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

Abstract

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...

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

“…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.…”

“…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.…”

The Fair Reward Problem: The Illusion of Success and How to Solve It

“…The power law distribution results from the cumulative effect of stochastic (lucky) proportional growth terms, which goes already a long way towards explaining pervasive inequality so that a small fraction of the population controls a large fraction of the resources. See quantitative verifications of this in growing social networks (Zhang and Sornette, 2011), competing electronic products (Hisano et al, 2011) and crypto-currency market shares .…”

A Common Construction Pattern of English Words and Chinese Characters

“…For example, in the English language, the frequency, , of encountering the th most common word is inversely proportional to rank order (namely, ), as indicated by Zipf's law [2], [3]. Besides linguistics [2]–[10], Zipfian type power laws (that include Zipf's law [2], [3] and its many extensions [4]–[25] given by with , where is a non-zero positive constant and is either rank orders or item's quantities that can be ranked, say, firm sizes [23]) have been observed and studied in many disciplines like physics [18]–[22], acoustics [11], biology [12], [13], economics or finance [15], [16], [24], sociology [17], [23], [25], and architectonics [14]. However, although many rankings can be described by Zipfian type power laws [2]–[25], many others can not, e.g., in communications [26] and linguistics [27], [28].…”