ArticlePDF Available

# Generalised Central Limit Theorems for Growth Rate Distribution of Complex Systems

Authors:

## Abstract and Figures

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.
Content may be subject to copyright.
A preview of the PDF is not available
... 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. ...
Article
Full-text available
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). ...
Preprint
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. ...
Preprint
Full-text available
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. ...
Article
Full-text available
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.
... 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. ...
Article
Full-text available
We propose a method to derive the stationary size distributions of a system, and the degree distributions of networks, using maximisation of the Gibbs-Shannon entropy. We apply this to a preferential attachment-type algorithm for systems of constant size, which contains exit of balls and urns (or nodes and edges for the network case). Knowing mean size (degree) and turnover rate, the power law exponent and exponential cutoff can be derived. Our results are confirmed by simulations and by computation of exact probabilities. We also apply this entropy method to reproduce existing results like the Maxwell-Boltzmann distribution for the velocity of gas particles, the Barabasi-Albert model and multiplicative noise systems.
... 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). ...
Article
Full-text available
Companies tend to publish financial reports in order to articulate strategies, disclose key performance measurements as well as summarise the complex relationships with external stakeholders as a result of their business activities. Therefore, any major changes to business models or key relationships will be naturally reflected within these documents, albeit in an unstructured manner. In this research, we automatically scan through a large and rich database, containing over 400,000 reports of companies in Japan, in order to generate structured sets of data that capture the essential features, interactions and resulting relationships among these firms. In doing so, we generate a citation type network where we empirically observe that node creation, annihilation and link rewiring to be the dominant processes driving its structure and formation. These processes prompt the network to rapidly evolve, with over a quarter of the interactions between firms being altered within every single calendar year. In order to confirm our empirical observations and to highlight and replicate the essential dynamics of each of the three processes separately, we borrow inspiration from ecosystems and evolutionary theory. Specifically, we construct a network evolutionary model where we adapt and incorporate the concept of fitness within our numerical analysis to be a proxy real measure of a company’s importance. By making use of parameters estimated from the real data, we find that our model reliably replicates degree distributions and motif formations of the citation network, and therefore reproducing both macro as well as micro, local level, structural features. This is done with the exception of the real frequency of bidirectional links, which are primarily formed as a result of an entirely separate and distinct process, namely the equity investments from one company into another.
Preprint
Full-text available
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.
Preprint
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.
Article
Full-text available
We develop an early warning system and subsequent optimal intervention policy to avoid the formation of disproportional dominance ("winner takes all," WTA) in growing complex networks. This is modeled as a system of interacting agents, whereby the rate at which an agent establishes connections to others is proportional to its already existing number of connections and its intrinsic fitness. We derive an exact four-dimensional phase diagram that separates the growing system into two regimes: one where the "fit get richer" and one where, eventually, the WTA. By calibrating the system's parameters with maximum likelihood, its distance from the unfavorable WTA regime can be monitored in real time. This is demonstrated by applying the theory to the eToro social trading platform where users mimic each other's trades. If the system state is within or close to the WTA regime, we show how to efficiently control the system back into a more stable state along a geodesic path in the space of fitness distributions. It turns out that the common measure of penalizing the most dominant agents does not solve sustainably the problem of drastic inequity. Instead, interventions that first create a critical mass of high-fitness individuals followed by pushing the relatively low-fitness individuals upward is the best way to avoid swelling inequity and escalating fragility.
Book
Cambridge Core - Econophysics and Financial Physics - The Rise and Fall of Business Firms - by S. V. Buldyrev
Article
Full-text available
The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form wi(t+1)=λ(t)wi(t)+aw¯(t)-bwi(t)w¯(t) is studied by computer simulations. The variables wi, i=1,...,N, are the individual system components and w¯(t)=(1/N)∑iwi(t) is their average. The parameters a and b are constants, while λ(t) is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution P(w,t) of the system components wi turns out to fulfill a Pareto power law P(w,t)~w-1-α. The time evolution of w¯(t) presents intermittent fluctuations parametrized by a Lévy-stable distribution with the same index α, showing an intricate relation between the distribution of the wi's at a given time and the temporal fluctuations of their average.
Article
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.
Article
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.
Article
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.
Article
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
Article
A great variety of complex phenomena in many scientific fields exhibit power-law behavior, reflecting a hierarchical or fractal structure. Many of these phenomena seem to be susceptible to description using approaches drawn from thermodynamics or statistical mechanics, particularly approaches involving the maximization of entropy and of Boltzmann-Gibbs statistical mechanics and standard laws in a natural way. The book addresses the interdisciplinary applications of these ideas, and also on various phenomena that could possibly be quantitatively describable in terms of these ideas.