Zipf's law and maximum sustainable growth

Malevergne, Yannick; Саичев, А. И.; Sornette, Didier

doi:10.1016/j.jedc.2013.02.004

Cited by 46 publications

(74 citation statements)

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“…The model suggests that a lognormal size distribution with a power law tail must always occur when elements of an economic system growth with a small mean growth rate and large growth rate fluctuations. Since size distributions similar to the bank size distribution can be often found in social systems (Bouchaud and Mezard 2000, Gabaix 2008, Malevergne et al 2013 it can be expected that its origin can be explained analogous to the presented model.…”

Section: _________________________ 3 Conclusionsupporting

confidence: 62%

Evolutionary Model of the Bank Size Distribution

Kaldasch

2014

Economics

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An evolutionary model of the bank size distribution is presented based on the exchange and creation of deposit money. In agreement with empirical results the derived size distribution is lognormal with a power law tail. The theory is based on the idea that the size distribution is the result of the competition between banks for permanent deposit money. The exchange of deposits causes a preferential growth of banks with a fitness that is determined by the competitive advantage to attract permanent deposits. While growth rate fluctuations are responsible for the lognormal part of the size distribution, treating the mean growth rate of banks as small, large banks benefit from economies of scale generating the Pareto tail. JEL G21 L11 E11

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Section: _________________________ 3 Conclusionsupporting

confidence: 62%

Evolutionary Model of the Bank Size Distribution

Kaldasch

2014

Economics

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“…The superposition of distributions has a neat interpretation: it corresponds to considering entities that are born successively. Combining the features of birth and stochastic proportional growth has been considered in [23,33,34]. Our model can be viewed as the simplest and purest incarnation of these mechanisms.…”

Section: Introductionmentioning

confidence: 99%

“…This allows us to examine dependence among entities in three particular different classes: complete independence; Kesten dependence (a dependence based on the Kesten process); and mixed dependence, combining both independence and Kesten dependence. Rather than studying the cross-section at a given time of the sizes of entities present in the system (as done, e.g., in [23,33,34]), we study the distribution of the sum of sizes of all entities in the system. This corresponds to the total capitalization of a country, when entities are firms, or to the total biomass of an ecosystem for biological populations.…”

Section: Introductionmentioning

confidence: 99%

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

Sousa

Takayasu

Sornette

et al. 2017

Entropy

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Abstract:We introduce a simple growth model in which the sizes of entities evolve as multiplicative random processes that start at different times. A novel aspect we examine is the dependence among entities. For this, we consider three classes of dependence between growth factors governing the evolution of sizes: independence, Kesten dependence and mixed dependence. We take the sum X of the sizes of the entities as the representative quantity of the system, which has the structure of a sum of product terms (Sigma-Pi), whose asymptotic distribution function has a power-law tail behavior. We present evidence that the dependence type does not alter the asymptotic power-law tail behavior, nor the value of the tail exponent. However, the structure of the large values of the sum X is found to vary with the dependence between the growth factors (and thus the entities). In particular, for the independence case, we find that the large values of X are contributed by a single maximum size entity: the asymptotic power-law tail is the result of such single contribution to the sum, with this maximum contributing entity changing stochastically with time and with realizations.

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“…It could be shown that, to a large extent, the characteristics of the distribution of initial sizes is irrelevant to the shape of the upper tail of the steady-state distribution of entities [30].…”

Section: Modelmentioning

confidence: 99%

“…Malevergne et al [30] also provided the explicit functional relationship between the power-law exponent and its key parameters, predicting deviations from Zipf's law due to various sources (see equation (4) below).…”

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confidence: 99%

Predicted and verified deviations from Zipf’s law in ecology of competing products

2011

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Zipf's power-law distribution is a generic empirical statistical regularity found in many complex systems. However, rather than universality with a single power-law exponent (equal to 1 for Zipf's law), there are many reported deviations that remain unexplained. A recently developed theory finds that the interplay between (i) one of the most universal ingredients, namely stochastic proportional growth, and (ii) birth and death processes, leads to a generic power-law distribution with an exponent that depends on the characteristics of each ingredient. Here, we report the first complete empirical test of the theory and its application, based on the empirical analysis of the dynamics of market shares in the product market. We estimate directly the average growth rate of market shares and its standard deviation, the birth rates and the "death" (hazard) rate of products. We find that temporal variations and product differences of the observed power-law exponents can be fully captured by the theory with no adjustable parameters. Our results can be generalized to many systems for which the statistical properties revealed by power law exponents are directly linked to the underlying generating mechanism. Power-law distributions constitute ubiquitous statistical features of many natural and social complex phenomena [1][2][3]. The probability distribution function p(s) of a random variable,describes a particularly slow decay with s of the probability p(s)ds that the random variable is found in one realization between s and s + ds. A power-law distribution is such that any of its moment of order q larger than the power-law exponent µ is mathematically infinite. Among power-law distributions, Zipf's law, corresponding to µ = 1, has been proposed as a fundamental characteristic for many systems [4-6]. Zipf's law implies that we no longer have finite means in infinite systems, or that the means are strongly system size dependent in the real world. Motivated by its apparent ubiquity and interesting features, many efforts have been made to attempt explaining the existence of power-law distributions. One of the general mechanisms to generate power-law distributions is embodied in the multiplicative stochastic growth models, having Gibrat's rule of proportional growth [7] as a key ingredient. Expressed in continuous time, Gibrat's rule is equivalent to the well known geometric Brownian motionwhere r denotes a drift, σ is the volatility (or standard deviation) and W (t) is a standard Wiener process. However, Gibrat's rule alone cannot generate a stable powerlaw distribution, since the solution of the process (2) leads to a non-stationary log-normal distribution. Since Herbert Simon's work [8][9][10] that extended previous attempts at explaining Zipf's law [4,[11][12][13]], a huge literature followed that is spread across a variety of disciplines [14][15][16][17][18][19][20][21][22][23]. This has led to the understanding that an apparently minor modification in the multiplicative process applying just when S(t) becomes small suffice...

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Zipf's law and maximum sustainable growth

Cited by 46 publications

References 47 publications

Evolutionary Model of the Bank Size Distribution

Evolutionary Model of the Bank Size Distribution

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

Predicted and verified deviations from Zipf’s law in ecology of competing products

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