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 Understanding scaling through history-dependent processes with collapsing sample space

2015 ◽  
Vol 112 (17) ◽  
pp. 5348-5353 ◽  
Author(s):  
Bernat Corominas-Murtra ◽  
Rudolf Hanel ◽  
Stefan Thurner
Keyword(s):  
Complex Systems ◽  
Diffusion Processes ◽  
Social Systems ◽  
Sample Space ◽  
Zipf’S Law ◽  
Scaling Exponent ◽  
Zipf's Law ◽  
Wide Range ◽  
Dependent Processes ◽  
Rank Distributions

History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space-reducing (SSR) processes necessarily lead to Zipf’s law in the rank distributions of their outcomes. We show that by adding noise to SSR processes the corresponding rank distributions remain exact power laws, p(x)∼x−λ, where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from α=2 to ∞. We discuss several applications showing how SSR processes can be used to understand Zipf’s law in word frequencies, and how they are related to diffusion processes in directed networks, or aging processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organized critical processes.

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 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 The Hierarchical Organization of Syntax

2021 ◽  
Author(s):  
Babak Ravandi ◽  
Valentina Concu
Keyword(s):  
Complex Systems ◽  
Complex Adaptive Systems ◽  
Adaptive Systems ◽  
Language Evolution ◽  
Hierarchical Organization ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Syntactic Structures ◽  
Complex Adaptive

Abstract Hierarchies are the backbones of complex systems and their analysis allows for a deeper understanding of their structure and how they evolve. We consider languages to be also complex adaptive systems. Hence, we analyzed the hierarchical organization of historical syntactic networks from German that were created from a corpus of texts from the 11th to 17th centuries. We tracked the emergence of syntactic structures in these networks and mapped them to specific communicative needs. We named these emerging structures communicative hierarchies. We hypothesise that the communicative needs of speakers are the organizational force of syntax. We propose that the emergence of these multiple communicative hierarchies is what shapes syntax, and that these hierarchies are the prerequisite to the Zipf's law. The emergence of communicative hierarchies indicates that the objective of language evolution is not only to increase the efficiency of transferring information. Language is also evolving to increase our capacity to communicate more sophisticated abstractions as we advance as a species.

Zipf's Law, Power Laws and Music Aesthetics

2011 ◽  
pp. 169-216 ◽  
Author(s):  
Bill Manaris ◽  
Patrick Roos ◽  
Dwight Krehbiel ◽  
Thomas Zalonis ◽  
J Armstrong
Keyword(s):  
Power Laws ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Music Aesthetics

Ibn Khaldun’s Theory of Social Change

1998 ◽  
Vol 15 (2) ◽  
pp. 25-45 ◽  
Author(s):  
Fida Mohammad
Keyword(s):  
Industrial Revolution ◽  
Natural World ◽  
Social Knowledge ◽  
Philosophical Tradition ◽  
Historical Process ◽  
Ibn Khaldun ◽  
Historical Processes ◽  
The Will ◽  
The Individual ◽  
Compare And Contrast

In this article I shall compare and contrast Ibn Khaldun’s ideas aboutsociohistorical change with those of Hegel, Marx, and Durkheim. I willdiscuss and elaborate Ibn Khaldun’s major ideas about historical andsocial change and compare them with three important figures of modemWestern sociology and philosophy.On reading Ibn Khaldun one should remember that he was living in thefourteenth century and did not have the privilege of witnessing the socialdislocation created by the industrial revolution. It is also very difficult tocategorize Ibn Khaldun within a single philosophical tradition. He is arationalist as well as an empiricist, a historicist as well as a believer inhuman agency in the historical process. One can see many “modem”themes in his thinking, although he lived a hundred years beforeMachiavelli.Lauer, who considers Ibn Khaldun the pioneer of modem sociologicalthought, has summarized the main points of his philosophy.’ In his interpretationof Ibn Khaldun, he notes that historical processes follow a regularpattern. However, whereas this pattern shows sufficient regularity, itis not as rigid as it is in the natural world. In this regard the position ofIbn Khaldun is radically different from those philosophies of history thatposit an immutable course of history determined by the will of divineprovidence or other forces. Ibn Khaldun believes that the individual isneither a completely passive recipient nor a full agent of the historicalprocess. Social laws can be discovered through observation and datagathering, and this empirical grounding of social knowledge represents adeparture from traditional rational and metaphysical thinking ...

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

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