Zipf's and Taylor's laws

@article{James2018ZipfsAT,
  title={Zipf's and Taylor's laws},
  author={Charlotte James and Sandro Azaele and Amos Maritan and Filippo Simini},
  journal={Physical Review E},
  year={2018}
}
Zipf's law states that the frequency of an observation with a given value is inversely proportional to the square of that value; Taylor's law, instead, describes the scaling between fluctuations in the size of a population and its mean. Empirical evidence of the validity of these laws has been found in many and diverse domains. Despite the numerous models proposed to explain the presence of Zipf's law, there is no consensus on how it originates from a microscopic process of individual dynamics… 

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References

SHOWING 1-10 OF 69 REFERENCES

Zipf's law and criticality in multivariate data without fine-tuning.

It is shown that Zipf's law arises generically for large systems, without fine-tuning parameters to a point, if there is a fluctuating unobserved variable (or variables) that affects the system, such as a common input stimulus that causes individual neurons to fire at time-varying rates.

Zipf’s Law Arises Naturally When There Are Underlying, Unobserved Variables

Theoretically, and empirically, this work provides a far simpler and more intuitive explanation of Zipf’s law, which at the same time considerably extends the class of models to which this explanation can apply.

Fluctuation scaling in complex systems: Taylor's law and beyond

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element

Zipf's law unzipped

It is argued that the reason that Zipf's law gives a good description of data from seemingly completely unrelated phenomena is that they can all be described as outcomes of a ubiquitous raison d'action.

Power laws, Pareto distributions and Zipf's law

When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the

Sample and population exponents of generalized Taylor’s law

This study shows that limited sampling hinders the anticipation of such transitions and provides estimates for the number of samples required to reveal early warning signals of abrupt biotic change and derives a generalized TL in terms of sample and population exponents bjk for the scaling of the kth vs. the jth cumulants.

Understanding scaling through history-dependent processes with collapsing sample space

It is demonstrated that sample-space-reducing (SSR) processes necessarily lead to Zipf's law in the rank distributions of their outcomes, and several applications showing how SSR processes can be used to understand Zipf’s law in word frequencies are discussed.

Zipf's Law for Cities in the Regions and the Country

The salient rank-size rule known as Zipf's law is not only satisfied for Germany's national urban hierarchy, but also for the city size distributions in single German regions. To analyze this

Dynamics of Text Generation with Realistic Zipf's Distribution

The model incorporates both features related to the general structure of languages and memory effects inherent to the production of long coherent messages in the communication process and gives support to the linguistic relevance of Zipf's law in human language.

Multiplicative processes and power laws

Takayasu, Sato, and Takayasu [Phys. Rev. Lett. 79, 966 (1997)] revisited the question of stochastic processes with multiplicative noise, which have been studied in several different contexts over the
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