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We introduce a solvable model of randomly growing systems consisted of many independent subunits. Scaling relations and growth rate distributions in the limit of infinite subunits are analyzed theoretically. Various types of scaling properties and distributions reported for growth rates of complex systems in wide fields can be derived from this basic physical model. Statistical data of growth rates for about 1 million business firms are analyzed as an example of randomly growing systems in the real-world. Not only scaling relations are consistent with the theoretical solution, the whole functional form of the growth rate distribution is fitted with a theoretical distribution having a power law tail.
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... Being referred as random multiplicative process or Kesten process [13][14][15], Equation (2) and its variations also appear in the modeling of growth of entities, either biological populations or in social context, e.g., companies and cities sizes [16][17][18][19][20]. The presence of the Kesten process in such growth models is explained by two basic ingredients: the Gibrat's law of proportional growth-the growth of an entity is proportional to its current size but with stochastic growth rates independent of it [21]-and a surviving mechanism to prevent the collapse to zero [14]. ...
... To check the transition from pure Kesten between-dependence (q = 0) to between-independence (q = 1), we choose the values q = 0.1, 0.25, 0.5, 0.75, and 0.9. Naturally, the exact values of probabilities differ for each value of parameter q (see zoom-in panel (b)) but bounds in Equation (20) are respected, including the same tail-index α = − log p log θ for all cases. ...
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The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.
... The successive differences are rescaled with the fluctuation width (i.e. scale factor) by stocks, where the scale factor is estimated by the root-median-square successive differences of the (detrended) recruitment series (Takayasu et al. 2014). The rescaled distributions with zero mean and unity width, when aggregated across 72 stocks, fit a symmetric Lévy-stable distribution with exponentα = 1.42 (maximum likelihood estimate); see Figure 1b (solid circles with solid line). ...
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Recruitment is calculated by summing random offspring-numbers entering the population, where the number of summands (i.e. spawning population size) is also a random process. A priori, it is not clear that individual reproductive variability would have a significant impact on aggregate measures for monitoring populations. Usually these variations are averaged out in a large population, and the aggregate output is merely influenced by population-wide environmental disturbances such as climate and fisheries. However, such arguments break down if the distribution of the individual offspring numbers is heavy-tailed. In a world with power-law offspring-number distribution with exponent $1<\alpha<2$, the recruitment distribution has a putative power-law regime in the tail with the same $\alpha$. The question is to what extent individual reproductive variability can have a noticeable impact on the recruitment under environmentally driven population fluctuations. This question is answered by considering the L\'evy-stable fluctuations as embedded in a randomly varying environment. I report fluctuation scaling and asymmetric fluctuations in recruitment of commercially exploited fish stocks throughout the North Atlantic. The linear scaling of recruitment standard deviation with recruitment level implies that the individual reproductive variability is dominated by population fluctuations. The totally asymmetric (skewed to the right) character is a sign of idiosyncratic variation in reproductive success.
... This suggests a simple picture emerging at the population level; however varying the growth rates among agents in the population will introduce complications. The most direct route to assess these effects follows by positing Gaussian distributions on growth rates, from the asymptotic behavior of growth rate distributions [22,26,27], and initial log resources across the population, from the solution to the FPE. ...
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Understanding the statistical dynamics of growth and inequality is a fundamental challenge to ecology and society. Recent analyses of wealth and income dynamics in contemporary societies show that economic inequality is very dynamic and that individuals experience substantially different growth rates over time. However, despite a fast growing body of evidence for the importance of fluctuations, we still lack a general statistical theory for understanding the dynamical effects of heterogeneneous growth across a population. Here we derive the statistical dynamics of correlated growth rates in heterogeneous populations. We show that correlations between growth rate fluctuations at the individual level influence aggregate population growth, while only driving inequality on short time scales. We also find that growth rate fluctuations are a much stronger driver of long-term inequality than earnings volatility. Our findings show that the dynamical effects of statistical fluctuations in growth rates are critical for understanding the emergence of inequality over time and motivate a greater focus on the properties and endogenous origins of growth rates in stochastic environments.
... Verifying previously hypothesized stylized facts and checking the mathematical consistency of independently known power-law exponents help to validate or refute possible theories regarding business firms. Numerous models [35,[48][49][50][51][52][53][54][55][56][57][58][59][60][61] have been suggested to explain only a few stylized facts relating to firms, and it appears that new criteria are needed to select the models empirically. Thus, our results for multivariate scaling should be incorporated into subsequent theoretical considerations. ...
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... Our example model also explains another scaling phenomenon, a 'tent-shaped' probability density for the aggregate growth rate g t , which often occurs in combination with a scalefree distribution in many real-world examples [22][23][24][25][26][27][28][29][30][31][32][33]. Tent-shaped growth rate probabilities are also generated by other preferential-attachment models like BA, but they are not produced by other families of models for scalefree distributions. ...
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... Recent developments in the field of complexity science have led to a renewed interest in social and economic activities [1][2][3] being captured as complex systems characterised properties such as small world [4] and scale-free [5]. It has previously been observed that business firm PLOS networks belong to a class of the complex systems interacting with others in distinct ways, accompanied by specific scaling relations [6][7][8][9][10][11][12][13][14][15]. A typical example of business firm networks is the inter-firm business transactions network, whereby nodes are firms that link through business transactions from customers and suppliers, producing a directional money flow (with the opposite direction being the goods/ service flows). ...
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