Data-driven stochastic simulation leading to the allometric scaling laws in complex systems

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Data-driven stochastic simulation leading to the allometric scaling laws in complex systems

Yuh Kobayashi, Hideki Takayasu, Shlomo Havlin, and Misako Takayasu

Phys. Rev. E 106, 064304 – Published 7 December 2022

Abstract

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.

Received 22 March 2022
Revised 15 September 2022
Accepted 19 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.064304

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Econophysics Scaling laws of complex systems Stochastic processes

Scale free & inhomogeneous networks Trade networks

Markovian processes Phase space dynamics Time series analysis

Networks

Authors & Affiliations

Yuh Kobayashi ^*

Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan

Hideki Takayasu ^†

Sony Computer Science Laboratories, Tokyo 141-0022, Japan

Shlomo Havlin^‡

Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

Misako Takayasu ^§

Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan

^*Also at Department of Mathematical Sciences, College of Science and Engineering, Aoyama Gakuin University, Sagamihara 252-5258, Japan; kobayashi@math.aoyama.ac.jp
^†Also at Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan.
^‡Also at Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan.
^§Corresponding author: takayasu.m.aa@m.titech.ac.jp

Article Text

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Issue

Vol. 106, Iss. 6 — December 2022

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Images

Figure 1
Dynamical characteristics of three-dimensional scaling laws between measures of firm size. (a) The three-dimensional scaling relationship is illustrated with a typical trajectory of firms in the phase space. Note the use of logarithmic scales. The orange line represents the scaling line determined from three scaling relations, $ℓ \propto k^{1.0}, s \propto k^{1.2}$ , and $s \propto ℓ^{1.2}$ , where $k, ℓ$ , and $s$ denote the number of trading partners, number of employees, and annual sales in million yen, respectively. Firms are relatively densely distributed on the line. Black dots connected by solid lines represent yearly locations of a hypothetical firm in the three-dimensional phase space. When a firm deviates from the scaling line, it usually moves back toward the line in the following years. (b) Distribution of the 9513 samples for the one-year three-dimensional displacement starting from the neighborhood $(\tilde{k}, ℓ, s) \approx (10, 10, 500)$ . The data points are plotted in the logarithmic scale. Growth rate $G_{x}$ for a size measure $x$ is defined by $x (t + 1) / x (t)$ . Red-dashed lines mark the axes where the growth rates of two of the variables are equal to unity. (c), (d) Distribution of the 1000 samples of one-year displacement starting from the neighborhoods $(\tilde{k}, ℓ, s) \approx (10, 1000, 500)$ and $(10, 1000, 5000)$ , respectively. Data outside the range indicated by cuboid box are truncated and not shown in the figures. Note that the distribution of growth rates strongly depends on the location in the phase space and is evidently not consistent with a multivariate normal distribution.
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Figure 2
Behavior of size distributions in the stochastic simulations presented as the change in cumulative distribution function (CDF) for (a) the annual sales, (b) employee count, and (c) normalized number of trading partners. Grey bold curves indicate the empirical distributions in 2016. Light blue and grey bars at the top right indicate the intervals for which a power-law and stretched exponential distributions fitted better to the empirical data in 2016. The ranges without empirical data are marked by broken bars. Red and blue dashed curves indicate the initial conditions of simulations for size distribution of $10^{6}$ firms, while pale to dark solid curves represent the simulated distribution at the time step $τ$ equal to 20, 100, 200, 500, and 1000. The initial distributions are log-normal for red, while the empirical samples in 2016 are raised to the power of 1.2 for blue. The simulated stationary distribution ( $τ > 500$ ) is remarkably consistent with the empirical distribution in 2016 for the annual sales. Although there are discrepancies between the simulated and empirical distributions for the other two size measures, the exponents of the power-law tails in the intermediate scales match well between the empirical data and simulation.
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Figure 3
Scaling relationships between pairs of system-size measures for the empirical data and simulated stationary state. Plots in the first row (a)–(c) are obtained from the empirical data in 2016, while those in the second row (d)–(f) are from the results of simulation at the time step $τ = 1000$ . Plots of 5-, 25-, 75-, and 95-percentiles of the $y$ axis variable conditional on the $x$ -axis variable are vertically shifted to collapse almost into the curve of the conditional median. The pairs are $(\tilde{k}, ℓ)$ for the left column [panels (a) and (d)], $(\tilde{k}, s)$ for the middle [panels (b) and (e)], and $(ℓ, s)$ for the right [panels (c) and (f)]. Dashed lines are placed at the same location in both panels of the same column. Gaps in the value of the conditional median are evident between the empirical data and simulation results for the pair $(\tilde{k}, ℓ)$ in panels (a) and (d) and $(\tilde{k}, s)$ in panels (b) and (e), while no apparent discrepancy exists in scaling exponents between the data and simulation. The plots in panels in the upper row are from Ref. [40] for comparison. Refer to Ref. [37] for details on the plotting method.
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Figure 4
Distributions of fluctuations from average relationships between pairs of system-size measures for the empirical and simulated firms. Plots in the first (a)–(c) and second (d)–(f) rows are obtained, respectively, from the empirical data in 2016 and the results of simulation at the time step $τ = 1000$ . Probability distribution functions (PDFs) of normalized size measures ( $y / 〈 y | x 〉$ ) conditional on eight intervals of $x$ that are equal in length in a logarithmic scale are plotted. Color gradation from blue to red indicates the smallest to largest values of $x$ . The $x$ values indicated by the colors are presented in the respective legend above the main plots. The pairs of system-size measures are the same as in Fig. 3. Dashed lines are placed at the same location in both panels of the same column. There is no obvious discrepancy between the empirical data and simulation results except for small values of normalized system size. The plots in panels in the upper row are from Ref. [40] for comparison. Refer to Ref. [37] for details on the plotting method.
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Figure 5
Nonstationary increase in data and degree normalization for the apparently growing interfirm trading network and its effect on the simulation results. (a) The number of firms and trading links observed in the data for every year between 1993 and 2017. The number of trading links increased by more than two-fold during the period. (b) 5-, 25-, 50-, 75-, 99-, 99.9-, and 99.99-percentiles of the degree distribution (the distribution of the number of trading partners, $k$ ) in years between 1993 and 2017, relative to the 1993 data. Paler colors mark lower percentiles. The curves for higher percentiles approximately match each other. (c), (d) Original and normalized degree distributions for every year between 1993 and 2016. CDFs for the range of degree between $10^{1}$ and $10^{3}$ are shown in the main plots, while the whole range are shown in the inset plots. Older data are plotted with paler red. A tick indicates two orders of magnitude in the inset plots. (e) The behavior of simulated distribution of the number of trading partners without normalization, shown as the change of CDF. Grey bold curve indicates the empirical distribution in 2016. Red dashed curve indicates the log-normal distribution for the initial condition, while pale to dark solid curves represent the simulated distribution at the time step $τ$ equal to 10, 20, 50, 100, 200, 500, and 1000. The simulated stationary distribution ( $τ > 500$ ) evidently disagrees with the empirical distribution in 2016.
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