Zipf's Law, Power Laws and Music Aesthetics

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.

Zipf's law, power laws and maximum entropy

New Journal of Physics ◽

10.1088/1367-2630/15/4/043021 ◽

2013 ◽

Vol 15 (4) ◽

pp. 043021 ◽

Cited By ~ 33

Author(s):

Matt Visser

Keyword(s):

Maximum Entropy ◽

Power Laws ◽

Zipf’S Law ◽

Zipf's Law

The logarithmic Zipf law in a general urn problem

ESAIM Probability and Statistics ◽

10.1051/ps/2020011 ◽

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.

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

10.31235/osf.io/gwha2 ◽

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.

Power laws, Pareto distributions and Zipf's law

Contemporary Physics ◽

10.1080/00107510500052444 ◽

2005 ◽

Vol 46 (5) ◽

pp. 323-351 ◽

Cited By ~ 2793

Author(s):

MEJ Newman

Keyword(s):

Power Laws ◽

Zipf’S Law ◽

Zipf's Law ◽

Pareto Distributions

Types, Tokens, and Hapaxes: A New Heap’s Law

Glottotheory ◽

10.1515/glot-2018-0014 ◽

2019 ◽

Vol 9 (2) ◽

pp. 113-129

Author(s):

Victor Davis

Keyword(s):

First Principles ◽

Scaling Law ◽

Written Language ◽

Zipf’S Law ◽

Academic Press ◽

Zipf's Law ◽

New Words ◽

Size Dependent ◽

Link Type ◽

Combinatorial Model

Abstract Heap’s Law https://dl.acm.org/citation.cfm?id=539986 Heaps, H S 1978 Information Retrieval: Computational and Theoretical Aspects (Academic Press). states that in a large enough text corpus, the number of types as a function of tokens grows as N = K{M^\beta } for some free parameters K, \beta . Much has been written http://iopscience.iop.org/article/10.1088/1367-2630/15/9/093033 Font-Clos, Francesc 2013 A scaling law beyond Zipf’s law and its relation to Heaps’ law (New Journal of Physics 15 093033)., http://iopscience.iop.org/article/10.1088/1367-2630/11/12/123015 Bernhardsson S, da Rocha L E C and Minnhagen P 2009 The meta book and size-dependent properties of written language (New Journal of Physics 11 123015)., http://iopscience.iop.org/article/10.1088/1742-5468/2011/07/P07013 Bernhardsson S, Ki Baek and Minnhagen 2011 A paradoxical property of the monkey book (Journal of Statistical Mechanics: Theory and Experiment, Volume 2011)., http://milicka.cz/kestazeni/type-token_relation.pdf Milička, Jiří 2009 Type-token & Hapax-token Relation: A Combinatorial Model (Glottotheory. International Journal of Theoretical Linguistics 2 (1), 99–110)., https://www.nature.com/articles/srep00943 Petersen, Alexander 2012 Languages cool as they expand: Allometric scaling and the decreasing need for new words (Scientific Reports volume 2, Article number: 943). about how this result and various generalizations can be derived from Zipf’s Law. http://dx.doi.org/10.1037/h0052442 Zipf, George 1949 Human behavior and the principle of least effort (Reading: Addison-Wesley). Here we derive from first principles a completely novel expression of the type-token curve and prove its superior accuracy on real text. This expression naturally generalizes to equally accurate estimates for counting hapaxes and higher n-legomena.

The Distribution of City Sizes in Turkey: A Failure of Zipf’s Law Due to Concavity

Regional Science Policy & Practice ◽

10.1111/rsp3.12449 ◽

2021 ◽

Author(s):

Hasan Engin Duran ◽

Andrzej Cieślik

Keyword(s):

Zipf’S Law ◽

Zipf's Law

Dynamical approach to Zipf's law

Physical Review Research ◽

10.1103/physrevresearch.3.013084 ◽

2021 ◽

Vol 3 (1) ◽

Author(s):

Giordano De Marzo ◽

Andrea Gabrielli ◽

Andrea Zaccaria ◽

Luciano Pietronero

Keyword(s):

Zipf’S Law ◽

Zipf's Law ◽

Dynamical Approach

Optimization of morpheme length: a cross-linguistic assessment of Zipf’s and Menzerath’s laws

Linguistics Vanguard ◽

10.1515/lingvan-2019-0076 ◽

2021 ◽

Vol 7 (s3) ◽

Author(s):

Matthew Stave ◽

Ludger Paschen ◽

François Pellegrino ◽

Frank Seifart

Keyword(s):

Structural Information ◽

Unit Length ◽

Zipf’S Law ◽

Zipf's Law ◽

Linguistic Structure ◽

Morphological Complexity ◽

Linguistic Assessment ◽

Linguistic Units

Abstract Zipf’s Law of Abbreviation and Menzerath’s Law both make predictions about the length of linguistic units, based on corpus frequency and the length of the carrier unit. Each contributes to the efficiency of languages: for Zipf, units are more likely to be reduced when they are highly predictable, due to their frequency; for Menzerath, units are more likely to be reduced when there are more sub-units to contribute to the structural information of the carrier unit. However, it remains unclear how the two laws work together in determining unit length at a given level of linguistic structure. We examine this question regarding the length of morphemes in spoken corpora of nine typologically diverse languages drawn from the DoReCo corpus, showing that Zipf’s Law is a stronger predictor, but that the two laws interact with one another. We also explore how this is affected by specific typological characteristics, such as morphological complexity.