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Brazilian drivers caught in traffic violations accumulate points in their official personal files. Here, we analyze the distribution probability of these data for the state of Rio de Janeiro where 4199 drivers accumulated 20 or more penalty points during one year.

We study a migration model, in which individuals migrate from one community to another. The choices of the source community i and the destination one j are proportional to some power of the population of and , respectively. Both analytical calculation and numerical simulation show that the population distribution of communities in stationary states is determined by the parameters α and β. The distribution is widely homogeneous with a characteristic size if α > β. Whereas, for α < β, the distribution is highly heterogeneous with the emergence of condensing phenomenon. Between the two regimes, α = β, the distribution gradually shifts from the nonmonotonous (α < 0) to scale-free (α > 0).

Complex network studies, as an interdisciplinary framework, span a large variety of subjects including social media. In social networks, several mechanisms generate miscellaneous structures like friendship networks, mention networks, tag networks, etc. Focusing on tag networks (namely, hashtags in twitter), we made a two-layer analysis of tag networks from a massive dataset of Twitter entries. The first layer is constructed by converting the co-occurrences of these tags in a single entry (tweet) into links, while the second layer is constructed converting the semantic relations of the tags into links. We observed that the universal properties of the real networks like small-world property, clustering and power-law distributions in various network parameters are also evident in the multilayer network of hashtags. Moreover, we outlined that co-occurrences of hashtags in tweets are mostly coupled with semantic relations, whereas a small number of semantically unrelated, therefore random links reduce node separation and network diameter in the co-occurrence network layer. Together with the degree distributions, the power-law consistencies of degree difference, edge weight and cosine similarity distributions in both layers are also appealing forms of Zipf’s law evident in nature.

We examine the statistical properties of data relating to clinical diagnoses extracted from the medical database of our hospital. Specifically, we analyze all diagnoses given to all patients (in- and out-patients). The data consisted of both data input to to computers as Japanese names, and the data input ICD10 (International Classification of Diseases 10th Revision). We adapted the Zipf approach to analyzing the frequencies of clinical diagnoses for these two data groups. We found that both group types have the inverse-power relationship between the rank order of diagnoses and the frequency of the appearance of these diagnoses. (This relationship is called Zipf's law, which is observed in natural language.) Though the reason why these sets follow Zipf's law is unknown, we speculate that the complex interaction between doctor and patient is the cause for adherence to Zipf's law.

An innovative method for measuring the temperature of texts is introduced by measuring deviation from a standard Maxwell-Boltzmann (M–B) distribution for a corpus of English words. The temperature of English texts for junior and senior high schools in Japan and grades one through 12 in the US are measured and found to be between 148 K and 87 K, when the temperature of ANC (American National Corpus) is assumed to be 100 K as a standard. The temperature of several books by famous authors (Einstein, Darwin, etc.) is also analyzed. The present technique also can be applied to many other topics. Application to mortality rate distribution in Japan is discussed as an example.

As the most fundamental empirical law, Zipf's law has been studied from many aspects. However, its meaning is still an open problem. Some models have been constructed to explain Zipf's law. In the letter, a concept of auxiliary information of an event is introduced, and a new entropy named non-symmetric entropy is defined. We prove that maximizing non-symmetric entropy leads naturally to Zipf's law under the special auxiliary information parameters. At the same time, if we take other auxiliary parameters we will get other distribution laws.

The scaling exponent of a hierarchy of cities used to be regarded as a fractional dimension. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was considered to be the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a ratio of the fractal dimension of a network of cities to the average dimension of city population. Accordingly, the Zipf exponent is the reciprocal of this dimension ratio. On a digital map, the Pareto exponent can be defined by the scaling relation between a map scale and the corresponding number of cities based on this scale. The cities of the United States of America in 1900, 1940, 1960, and 1980 and Indian cities in 1981, 1991, and 2001 are utilized to illustrate the geographical spatial meaning of Pareto's exponent. The results suggest that the Pareto exponent of city-size distributions is a dimension ratio rather than a fractal dimension itself. This conclusion is revealing for scientists to understand Zipf's law on the rank-size pattern and the fractal structure of hierarchies of cities.

The difference between the inverse power function and the negative exponential function is significant. The former suggests a complex distribution, while the latter indicates a simple distribution. However, the association of the power-law distribution with the exponential distribution has been seldom researched. This paper is devoted to exploring the relationships between exponential laws and power laws from the angle of view of urban geography. Using mathematical derivation and numerical experiments, I reveal that a power-law distribution can be created through a semi-moving average process of an exponential distribution. For the distributions defined in a one-dimension space (e.g. Zipf's law), the power exponent is 1; while for those defined in a two-dimension space (e.g. Clark's law), the power exponent is 2. The findings of this study are as follows. First, the exponential distributions suggest a hidden scaling, but the scaling exponents suggest a Euclidean dimension. Second, special power-law distributions can be derived from exponential distributions, but they differ from the typical power-law distributions. Third, it is the real power-law distributions that can be related with fractal dimension. This study discloses an inherent link between simplicity and complexity. In practice, maybe the result presented in this paper can be employed to distinguish the real power laws from spurious power laws (e.g. the fake Zipf distribution).

Random-text models have been proposed as an explanation for the power law relationship between word frequency and rank, the so-called Zipf's law. They are generally regarded as null hypotheses rather than models in the strict sense. In this context, recent theories of language emergence and evolution assume this law as a priori information with no need of explanation. Here, random texts and real texts are compared through (a) the so-called lexical spectrum and (b) the distribution of words having the same length. It is shown that real texts fill the lexical spectrum much more efficiently and regardless of the word length, suggesting that the meaningfulness of Zipf's law is high.

Here, we test Neutral models against the evolution of English word frequency and vocabulary at the corpus scale, as recorded in annual word frequencies from three centuries of English language books. Against these data, we test both static and dynamic predictions of two neutral models, including the relation between corpus size and vocabulary size, frequency distributions, and turnover within those frequency distributions. Although a commonly used Neutral model fails to replicate all these emergent properties at once, we find that modified two-stage Neutral model does replicate the static and dynamic properties of the corpus data. This two-stage model is meant to represent a relatively small corpus of English books, analogous to a ‘canon’, sampled by an exponentially increasing corpus of books among the wider population of authors. More broadly, this model — a smaller neutral model within a larger neutral model — could represent more broadly those situations where mass attention is focused on a small subset of the cultural variants.

A cellular automaton model called the Firm Dynamics Model (FDM) is introduced to simulate the dynamics of firms within an economy. The model includes the growth of firms and their mergers and exits. The main objective is to compare the size-frequency distributions in the model with the empirical firm size distributions of several countries (USA, UK, Spain and Sweden). The empirical size distributions were assembled from business censuses and additional information on the country’s largest companies in terms of the number of employees. For the four datasets analyzed here, the firm size distribution is compatible with a power law of the Pareto type with an exponent of close to two (for the probability density). For its part, the model delivers two different size-frequency distributions depending on the type of merger that firms can undergo: the friendly-merger version gives rise to subcritical distributions with an exponential tail, whereas the aggressive-merger version produces power-law distributions. The simulation model was run with underlying lattices in one, two and three dimensions in order to compare the simulated power-law exponent with the empirical one. The best agreement was obtained with the two-dimensional aggressive-merger model version, for which the power-law exponent is $2.19\pm 0.01$, as compared with an empirical exponent of $2.1\pm 0.1$ (average over the four datasets). Further simulations with the model on a Bethe lattice confirm that the two-dimensional model provides the best fit to the empirical exponent.

The principle of least effort (PLE) is believed to be a universal rule for living systems. Its application to the derivation of the power law probability distributions of living systems has long been challenging. Recently, a measure of efficiency was proposed as a tool of deriving Zipf’s and Pareto’s laws directly from the PLE. This work is a further investigation of this efficiency measure from a mathematical point of view. The aim is to get further insight into its properties and usefulness as a metric of performance. We address some key mathematical properties of this efficiency such as its sign, uniqueness and robustness. We also look at the relationship between this measure and other properties of the system of interest such as inequality and uncertainty, by introducing a new method for calculating nonnegative continuous entropy.

In this paper, we first study the stability and convergence property of fuzzy logic networks (FLN). A random approach is adopted to simulate the convergence speed and steady-state properties based on four fuzzy logical functions. The simulation results show that MV logical function causes the system to be on the edge of chaos when the number of nodes increases. Thus this logical function is more useful to infer real complex networks, such as gene regulatory networks. Then, a novel gene regulatory network inference algorithm based on the FLN theory is proposed and tested. The algorithm uses fuzzy logical functions to model gene regulatory relationships, and the degrees of regulations are represented as the length of accumulated distances during a period of time intervals. Based on the assumption that the distribution of connectivity in yeast protein-protein networks follows the Zipf's law, the criteria for algorithm parameter quantifications are deduced. One unique feature of this algorithm is that it makes limited a priori assumptions concerning the modeling; hence the algorithm is categorized as a data-driven algorithm. The algorithm was applied to the S. pombe time-series dataset and 407 cell cycle regulated genes were used. The algorithm inferred 59 functionally verified regulations, 47 regulations involving genes with unknown functions, and 19 dubious regulations. The 125 regulatory pairs involve 108 genes, and the average connectivity of the inferred network, 1.157, confirms the Zipf's law.

Chains of the CME Group Time and Sales E-mini S&P 500 futures tick prices and their a-b-c-d-increments are studied. A discrete probability distribution based on the Hurwitz zeta function and Dirichlet series is suggested for the price increments. The randomness of the ticks is discussed using the notions of typicalness, chaoticness, and stochasticness introduced by Kolmogorov and Uspenskii and developed by predecessors, them, and pupils. They define randomness in terms of the theory of algorithms.

The purpose of this paper is to highlight certain newly discovered social phenomena that accord with Zipf's law, in addition to the famous natural and social phenomena including word frequencies, earthquake magnitude, city size, income¹etc. that are already known to follow it. These phenomena have recently been discovered within the transaction amount (payments or receipts) distributions within two different Community Currencies (CC) that had been initiated as social experiments. One is a local CC circulating in a specific geographical area, such as a town. The other is a virtual CC used among members who belong to a certain community of interest (COI) on the Internet. We conducted two empirical studies to estimate the economic vitalization effects they had on their respective local economies. The results we found were that the amount of transactions (payments and receipts) of the two CCs was distributed according to a power-law distribution with a unity rank exponent. In addition, we found differences between the two CCs with regard to the shapes of their distribution over a low-transaction range. The result may originate from the difference in methods of issuing CCs or in the magnitudes of the minimum-value unit; however, this result calls for further investigation.

We propose and sketch a novel approach toward the study of behavioral plasticity. When one encounters a new animal behavior, one formally describes this behavioral pattern, however, one confronts with more new behavioral patterns as the observation proceeds. As a result, the question arises how the hierarchy in behavioral pattern is originated and/or is changed. Focusing on the relationship between before and after the emergence of new behavioral pattern, we explain emergency a newly behavior and the origin of hierarchy of behavior. In particular, we illustrate the changing shell in experiments by hermit crabs, and the usage of a cart in experiments on the transportation of foods by ants. We observe a particular time-series sequence of hermit crabs and ants behaviors following a 1/f or Zipf's law. The behaviors following the 1/f or Zipf's law manifest emergence as logical jump on the part of object.