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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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All content in this area was uploaded by Misako Takayasu on Jan 29, 2014

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... This is a natural result, as the empir- ical distribution of yearly displacements in the logarithms of the single measures of the system size has been characterized with "tent-shaped" or slightly heavier-tailed than double exponential Laplace distribution for commercial and cultural organizations [35,54,70,71]. Note that these empirical distributions have been approximated differently in previous studies [70,72,73]. Because the distribution of displacements from around a point cannot be appropriately fitted by a Gaussian distribution, the system dynamic may not be approximated by a simple diffusion equation. ...

... We propose that a key step to such a model would be to determine mathematical formulations for the transition probabilities, which are denoted by g(t ) conditional on x(t ) (see also Fig. 1). Growth rate distributions of a single-size measure have been studied for decades [35,54,70,72,73,79]. However, there are insufficient studies on the conditional probability distribution of growth rate that depends on the location on a multidimensional phase space. ...

We propose a data-driven stochastic method that allows the simulation of a complex system's long-term evolution. Given a large amount of historical data on trajectories in a multi-dimensional phase space, our method simulates the time evolution of a system based on a random selection of partial trajectories in the data without detailed knowledge of the system dynamics. We apply this method to a large data set of time evolution of approximately one million business firms for a quarter century. Accordingly, from simulations starting from arbitrary initial conditions, we obtain a stationary distribution in three-dimensional log-size phase space, which satisfies the allometric scaling laws of three variables. Furthermore, universal distributions of fluctuation around the scaling relations are consistent with the empirical data.

... 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. ...

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.

... In this model, the multiplicative stochastic variable has the meaning of growth rate, and it is known that the power law exponent is determined uniquely from the distribution of growth rate by solving the equation ⟨b α ⟩ = 1, where b denotes growth rate, ⟨·⟩ represents the average, and α is the power law exponent 4 . There have been many studies on the growth rates of business firms, and a typical growth rate distribution is known as the Laplace distribution which is also called the tent-shaped distribution [11][12][13][14][15][16] . Similar statistical properties of growth rates are also found in other fields of sciences [17][18][19][20][21][22] , such as microbial communities, tropical forests, and urban populations 19 , implying that the growth rate statistics show universal properties. ...

We analyze the time series of hashtag numbers of social media data. We observe that the usage distribution of hashtags is characterized by a fat-tailed distribution with a size-dependent power law exponent and we find that there is a clear dependency between the growth rate distributions of hashtags and size of hashtags usage. We propose a generalized random multiplicative process model with a theory that explains the size dependency of the fat-tailed distribution. Numerical simulations show that our model reproduces these size-dependent properties nicely. We expect that our model is useful for understanding the mechanism of fat-tailed distributions in various fields of science and technology.

... This suggests a simple picture emerging at the population level; however, covarying the growth rates among agents in the population will introduce some 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 [26,30,31], and initial log resources across the population, from the solution to the FPE. ...

Understanding the statistical dynamics of growth and inequality is a fundamental challenge to ecology and society. Recent analyses of wealth and income in contemporary societies show that economic inequality is very dynamic and that individuals experience substantially different wealth 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 heterogeneous growth across a population. Here we derive the statistical dynamics of correlated wealth 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 income 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.

... 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). ...

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. ...

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. ...

Although the sizes of business firms have been a subject of intensive research, the definition of a “size” of a firm remains unclear. In this study, we empirically characterize in detail the scaling relations between size measures of business firms, analyzing them based on allometric scaling. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994–2015), we examined up to the trivariate relations between corporate size measures: annual sales, capital stock, total assets, and numbers of employees and trading partners. The data were examined using a multivariate generalization of a previously proposed method for analyzing bivariate scalings. We found that relations between measures other than the capital stock are marked by allometric scaling relations. Power–law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that appeared to correlate with national economic conditions. We established theoretical relations between the exponents. We expect these results to allow direct estimation of the effects of using alternative size measures of business firms in regression analyses, to facilitate the modeling of firms, and to enhance the current theoretical understanding of complex systems.

Recently, graph collaborative filtering methods have been proposed as an effective recommendation approach, which can capture users' preference over items by modeling the user-item interaction graphs. In order to reduce the influence of data sparsity, contrastive learning is adopted in graph collaborative filtering for enhancing the performance. However, these methods typically construct the contrastive pairs by random sampling, which neglect the neighboring relations among users (or items) and fail to fully exploit the potential of contrastive learning for recommendation. To tackle the above issue, we propose a novel contrastive learning approach, named Neighborhood-enriched Contrastive Learning, named NCL, which explicitly incorporates the potential neighbors into contrastive pairs. Specifically, we introduce the neighbors of a user (or an item) from graph structure and semantic space respectively. For the structural neighbors on the interaction graph, we develop a novel structure-contrastive objective that regards users (or items) and their structural neighbors as positive contrastive pairs. In implementation, the representations of users (or items) and neighbors correspond to the outputs of different GNN layers. Furthermore, to excavate the potential neighbor relation in semantic space, we assume that users with similar representations are within the semantic neighborhood, and incorporate these semantic neighbors into the prototype-contrastive objective. The proposed NCL can be optimized with EM algorithm and generalized to apply to graph collaborative filtering methods. Extensive experiments on five public datasets demonstrate the effectiveness of the proposed NCL, notably with 26% and 17% performance gain over a competitive graph collaborative filtering base model on the Yelp and Amazon-book datasets respectively. Our code is available at: https://github.com/RUCAIBox/NCL.

This paper studies the scaling properties of recruitment fluctuations in randomly varying environments, for abundant marine species with extreme reproductive behavior. Fisheries stock-recruitment data from the North Atlantic display fluctuation scaling, a proportionality between the standard deviation and the average recruitment among stocks. The proportionality covers over five orders of magnitude in the range studied. A linear-scaling behavior can be a sign of a universal distribution of the normalized data across stocks. In light of this conjecture, it is demonstrated that the L\'evy-stable model offers a better effective description of the recruitment distribution than the log-normal model. Care is devoted to the problem of random sums of random variables. Recruitment is calculated by summing random offspring numbers with infinite variance, where the number of summands (i.e. spawning population size) is also a random process with infinite variance.

Significance
Proportional growth is the driver behind the dynamics of a large class of complex networks. However, if left uncontrolled a few agents may become so dominant that their actions compromise the entire system. We present a framework that monitors the system’s distance from such imbalanced states. When the system approaches an imbalanced state, we show how to structure an optimal, cost-efficient intervention policy. Focusing only on either helping the least-fit agents or punishing the most dominant ones in isolation turns out to be inefficient. Instead, our results call for a more wholistic approach, with important implications for the structure of regulatory matters such as antitrust policies, taxation law, subsidies, or development aid.

Econophysics is an emerging interdisciplinary field that takes advantage of the concepts and methods of statistical physics to analyse economic phenomena. This book expands the explanatory scope of econophysics to the real economy by using methods from statistical physics to analyse the success and failure of companies. Using large data sets of companies and income-earners in Japan and Europe, a distinguished team of researchers show how these methods allow us to analyse companies, from huge corporations to small firms, as heterogeneous agents interacting at multiple layers of complex networks. They then show how successful this approach is in explaining a wide range of recent findings relating to the dynamics of companies. With mathematics kept to a minimum, the book is not only a lively introduction to the field of econophysics but also provides fresh insights into company behaviour. © Hideaki Aoyama, Yoshi Fujiwara, Yuichi Ikeda, Hiroshi Iyetomi and Wataru Souma 2010.

Levy and Solomon have found that random multiplicative processes wt = λ1λ2...λt (with λj > 0) lead, in the presence of a boundary constraint, to a distribution P(wt) in the form of a power law wt-(1+μ) . We provide a simple exact physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic (t → ∞) distribution of Wt and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable 1/√t=(log wt -(log wt) 2) the two necessary and sufficient conditions for P(wt) to be a power law are that (logAj) < 0 (corresponding to a drift wt → 0) and that tu< not be allowed to become too small. We discuss several models, previously thought unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable log wt undergoes a random walk repelled from -∞, which we describe by a Fokker-PIanck equation. 3) For all these models, we obtain the exact result that μ is solution of (λμ) = 1 and thus depends on the distribution of λ. 4) For finite t, the power law is cut-off by a log-normal tail, reflecting the fact that the random walk has not the time to scatter off the repulsive force to diffusively transport the information far in the tail.

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Part 1 Theory: probability elementary introduction to the theory of stable laws characteristic functions probability densities integral transformations special functions and equations multivariate stable laws simulations estimation. Part 2 Applications: some probabilistic models correlated systems and fractals anomalous diffusion and chaos physics radiophysics astrophysics and cosmology stochastic algorithms financial applications miscellany. Appendix.

Time series outliers and their impactClassical estimates for AR modelsClassical estimates for ARMA modelsM-estimates of ARMA modelsGeneralized M-estimatesRobust AR estimation using robust filtersRobust model identificationRobust ARMA model estimation using robust filtersARIMA and SARIMA modelsDetecting time series outliers and level shiftsRobustness measures for time seriesOther approaches for ARMA modelsHigh-efficiency robust location estimatesRobust spectral density estimationAppendix A: heuristic derivation of the asymptotic distribution of M-estimates for ARMA modelsAppendix B: robust filter covariance recursionsAppendix C: ARMA model state-space representationProblems