Zipf's Law Research Articles

Zipf's Law of Vasovagal Heart Rate Variability Sequences.

Cardiovascular self-organized criticality (SOC) has recently been demonstrated by studying vasovagal sequences. These sequences combine bradycardia and a decrease in blood pressure. Observing enough of these sparse events is a barrier that prevents a better understanding of cardiovascular SOC. Our primary aim was to verify whether SOC could be studied by solely observing bradycardias and by showing their distribution according to Zipf’s law. We studied patients with vasovagal syncope. Twenty-four of them had a positive outcome to the head-up tilt table test, while matched patients had a negative outcome. Bradycardias were distributed according to Zipf’s law in all of the patients. The slope of the distribution of vasovagal sequences and bradycardia are slightly but significantly correlated, but only in cases of bradycardias shorter than five beats, highlighting the link between the two methods (r = 0.32; p < 0.05). These two slopes did not differ in patients with positive and negative outcomes, whereas the distribution slopes of bradycardias longer than five beats were different between these two groups (−0.187 ± 0.004 and −0.213 ± 0.006, respectively; p < 0.01). Bradycardias are distributed according to Zipf’s law, providing clear insight into cardiovascular SOC. Bradycardia distribution could provide an interesting diagnosis tool for some cardiovascular diseases.

Brevity is not a universal in animal communication: evidence for compression depends on the unit of analysis in small ape vocalizations.

Evidence for compression, or minimization of code length, has been found across biological systems from genomes to human language and music. Two linguistic laws—Menzerath's Law (which states that longer sequences consist of shorter constituents) and Zipf's Law of abbreviation (a negative relationship between signal length and frequency of use)—are predictions of compression. It has been proposed that compression is a universal in animal communication, but there have been mixed results, particularly in reference to Zipf's Law of abbreviation. Like songbirds, male gibbons (Hylobates muelleri) engage in long solo bouts with unique combinations of notes which combine into phrases. We found strong support for Menzerath's Law as the longer a phrase, the shorter the notes. To identify phrase types, we used state-of-the-art affinity propagation clustering, and were able to predict phrase types using support vector machines with a mean accuracy of 74%. Based on unsupervised phrase type classification, we did not find support for Zipf's Law of abbreviation. Our results indicate that adherence to linguistic laws in male gibbon solos depends on the unit of analysis. We conclude that principles of compression are applicable outside of human language, but may act differently across levels of organization in biological systems.

Efficient Communication in Written and Performed Music.

Since its inception, Shannon's information theory has attracted interest for the study of language and music. Recently, a wide range of converging studies have shown how efficient communication pervades language, from phonetics to syntax. Efficient principles imply that more resources should be assigned to highly informative items. For instance, average information content was shown to be a better predictor of word length than frequency, revisiting the famous Zipf's law. However, in spite of the success of the efficient communication framework in the study of language and speech, very little work has investigated its relevance in the analysis of music. Here, we examine the organization of harmonic information in two large corpora of Western music, one made of MIDI files directly sequenced from scores, and the other made of MIDI recordings of live performances of highly skilled piano players. We show that there is a clear positive relationship between (contextual) information content of harmonic sequences and two essential musical properties, namely duration and loudness: the more unexpected a harmonic event is, the longer and the louder it is.

Quantifying urban areas with multi-source data based on percolation theory

Quantifying urban areas is crucial for addressing associated urban issues such as environmental and sustainable problems. Remote sensing data, especially the nighttime light images, have been widely used to delineate urbanized areas across the world. Meanwhile, some emerging urban data, such as volunteered geographical information (e.g., OpenStreetMap) and social sensing data (e.g., mobile phone and social media), have also shown great potential in revealing urban boundaries and dynamics. However, consistent and robust methods to quantify urban areas from these multi-source data have remained elusive. Here, we propose a percolation-based method to extract urban areas from these multi-source urban data. We derive the optimal urban/non-urban threshold by considering the critical nature of urban systems with the support of the percolation theory. Furthermore, we apply the method with three open-source datasets – population, road, and nighttime light – to 28 countries. We show that the proposed method captures the similar urban characteristics in terms of urban areas from multi-source data, and Zipf's law holds well in most countries. The accuracy of the derived urban areas by different datasets has been validated with the Landsat-based reference data in 10 cities, and the accuracy can be further improved through data fusion (κ = 0.69–0.85, mean κ = 0.78). Our study not only provides an efficient method to quantify urban areas with open-source data, but also deepens the understanding of urban systems and sheds some light on multi-source data fusion in geographical fields.

A cardiodynamic model for power laws of heartbeats in aging and congestive heart failure

The output of a healthy physiological system exhibits complex fluctuation. Nonlinear analysis, such as power-law characteristics, shows the potential for detecting changes in the biological complexity of disease and aging. This paper characterized the heart rate variability (HRV) of aging and patients with congestive heart failure (CHF) by three types of distribution: Zipf's law, Heaps' law, and frequency distribution. All data analysis and modeling are based on a constructed sequence, that is, the monotonous increase to monotonous decrease amplitude ratios as derived from heartbeat interval data. The experimental result shows a significant decrease of HRV from healthy young people to healthy elderly to CHF patients. We proposed a model by taking account of the “rich-get-richer” theory in experimental observations, which successfully reproduced three types of distribution characterizing the constructed ratio sequences as obtained from the analysis of measured cardiac data. This work provides insight into the dynamic mechanism of cardiac data underlying the regulation of autonomic nerve.

Identifying IGH disease clones for MRD monitoring in childhood B-cell acute lymphoblastic leukemia using RNA-Seq.

Identifying patient-specific clonal IGH/TCR junctional sequences is critical for minimal residual disease (MRD) monitoring. Conventionally these junctional sequences are identified using laborious Sanger sequencing of excised heteroduplex bands. We found that the IGH is highly expressed in our diagnostic B-cell acute lymphoblastic leukemia (B-ALL) samples using RNA-Seq. Therefore, we used RNA-Seq to identify IGH disease clone sequences in 258 childhood B-ALL samples for MRD monitoring. The amount of background IGH rearrangements uncovered by RNA-Seq followed the Zipf's law with IGH disease clones easily identified as outliers. Four hundred and ninety-seven IGH disease clones (median 2, range 0-7 clones/patient) are identified in 90.3% of patients. High hyperdiploid patients have the most IGH disease clones (median 3) while DUX4 subtype has the least (median 1) due to the rearrangements involving the IGH locus. In all, 90.8% of IGH disease clones found by Sanger sequencing are also identified by RNA-Seq. In addition, RNA-Seq identified 43% more IGH disease clones. In 69 patients lacking sensitive IGH targets, targeted NGS IGH MRD showed high correlation (R = 0.93; P = 1.3 × 10-14), better relapse prediction than conventional RQ-PCR MRD using non-IGH targets. In conclusion, RNA-Seq can identify patient-specific clonal IGH junctional sequences for MRD monitoring, adding to its usefulness for molecular diagnosis in childhood B-ALL.

Spatio-temporal analysis of irrigation water use coefficients in China

Water shortage is a major problem for agriculture in many countries. Agricultural water accounts for more than 60% of total water consumption in China. Improving agricultural strategies in using irrigation water more effectively and efficiently is important to alleviate water shortages and associated environmental problems. In this study, the spatiotemporal distribution differences of irrigation water use coefficient in large, medium and small-scale irrigation districts and pure well irrigation districts in the 31 provinces of China (excluding Hong Kong, Macao and Taiwan) were analyzed by means such as the Zipf's law and spatial autocorrelation. The results showed that the national irrigation water use coefficient had increased from 0.501 in 2010 to 0.542 in 2016. By size of irrigation area, the irrigation water use coefficient in small irrigation areas was higher than that of large- and medium-scale irrigation districts. Regionally, the irrigation water use coefficient in the eastern region was significantly higher than that in the western region, and the coefficient in northern region was higher than that in the southern region. The spatial spillover effect of the national irrigation water use coefficient gradually weakened, and the spatial spillover effect of large and medium-scale irrigation districts was not obvious. The spatial spillover effect of pure well irrigation districts varied greatly over time. All these results give a spatiotemporal overview of agricultural water use in China, which provides a direction for improving irrigation water coefficient in China.

Do penguins' vocal sequences conform to linguistic laws?

Information compression is a general principle of human language: the most frequent words are shorter in length (Zipf's Law of Brevity) and the duration of constituents decreases as the size of the linguistic construct increases (Menzerath-Altmann Law). Vocal sequences of non-human primates have been shown to conform to both these laws, suggesting information compression might be a more general principle. Here, we investigated whether display songs of the African penguin, which mediate recognition, intersexual mate choice and territorial defence, conform with these laws. Display songs are long, loud sequences combining three types of syllables. We found that the shortest type of syllable was the most frequent (with the shortest syllable being repeated stereotypically, potentially favouring signal redundancy in crowded environments). We also found that the average duration of the song's constituents was negatively correlated with the size of the song (a consequence of increasing the relative number of the shortest syllable type, rather than reducing the duration across all syllable types, thus preserving the communication of size-related information in the duration of the longest syllable type). Our results provide the first evidence for conformity to Zipf's and Menzerath-Altmann Laws in the vocal sequences of a non-primate species, indicating that these laws can coexist with selection pressures specific to the species' ecology.

Does China's Urban Development Satisfy Zipf's Law? A Multiscale Perspective from the NPP-VIIRS Nighttime Light Data.

Currently, whether the urban development in China satisfies Zipf’s law across different scales is still unclear. Thus, this study attempted to explore whether China’s urban development satisfies Zipf’s law across different scales from the National Polar-Orbiting Partnership’s Visible Infrared Imaging Radiometer Suite (NPP-VIIRS) nighttime light data. First, the NPP-VIIRS data were corrected. Then, based on the Zipf law model, the corrected NPP-VIIRS data were used to evaluate China’s urban development at multiple scales. The results showed that the corrected NPP-VIIRS data could effectively reflect the state of urban development in China. Additionally, the Zipf index (q) values, which could express the degree of urban development, decreased from 2012 to 2018 overall in all provinces, prefectures, and counties. Since the value of q was relatively close to 1 with an R2 value > 0.70, the development of the provinces and prefectures was close to the ideal Zipf’s law state. In all counties, q > 1 with an R2 value > 0.70, which showed that the primate county had a relatively stronger monopoly capacity. When the value of q < 1 with a continuous declination in the top 2000 counties, the top 250 prefectures, and the top 20 provinces in equilibrium, there was little difference in the scale of development at the multiscale level with an R2 > 0.90. The results enriched our understanding of urban development in terms of Zipf’s law and had valuable implications for relevant decision-makers and stakeholders.

The evolution of Zipf's Law for U.S. cities

AbstractExploiting the hierarchical structure of cities and based on a dataset for U.S. cities between 1840 and 2016, the aim of this paper is to analyze the evolution of the U.S. city size distribution. For that purpose we estimate a general three‐parameter Zipf model, which can be traced back to Mandelbrot (1982), and validate our results by means of the hierarchical scaling law. Especially in the second half of the twentieth century, we find a pronounced departure from the exact Zipf's law. The city size distribution has become more equally distributed over time. Besides, the applied estimation method reveals evidence for leading cities dominating the remaining largest cities. Thus, the growing equality of the city sizes can be explained rather by growing smaller cities than by a loss of importance of the largest ones.

Combining deep learning with token selection for patient phenotyping from electronic health records

Artificial intelligence provides the opportunity to reveal important information buried in large amounts of complex data. Electronic health records (eHRs) are a source of such big data that provide a multitude of health related clinical information about patients. However, text data from eHRs, e.g., discharge summary notes, are challenging in their analysis because these notes are free-form texts and the writing formats and styles vary considerably between different records. For this reason, in this paper we study deep learning neural networks in combination with natural language processing to analyze text data from clinical discharge summaries. We provide a detail analysis of patient phenotyping, i.e., the automatic prediction of ten patient disorders, by investigating the influence of network architectures, sample sizes and information content of tokens. Importantly, for patients suffering from Chronic Pain, the disorder that is the most difficult one to classify, we find the largest performance gain for a combined word- and sentence-level input convolutional neural network (ws-CNN). As a general result, we find that the combination of data quality and data quantity of the text data is playing a crucial role for using more complex network architectures that improve significantly beyond a word-level input CNN model. From our investigations of learning curves and token selection mechanisms, we conclude that for such a transition one requires larger sample sizes because the amount of information per sample is quite small and only carried by few tokens and token categories. Interestingly, we found that the token frequency in the eHRs follow a Zipf law and we utilized this behavior to investigate the information content of tokens by defining a token selection mechanism. The latter addresses also issues of explainable AI.

Male gibbon loud morning calls conform to Zipf's law of brevity and Menzerath's law: insights into the origin of human language

The study of vocal communication in nonhuman primates, especially apes, offers critical insight into the origins of human language. Although human language represents a highly derived and complex form of communication, researchers have found that the organization of language follows a series of common statistical patterns, known as ‘linguistic laws’. Zipf's law of brevity and Menzerath's law are pervasive across human languages, and these laws have been identified in the communication of a small number of primate species. What remains less clear is whether these two laws also affect long-distance vocal communication in primates. Here, we provide evidence that the long-distance morning calls of male gibbons (cao vit gibbon, Nomascus nasutus, and western black-crested gibbon, Nomascus concolor ) follow both Zipf's law of brevity and Menzerath's law. We found that notes of male gibbon calls conform to Zipf's law of brevity, with the most common notes being shortest in duration. Similarly, longer sequences are made up of shorter calls on average, consistent with Menzerath's law; we also found a shortening of specific note type duration and an increase in proportion of shorter call types in longer sequences, which may underpin the emergence of this law. Our findings support the generality of these two linguistic laws beyond human language and provide evidence for compression at two levels of organizations (how frequently different note types are used, and how vocal sequences are constructed) in a long-range communication system. • Male gibbon morning calls conform to Zipf's law of brevity and Menzerath's law. • The most common notes being shortest in duration. • Longer sequences are made up of shorter calls on average.

A Tale of Two Dispersions: Wage and Firm Size

The traditional literature treat wage dispersion and firm dynamics, which are closely connected to each other, in isolation. This paper delivers a unified treatment to wage dispersion and firm-size distribution by developing a real-option-theoretic approach. The model is tractable with analytic solution, generating the following testable implications. Firstly, the distribution of firm size is a uni-modal, right-skewed with a Paretian tail, which is in line the empirical findings, in particular the Zipf Law. So is that of wage dispersion. Secondly, as in Mortensen and Pissarides (1994), the incumbents prefer to preserve the pattern of labor hoarding rather than exiting the market when hit by (not too severely) negative productivity shock. Thirdly, in addition to the effect in standard search and matching theory, the labor market tightness is also found to produce additional transition mechanisms to the unemployment rate. Fourthly, the model predicts that, the larger the firm is, the longer the firm will survive at the market.

EFFECTIVENESS OF THE BINARY SEARCH METHOD IN DATABASE FILES IN THE CASE OF A GENERALIZED DISTRIBUTION OF PROBABILITIES OF ACCESS TO RECORDS.

The article presents a formula for calculating the mathematical expectation of the number of comparisons needed to nd a record in a le in the case of generalized distribution of the probabilities of accessing records. We compare the eectiveness of the binary search method in the case of generalized distribution of probabilities of access to records and Zipf's law.

EFFECTIVENESS OF THE BINARY SEARCH METHOD IN DATABASE FILES IN THE CASE OF A GENERALIZED DISTRIBUTION OF PROBABILITIES OF ACCESS TO RECORDS.

The article presents a formula for calculating the mathematical expectation of the number of comparisons needed to nd a record in a le in the case of generalized distribution of the probabilities of accessing records. We compare the eectiveness of the binary search method in the case of generalized distribution of probabilities of access to records and Zipf's law.

Investigation of urban regularities for Croatia in the period from 1857 to 2011

Two main regularities in the field of urban economics are Zipf’s law and Gibrat’s law. Zipf’s law states that distribution of largest cities should obey the Pareto rank-size distribution while Gibrat’s law states that proportionate growth of cities is independent of its size. These two laws are interconnected and therefore are often considered together. The objective of this paper is the investigation of urban regularities for Croatia in the period from 1857 to 2011. In order to estimate and evaluate the structure of Croatian urban hierarchy, Pareto or Zipf’s coefficients are calculated. The results have shown that the coefficient values for the largest settlements in different years are close to one, indicating that the Croatian urban hierarchy system follows the rank-size distribution and therefore obeys Zipf's law. The independence of city growth regarding the city size is tested using penal unit roots. Results for Gibrat's law testing using panel unit root tests have shown that there is a presence of unit root in growth of settlements therefore leading to the acceptance of Gibrat’s law.

An Application of Zipf's Law for Prose and Verse Corpora Neutrality for Hindi and Marathi Languages

Availability of the text in different languages has become possible, as almost all websites have offered multilingual option. Hindi is considered as official language in one of the states of India. Hindi text analysis is dominated by the corpus of stories and poems. Before performing any text analysis token extraction is an important step and supports many applications like text summarization , categorizing text and so on. Token extraction is a part of Natural language processing (NLP). NLP includes many steps such as preprocessing the corpus, lemmatization and so on. In this paper the tokens are extracted by two methods and on two corpora. BaSa, a context-based term extraction technique having different NLP activities, e.g. Term Frequency Inverse Document Frequency (TF-IDF) and Zipf ‘s law are used to count and compare extracted tokens. Further token comparison between both of the methods is achieved. The corpus contains proses and verses of Hindi as well as the Marathi language. Common tokens from corpora of verses and proses of Marathi as well as Hindi are identified to prove that both of them behave same as per as NLP activities are concerened. The betterment of BaSa over Zipf’s law is proved. Hindi Corpus includes 820 stories and 710 poems and Marathi corpus includes 610 stories and 505 poems.

Optimal Pricing for Peer-to-Peer Sharing With Network Externalities

In this paper, we analyse how a peer-to-peer sharing platform should price its service to maximize profit, when user participation increases the value of the service to others by causing positive externalities. Modelling the service as an excludable public good, we propose a bounded utility model to capture many infrastructure sharing applications with bounded network value, in which complete coverage generates finite user valuation (e.g., WiFi or hotspot). Unbounded utility models are used to capture the large-scale user interactions in social media, where the network value follows Metcalfe's or Zipf's law. For these utility models, we analyze the optimal pricing schemes in the case of heterogeneous users under complete and incomplete information of users' service valuations. We propose the concept of `price of information' (PoI) to characterize the profit loss due to lack of information, and present asymptotic PoI bounds for different utility models. We also show that the difficult-to-implement differentiated pricing scheme, which is optimal under incomplete user information, can be replaced by a simple uniform price scheme that is asymptotic optimal. Finally, we extend our pricing schemes to a two-sided market by including a new group of `pure' service users who do not contribute to the public good, and show that the platform may charge zero price to the original group of users in order to attract this pure user group.

The logarithmic Zipf law in a general urn problem

The origin of power-law behavior (also known variously as Zipf’s law) has been a topic of debate in the scientific community for more than a century. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. In a highly cited article, Mark Newman [Contemp. Phys. 46 (2005) 323–351] reviewed some of the empirical evidence for the existence of power-law forms, however underscored that even though many distributions do not follow a power law, quite often many of the quantities that scientists measure are close to a Zipf law, and hence are of importance. In this paper we engage a variant of Zipf’s law with a general urn problem. A collector wishes to collect m complete sets of N distinct coupons. The draws from the population are considered to be independent and identically distributed with replacement, and the probability that a type-j coupon is drawn is denoted by pj, j = 1, 2, …, N. Let Tm(N) the number of trials needed for this problem. We present the asymptotics for the expectation (five terms plus an error), the second rising moment (six terms plus an error), and the variance of Tm(N) (leading term) as N →∞, when pj = aj / ∑j=2N+1aj, where aj = (ln j)−p, p &gt; 0. Moreover, we prove that Tm(N) (appropriately normalized) converges in distribution to a Gumbel random variable. These “log-Zipf” classes of coupon probabilities are not covered by the existing literature and the present paper comes to fill this gap. In the spirit of a recent paper of ours [ESAIM: PS 20 (2016) 367–399] we enlarge the classes for which the Dixie cup problem is solved w.r.t. its moments, variance, distribution.

By Force of Habit: Self-Trapping in a Dynamical Utility Landscape

Historically, rational choice theory has focused on the utility maximization principle to describe how individuals make choices. In reality, there is a computational cost related to exploring the universe of available choices and it is often not clear whether we are truly maximizing an underlying utility function. In particular, memory effects and habit formation may dominate over utility maximization. We propose a stylized model with a history-dependent utility function where the utility associated to each choice is increased when that choice has been made in the past, with a certain decaying memory kernel. We show that self-reinforcing effects can cause the agent to get stuck with a choice by sheer force of habit. We discuss the special nature of the transition between free exploration of the space of choice and self-trapping. We find in particular that the trapping time distribution is precisely a Zipf law at the transition, and that the self-trapped phase exhibits super-aging behavior.