The logarithmic Zipf law in a general urn problem

2020 ◽  
Vol 24 ◽  
pp. 275-293
Author(s):  
Aristides V. Doumas ◽  
Vassilis G. Papanicolaou
Keyword(s):  
Social Sciences ◽  
Power Law ◽  
Random Variable ◽  
Power Laws ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
The Social ◽  
Leading Term ◽  
Zipf Law ◽  
Highly Cited

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 > 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.

scholarly journals ZIPF'S LAW AND RANDOM TEXTS

2002 ◽  
Vol 05 (01) ◽  
pp. 1-6 ◽  
Author(s):  
RAMON FERRER i CANCHO ◽  
RICARD V. SOLÉ
Keyword(s):  
Word Frequency ◽  
Power Law ◽  
Word Length ◽  
A Priori ◽  
Zipf’S Law ◽  
Strict Sense ◽  
A Priori Information ◽  
Zipf's Law ◽  
Priori Information ◽  
Random Text

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.

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

2020 ◽  
Vol 17 (4) ◽  
pp. 1460 ◽  
Author(s):  
Yizhen Wu ◽  
Mingyue Jiang ◽  
Zhijian Chang ◽  
Yuanqing Li ◽  
Kaifang Shi
Keyword(s):  
Urban Development ◽  
Infrared Imaging ◽  
Multiple Scales ◽  
Decision Makers ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Nighttime Light Data ◽  
Nighttime Light ◽  
Zipf Law ◽  
The Ideal

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.

scholarly journals Zipf's Law, Hierarchical Structure, and Cards-Shuffling Model for Urban Development

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Yanguang Chen
Keyword(s):  
Complex Systems ◽  
Hierarchical Structure ◽  
Spatial Dynamics ◽  
Social Institutions ◽  
Power Laws ◽  
Natural World ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Cascade Structure ◽  
The Individual

Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world and social institutions. Where there is a hierarchy with cascade structure, there is a Zipf's rank-size distribution, andvice versa. However, we have no theory to explain the spatial dynamics associated with Zipf's law of cities. In this paper, a new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random distribution of cities. The hierarchical structure can be described with a set of exponential functions that are identical in form to Horton-Strahler's laws on rivers and Gutenberg-Richter's laws on earthquake energy. From the exponential models, we can derive four power laws including Zipf's law indicative of fractals and scaling symmetry. A card-shuffling model is built to interpret the relation between Zipf's law and hierarchy of cities. This model can be expanded to illuminate the general empirical power-law distributions across the individual physical and social sciences, which are hard to be comprehended within the specific scientific domains. This research is useful for us to understand how complex systems such as networks of cities are self-organized.

scholarly journals Zipf's law, power laws and maximum entropy

2013 ◽  
Vol 15 (4) ◽  
pp. 043021 ◽  
Author(s):  
Matt Visser
Keyword(s):  
Maximum Entropy ◽  
Power Laws ◽  
Zipf’S Law ◽  
Zipf's Law

scholarly journals Empirical analysis of Zipf’s law, power law, and lognormal distributions in medical discharge reports

2021 ◽  
Vol 145 ◽  
pp. 104324
Author(s):  
Juan C Quiroz ◽  
Liliana Laranjo ◽  
Catalin Tufanaru ◽  
Ahmet Baki Kocaballi ◽  
Dana Rezazadegan ◽  
...  
Keyword(s):  
Empirical Analysis ◽  
Power Law ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Lognormal Distributions

scholarly journals Zipf's law and phase transition

2014 ◽  
Vol 28 (11) ◽  
pp. 1450088 ◽  
Author(s):  
K. Lukierska-Walasek ◽  
K. Topolski
Keyword(s):  
Phase Transition ◽  
Zipf’S Law ◽  
Statistical Distributions ◽  
Zipf's Law ◽  
Potts Models ◽  
Fixed Domain ◽  
Zipf Law

In this paper, we describe the link between the Zipf law and statistical distributions for the Fortuin–Kasteleyn clusters in Ising as well as Potts models. From these results, it is seen that Zipf's law can be a criterion of a phase transition, but it does not determine its order. We present the corresponding histograms for fixed domain configurations.

scholarly journals Power Laws, Prices and National Accounts: Are There Any Irregularities?

2020 ◽  
Author(s):  
Ciprian Florin Pater ◽  
Deni Mazrekaj
Keyword(s):  
Data Quality ◽  
Price Index ◽  
Consumer Price Index ◽  
Power Laws ◽  
Zipf’S Law ◽  
Benford’S Law ◽  
National Accounts ◽  
Zipf's Law ◽  
High Data ◽  
Benford's Law

Many economic regularities have been found to adhere to power laws. In this paper, we apply Benford’s law to consumer price index data from Norway and Zipf’s law on a Norwegian report about the history of Norwegian national accounts. Norway is a particularly interesting country to study as it scores among the highest-ranked countries on data quality. We find that the consumer price index adheres to Benford’s law, showing high data quality. On the other hand, our results do indicate that the report does not adhere to Zipf’s law.

scholarly journals Re-evaluating Phoneme Frequencies

2020 ◽  
Vol 11 ◽  
Author(s):  
Jayden L. Macklin-Cordes ◽  
Erich R. Round
Keyword(s):  
Maximum Likelihood ◽  
Power Law ◽  
Power Laws ◽  
Linguistic Variables ◽  
Zipf's Law ◽  
Causal Processes ◽  
Frequency Structure ◽  
Future Work ◽  
Australian Languages ◽  
Over Time

Causal processes can give rise to distinctive distributions in the linguistic variables that they affect. Consequently, a secure understanding of a variable's distribution can hold a key to understanding the forces that have causally shaped it. A storied distribution in linguistics has been Zipf's law, a kind of power law. In the wake of a major debate in the sciences around power-law hypotheses and the unreliability of earlier methods of evaluating them, here we re-evaluate the distributions claimed to characterize phoneme frequencies. We infer the fit of power laws and three alternative distributions to 166 Australian languages, using a maximum likelihood framework. We find evidence supporting earlier results, but also nuancing them and increasing our understanding of them. Most notably, phonemic inventories appear to have a Zipfian-like frequency structure among their most-frequent members (though perhaps also a lognormal structure) but a geometric (or exponential) structure among the least-frequent. We compare these new insights the kinds of causal processes that affect the evolution of phonemic inventories over time, and identify a potential account for why, despite there being an important role for phonetic substance in phonemic change, we could still expect inventories with highly diverse phonetic content to share similar distributions of phoneme frequencies. We conclude with priorities for future work in this promising program of research.

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