable to the case of computer- and Internet-mediated communication. We then present a simple but very general system-theoretic model of the communication process, propose its mathematical interpretation, and derive several formulas, which qualitatively and quantitatively accord with data obtained on-line. The devised theoretical results generalize and correct the Zipf–Mandelbrot law and can be used in information system design. At the paper’s end, we give some conclusions and draw implications for future work.

Some coding properties of speech

Some important properties of speech are considered from the point of view of the theory of error correcting codes. It has found experimentally that the properties of Russian words encoded in terms of phonemes are largely similar to the properties of the so-called prefix codes. In the prefix codes, no code word is a prefix of another word. According to the theory of coding, for any prefix code exists an algorithm of unambiguous decoding where no pause or special symbol-delimeter separates code words. Apparently, word segmentation in the continuous speech signal is provided mainly by the use of the prefix property. Phoneme probability in Russian follows the Mandelbrot law. This finding is evidence in favour of the assumption that the probability is determined by the “complexity” or “expense” of phoneme generation. Speech recognition for a large vocabulary requires much time for access to word templates. Thus, a preliminary sorting of the templates is necessary to restrict a number of candidates for final recognition. The preliminary sorting can be executed by means of word coding by few phonemic cues. Auditory experiments with speech masked by white noise have revealed the most reliable cues. These cues are “vowel, voiced, nasal, fricative”. About 150 templates were left after the fast sorting procedure for approximately 100,000 templates in the vocabulary of 10,000 of the most frequent English words. Speech recognition rate obtained by an automatic recognition system must be compared with potentially achievable rate. The potential rate of word recognition for various S/ N ratios can be computed with the use of methods developed in the theory of coding. It can be argued that an optimal machine for automatic speech recognition should find robust the same cues which humans find robust. The potential rate for words encoded in terms of independent distinctive features is closer to the subjective reliability of word perception than the rate for words encoded by phonemes.

Beyond the Zipf–Mandelbrot law in quantitative linguistics

In this paper the Zipf–Mandelbrot law is revisited in the context of linguistics. Despite its widespread popularity the Zipf–Mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus. In particular, we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation. Finally, it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by C. Tsallis.

Microstructures of Inhomogeneous, Discontinuous Metal Films: Computer Simulations and Statistical Description

The ranges of statistical description of the systems may be determined on the basis of inverse power law (Mandelbrot law). The slope of the straight line representing the power law in a double-logarithmic plot, determined as -1/μ (μ being a critical exponent), characterizes the distribution of elements in the microstructure. In this work the inverse power law is used to describe the microstructure and the statistical distribution of discontinuous metal films on dielectric substrates for μ < 2 (second and third case). The inhomogeneous films were obtained experimentally and using the computer simulations. The ranges of critical exponent μ, for which the Poisson, or Pareto's distribution describes the microstructures, were determined. The statistics determined on the basis of microscopic examination is compared with that for the simulated microstructure.

Microstructures of island metal films—the computer simulations and statistical description

The ranges of statistical description of the systems may be determined on the basis of inverse power law (Mandelbrot law). The slope of the straight line representing the power law in a double-logarithmic plot, determined as (−1/μ) (μ being a critical exponent), characterizes the distribution of elements in the microstructure. For μ>2, two parameters of distribution (the mean value and the variance) are finite, Gaussian or log-normal distribution describes the system. When 1<μ<2 only the mean value is finite. Therefore, for a description of such system, a statistical distribution with one parameter may be used. For μ<1, both the mean value and the variance are infinite. The objects of such microstructure are distributed according to the Lévy stable distribution. In this paper the inverse power law is used to describe the microstructure and the statistical distribution of discontinuous metal films on dielectric substrates for μ>2 (first case). The homogeneous films were obtained experimentally and using the computer simulations. The ranges of critical exponent μ, for which the Gaussian or log-normal distribution describes the microstructure, were determined. The statistics determined on the basis of microscopic examination is compared with that for the simulated microstructure.

Regularities in football goal distributions

Besides complexities concerning football championships, some regularities are identified in them. These regularities refer to goal distributions by goal-players and by games. In particular, the goal distribution by goal-players is well adjusted by the Zipf–Mandelbrot law, suggesting a connection with an anomalous decay.

Statistical description of systems on the basis of the Mandelbrot law: discontinuous metal films on dielectric substrates

The inverse power law has been used to describe the structures and the statistical distributions of discontinuous films of Au, Cu, and Mn on dielectric substrates. To this end, the rank of an island (k) was connected with its area (X) for the films with different coverage coefficients (metal contents) p. The dependencies of the island areas on the rank orders in a double-logarithmic plot are straight lines according to the Mandelbrot law. The slope of the straight line, determined as -1/µ, characterizes the distributions of the islands. One can distinguish three ranges of the critical index µ. For µ>2, the moments m1, m2 are finite; if the number of islands is large, the distribution of island areas tends towards a Gaussian or log-normal one. The films structures exhibit Euclidean character. When 1<µ<2, the moment m1 is finite, m2 is infinite; the islands become irregular in shape, and their areas may be described using Poisson distributions. For µ<1, the `hierarchization' of X is more pronounced, the moments m1, m2 are infinite, and the island areas are distributed according to a Lévy stable distribution (for example, Pareto's distribution). At the percolation threshold (phase transition), the infinite cluster appears and the inverse power law is no longer valid.