Zipf Distribution Research Articles

Core principles of melodic organisation emerge from transmission chains with random melodies

Music is a product of both biological and cultural evolution. Cultural transmission is the engine of cultural evolution and may play a role in the establishment of musical universals. Here, we examined how transmission dynamics can shape melodic features in music. Specifically, we tested whether random melodic seeds, in their transformation, take on properties known to characterise music within or even across cultures. Using an iterated learning paradigm, we investigated the transmission of random melodic seeds through a chain of non-musician participants (N = 64). We found that melodies reproduced vocally between “generations” became more similar to known musical scales, exhibited a predominance of consonant intervals, and reduced the number of scale degrees used. Additionally, we observed the previously documented tendency for large intervals to be followed by a change in direction, as well as features common to both music and speech including phrase-final lengthening and the Zipfian distribution of signalling units. As participants' vocalisations converged towards greater memorability, they exhibited decreased entropy, and their contours became smoother and more consistent. Finally, certain short melodic patterns became prominent motifs within the incipient musical “traditions” simulated by the chains. These emerging features may reflect a process shaped by (i) cognitive bottlenecks such as learnability; (ii) statistical properties of the processes and structures involved in inter-generational vocal transmission; but also by (iii) idiosyncratic cultural artefacts specific to the lab samples employed. Overall, our results demonstrate that fundamental aspects of melodic structure emerge naturally through the process of cultural transmission, as simulated in the lab.

Beyond Zipf’s law: Exploring the discrete generalized beta distribution in open-source repositories

Rank-size distributions, such as Zipf’s Law, have been instrumental in providing insights into the emergence of hierarchies across diverse systems, from linguistic corpuses to urban structures. However, the application of Zipf’s Law reveals limitations, particularly in its focus on distribution tails, sometimes overlooking a large proportion of the data which might play a pivotal role in system dynamics. Yet, fitting rank-size distributions other than a straight line on the log–log scale requires caution. In this study, we re-evaluate the utility of rank-size distributions by contrasting the traditional Zipf’s Law with the Discrete Generalized Beta Distribution (DGBD). We show the need of cautious fitting techniques for rank distributions, including the use of binning to prevent overfitting to data tails. Through both analytical derivation and empirical validation on commit data of open-source repositories, we show that DGBD consistently improves over Zipf distribution for concave rank distributions of large datasets (N≥100). This approach contributes to the advancement of methodologies for analyzing hierarchical systems.

Gender and Education: Their Role in the Zipfian Distribution of Speech Acts

ABSTRACT Speech acts, as basic communication units in pragmatics, are highly correlated to speakers’ communicative intentions. It is a worthwhile goal to explore whether they obey some linguistic laws that reflect people’s cognitive mechanisms, e.g. Zipf’s Law. However, few studies have examined whether the Zipf distribution can capture the frequencies of speech acts, and whether the parameters of it can show how speakers’ attributes (e.g. gender and education level) relate to the use of speech acts. As a preliminary attempt to bridge this gap, the current study finds from the data sample of the Switchboard Dialog Act Corpus that: (1) the Zipf distribution well captures the distribution of speech acts; (2) the parameter a can distinguish the dialogs between speakers of different genders, and it shows a regularity of the highest frequency of an SA in a dialigue to some degree; (3) the parameter b can differentiate speakers of different education levels – people with college-level education may have higher expressivity of SAs compared with those with higher-than-college-level education. This study shows that Zipf’s Law can be extended to the analysis of pragmatic phenomena, which may become part of the synergetic system of language through the approach of Quantitative Linguistics.

Purported Self-Organized Criticality of the Cardiovascular Function: Methodological Considerations for Zipf's Law Analysis.

Self-organized criticality is a universal theory for dynamical systems that has recently been applied to the cardiovascular system. Precise methodological approaches are essential for understanding the dynamics of cardiovascular self-organized criticality. This study examines how the duration and quality of data recording affect the analysis of cardiovascular self-organized criticality, with a focus on the beat-by-beat heart rate variability time series obtained from seven healthy subjects in a standing position. Drawing a Zipf diagram, we evaluated the distribution of cardiovascular events of bradycardia and tachycardia. We identified tipping points for the distribution of both bradycardia and tachycardia events. By varying the recording durations (1, 2, 5, 10, 20, 30, and 40 min) and sampling frequencies (500, 250, and 100 Hz), we investigated their influence on the observed distributions. While shorter recordings can effectively capture cardiovascular events, they may underestimate the variables describing their distribution. Additionally, the tipping point of the Zipf distribution differs between bradycardia and tachycardia events. Comparisons of the distribution of bradycardia and tachycardia events should be conducted using long data recordings. Utilizing devices with lower sampling frequencies may compromise data fidelity. These insights contribute to refining experimental protocols and advancing our understanding of the complex dynamics underlying cardiovascular regulation.

Zipf’s Law for Speech Acts in Spoken English

ABSTRACT We, the Editors and Publisher of the Journal of Quantitative Linguistics, have retracted the following article: Zipf’s Law for Speech Acts in Spoken English by Da Qi and Hua Wang (DOI: 10.1080/09296174.2023.2202470). Since publication, concerns have been raised about the fact there is significant overlap in the discussion section and references of this article with the following article, published in Springer’s Psychonomic Bulletin & Review: Zipf’s Law Revisited: Spoken Dialog, Linguistic Units, Parameters, and the Principle of Least Effort, by Guido M. Linders and Max M. Louwerse (DOI: 10.3758/s13423-022-02142-9). This overlap has been verified by the journal Editorial team and as this is a breach of our Editorial Policies, we are retracting the article from the journal. However, given the originality of the authors’ core research idea is not in doubt, we have given the authors the option to submit a revised version of their article. The authors have been informed of this and they have agreed to retract the article. We have been informed in our decision-making by our policy on publishing ethics and integrity and the COPE guidelines on retractions. The retracted article will remain online to maintain the scholarly record, but it will be digitally watermarked on each page as ‘Retracted’.

Energy efficiency of cache collaboration in core content distribution networks

With escalating demands for high-definition video, cache collaboration allows neighbor nodes to share locally stored content in order to reduce download traffic. High energy consumption associated with content delivery remains a concern for Content Distribution Networks (CDNs). Therefore, this paper proposes cluster-based collaborative caching in a core network employing IP over WDM. The aim is to allow sets of core caches to fully share content while minimizing power. A Mixed Integer Linear Programming (MILP) model is used to form energy-efficient cache clusters. The energy consumption of the network is evaluated under different cluster sizes to find the optimum size that minimizes energy. To evaluate the influence of content popularity distribution, a heavy-tailed Zipf distribution and an Equal popularity distribution are evaluated. In addition, the work investigates the influence of downlink traffic behavior and power consumption parameters on optimum cluster sizes. Attained results reveal that maximum savings in energy consumption introduced by cluster-based collaborative caching are up to 34.3% and 21.8% under the Zipf and Equal distribution, respectively. Cache collaboration is not recommended when all core nodes contain fully replicated content servers. Results also show that power consumption parameters do not influence cluster formation. It is recommended keeping cache collaboration in the core network simple, so as to reduce intra-cluster communication.

Learning to Sketch: A Neural Approach to Item Frequency Estimation in Streaming Data.

Recently, there has been a trend of designing neural data structures to go beyond handcrafted data structures by leveraging patterns of data distributions for better accuracy and adaptivity. Sketches are widely used data structures in real-time web analysis, network monitoring, and self-driving to estimate item frequencies of data streams within limited space. However, existing sketches have not fully exploited the patterns of the data stream distributions, making it challenging to tightly couple them with neural networks that excel at memorizing pattern information. Starting from the premise, we envision a pure neural data structure as a base sketch, which we term the meta-sketch, to reinvent the base structure of conventional sketches. The meta-sketch learns basic sketching abilities from meta-tasks constituted with synthetic datasets following Zipf distributions in the pre-training phase and can be quickly adapted to real (skewed) distributions in the adaption phase. The meta-sketch not only surpasses its competitors in sketching conventional data streams but also holds good potential in supporting more complex streaming data, such as multimedia and graph stream scenarios. Extensive experiments demonstrate the superiority of the meta-sketch and offer insights into its working mechanism.

Discrete Extremes

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable are commonly modeled using the generalized Pareto distribution, a peak-over-threshold method that often gives good results in practice. When data is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

Per-capita change rates of the Phanerozoic Earth-life system exhibited Zipf distributions

The Zipf distribution is widespread in physical and biological systems on the modern Earth; whether such distributions existed in the ancient Earth-life system, however, remains understudied. Here I demonstrate that per-capita change rates (PCCRs) of the atmospheric CO2 level, global average temperature, genus richness, and body size during the Phanerozoic exhibited Zipf distributions. Statistical analyses, including the goodness-of-fit test, likelihood-ratio test, and complementary cumulative distribution function (CCDF), support these Zipf distributions. Moreover, correlation analyses show that these environmental variables and biological metrics are correlated while their PCCRs are not, suggesting that Zipf distributions in PCCRs are unlikely to directly connect to each other; instead, there probably existed a set of common but unknown factors responsible for these systematic Zipf distributions in PCCRs. The results in this study offer new perspectives on the evolutionary patterns of the ancient Earth-life system.

Elastically accelerating lookup on virtual SDN flow tables for software-defined cloud gateways

In recent years, Software-Defined Networking has been gradually applied to cloud gateways to provide efficient, reliable and flexible data transmission services for various large-scale cloud platforms. However, massive concurrent accessing produces huge network traffic from tenants to cloud platforms, which aggregates at cloud gateways and brings serious performance bottlenecks regarding packet classification. To solve this problem, this paper proposes an elastically accelerated lookup method of virtual SDN flow tables for software-defined cloud gateways. The method caches active exact flows in virtue of network traffic locality, enabling most packets to directly hit the cache and bypass tuple space search, which significantly accelerates flow table lookup. Focusing on network traffic jitters, the cache adaptively adjusts its capacity according to the dynamic changes of the number of active exact flows to maintain high cache hit rates, aiming to achieve elastic acceleration of flow table lookup. Furthermore, we theoretically derive the performance metrics of our proposed method such as cache hit rates, cache yield rates and average search length, based on the Zipf distribution model of network traffic. Eventually, we evaluate the performance of our proposed elastically accelerated lookup method by experiments with real network traffic traces. Experimental results indicate that our proposed method outperforms existing cache-accelerated methods with stable cache hit rates around 80% and the speedup of average search length up to 2.84 even under network traffic jitters.

Consciousness influences the enhancement of visual statistical learning in Zipfian distributions.

It has been reported that visual statistical learning (VSL) is facilitated in skewed distributions. However, it remains unclear whether enhancement of VSL in Zipfian distributions is due to consciousness of the regularities presented at high frequency. This study addressed this issue. We measured participants' subjective confidence in regularities and awareness of regularities during familiarization by combining a previously reported procedure for VSL with a postdecision wagering task and posttest questionnaire. The results demonstrated that Zipfian distribution enhanced not only VSL but also metacognitive sensitivity, particularly for high-frequency regularities, as the effects of consciousness on VSL were limited to high-frequency regularities. Moreover, the results indicated that awareness during familiarization mediated VSL enhancement in the Zipfian distribution. These results suggest that VSL for events with high-frequency regularities plays an important role in the cognition of events with low-frequency regularities via awareness. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

The Dimensionality of Genetic Information

This paper investigates the dimensionality of genetic information from the perspective of optimal representation. Recently it has been shown that optimal coding of information is in terms of the noninteger dimension of e, which is accompanied by the property of scale invariance. Since Nature is optimal, we should see this dimension reflected in the organization of the genetic code. With this as background, this paper investigates the problem of the logic behind the nature of the assignment of codons to amino acids, for they take different values that range from 1 to 6. It is shown that the non-uniformity of this assignment, which goes against mathematical coding theory that demands a near uniform assignment, is consistent with noninteger dimensions. The reason why the codon assignment for different amino acids varies is because uniformity is a requirement for optimality only in a standard vector space, and is not so in the noninteger dimensional space. It is noteworthy that there are 20 different covering regions in an e-dimensional information space, which is equal to the number of amino acids. The problem of the visualization of data that originates in an e-dimensional space but examined in a 3-dimensional vector space is also discussed. It is shown that the assignment of the codons to the amino acids is fractal-like that is well modeled by the Zipf distribution which is a power law. It is remarkable that the Zipf distribution that holds for the letter frequencies of words in a natural language also applies to the rank order of triplets in the code for amino acids.

Meta-Sketch: A Neural Data Structure for Estimating Item Frequencies of Data Streams

To estimate item frequencies of data streams with limited space, sketches are widely used in real applications, including real-time web analytics, network monitoring, and self-driving. Sketches can be viewed as a model which maps the identifier of a stream item to the corresponding frequency domain. Starting from the premise, we envision a neural data structure, which we term the meta-sketch, to go beyond the basic structure of conventional sketches. The meta-sketch learns basic sketching abilities from meta-tasks constituted with synthetic datasets following Zipf distributions in the pre-training phase, and can be fast adapted to real (skewed) distributions in the adaption phase. Extensive experiments demonstrate the performance gains of the meta-sketch and offer insights into our proposals.

Use of locative postposition-verb construction in Korean: analysis of L1-Korean corpora and L2-Korean textbooks

A usage-based account argues that the actual experience of language use greatly affects the course of language acquisition. On the basis of this premise, this study investigates how language textbooks, as a main input source for L2 learners, reflect the properties of a target language. We focus on a Korean locative postposition–verb construction, consisting of one of the three locative postpositions (– ey, – eyse and –( u) lo) and particular sets of verbs. For this purpose, we adopt representative L1 written/spoken corpora in Korean and two L2-Korean textbook corpora, and analysed three aspects of them: frequency of these postpositions and verbs, their association strength, and change in use of the construction’s components across the corpora. We find that ( i) verb types co-occurring with each postposition follow the Zipfian distribution in both datasets, ( ii) L1 corpora and L2 textbooks are inconsistent with their manifestation of postposition–verb associations within the construction, and ( iii) the two textbook types differ in verb use co-occurring with the postpositions. The implications of this study’s findings are discussed in terms of the acquisitional benefits of constructional frames and using corpora for L2 learning and teaching

Optimal Delay-Outage Analysis for Noise-Limited Wireless Networks With Caching, Computing, and Communications

Performance assessment and optimization for networks jointly performing caching, computing, and communication (3C) has recently drawn significant attention because many emerging applications require 3C functionality. However, studies in the literature mostly focus on the particular algorithms and setups of such networks, while their theoretical understanding and characterization has been less explored. To fill this gap, this paper conducts the asymptotic (scaling-law) analysis for the delay-outage tradeoff of noise-limited wireless edge networks with joint 3C. In particular, assuming the user requests for different tasks following a Zipf distribution, we derive the analytical expression for the optimal caching policy. Based on this, we next derive the closed-form expression for the optimum outage probability as a function of delay and other network parameters for the case that the Zipf parameter is smaller than 1. Then, for the case that the Zipf parameter is larger than 1, we derive the closed-form expressions for upper and lower bounds of the optimum outage probability. We provide insights and interpretations based on the derived expressions. Computer simulations validate our analytical results and insights.

Zipfian Distributions in Child-Directed Speech.

Across languages, word frequency and rank follow a power law relation, forming a distribution known as the Zipfian distribution. There is growing experimental evidence that this well-studied phenomenon may be beneficial for language learning. However, most investigations of word distributions in natural language have focused on adult-to-adult speech: Zipf's law has not been thoroughly evaluated in child-directed speech (CDS) across languages. If Zipfian distributions facilitate learning, they should also be found in CDS. At the same time, several unique properties of CDS may result in a less skewed distribution. Here, we examine the frequency distribution of words in CDS in three studies. We first show that CDS is Zipfian across 15 languages from seven language families. We then show that CDS is Zipfian from early on (six-months) and across development for five languages with sufficient longitudinal data. Finally, we show that the distribution holds across different parts of speech: Nouns, verbs, adjectives and prepositions follow a Zipfian distribution. Together, the results show that the input children hear is skewed in a particular way from early on, providing necessary (but not sufficient) support for the postulated learning advantage of such skew. They highlight the need to study skewed learning environments experimentally.

Zipf’s Law, Benford’s Law, and Pareto Rule

From a basic probabilistic argumentation, the Zipfian distribution and Benford’s law are derived. It is argued that Zipf’s law fits to calculate the rank probabilities of identical indistinguishable objects and that Benford’s distribution fits to calculate the rank probabilities of distinguishable objects. i.e. in the distribution of words in long texts all the words in a given rank are identical, therefore, the rank distribution is Zipfian. In logarithmic tables, the objects with identical 1st digits are distinguishable as there are many different digits in the 2nd, 3rd… places, etc., and therefore the distribution is according to Benford’s law. Pareto 20 - 80 rule is shown to be an outcome of Benford’s distribution as when the number of ranks is about 10 the probability of 20% of the high probability ranks is equal to the probability of the rest of 80% low probability ranks. It is argued that all these distributions, including the central limit theorem, are outcomes of Planck’s law and are the result of the quantization of energy. This argumentation may be considered a physical origin of probability.

The meaning distributions on different levels of granularity

The meaning distributions of certain linguistic forms generally follow a Zipfian distribution. However, since the meanings can be observed and classified on different levels of granularity, it is thus interesting to ask whether their distributions on different levels can be fitted by the same model and whether the parameters are the same. In this study, we investigate three quasi-prepositions in Shanghainese, a dialect of Wu Chinese, and test whether the meaning distributions on two levels of granularity can be fitted by the same model and whether the parameters are close. The results first show that the three models proposed by modern quantitative linguists can both achieve a good fit for all cases, while both the exponential (EXP) model and the right-truncated negative binomial (RTBN) models behave better than the modified right-truncated Zipf-Alekseev distribution (MRTZA), in terms of the consistency of the goodness of fit, parameter change, rationality, and simplicity. Second, the parameters of the distributions on the two levels and the curves are not exactly the same or even close to each other. This has supported a weak view of the concept of ‘scaling’ in complex sciences. Finally, differences are found to lie between the distributions on the two levels. The fine-grained meaning distributions are more right-skewed and more non-linear. This is attributed to the openness of the categories of systems. The finer semantic differentiation behaves like systems with open set of categories, while the coarse-grained meaning distribution resembles those having a close set of few categories.

Revisiting the gravity laws of inter-city mobility in megacity regions

Inter-city mobility is one of the most important issues in the UN Sustainable Development Goals, as it is essential to access the regional labour market, goods and services, and to constrain the spread of infectious diseases. Although the gravity model has been proved to be an effective model to describe mobility among settlements, knowledge is still insufficient in regions where dozens of megacities interact closely and over 100 million people reside. In addition, the existing knowledge is limited to overall population mobility, while the difference in inter-city travel with different purposes is unexplored on such a large geographic scale. We revisited the gravity laws of inter-city mobility using the 2.12 billion trip chains recorded by 40.48 million mobile phone users’ trajectories in the Jing-Jin-Ji Region, which contains China’s capital Beijing. Firstly, unlike previous studies, we found that non-commuting rather than commuting is the dominant type of inter-city mobility (89.3%). Non-commuting travellers have a travel distance 42.3% longer than commuting travellers. Secondly, we developed more accurate gravity models for the spatial distribution of inter-city commuting and non-commuting travel. We also found that inter-city mobility has a hierarchical structure, as the distribution of inter-city travel volume follows Zipf’s law. In particular, the hierarchy of non-commuting travel volume among the cities is more in line with an ideal Zipf distribution than commuting travel. Our findings contribute to new knowledge on basic inter-city mobility laws, and they have significant applications for regional policies on human mobility.

An Efficient Video Encryption Algorithm Based on the Pseudorandom Number Generator of Zipf Distribution

In this study, a pioneer selective video encryption (PSVE) algorithm is proposed based on the pseudorandom number generator (PRNG) of the Zipf distribution (Z-PRNG). It is a general algorithm with high efficiency and security. The encryption process is completely separable from the video coding process. In the PSVE algorithm, Z-PRNG is designed based on the 3D SCL-HMC hyperchaotic map. Firstly, encapsulated byte sequence payloads (EBSPs) are extracted from the video bitstream. Secondly, random numbers of the Zipf distribution are generated by Z-PRNG, and they are used to randomly select encrypted data from each EBSP. Lastly, the extracted data are encrypted by AES-CTR to obtain the encrypted video. Compared with existing algorithms, the encryption position is more flexible, and the key space is further enhanced. The high efficiency video coding (HEVC) video and the advanced video coding (AVC) video are taken as examples to test the PSVE algorithm. The analysis results show that the proposed scheme can effectively resist common attacks, and its time complexity is much less than most existing algorithms.