Figure 5
Empirical three laws for the network money transport layer. The light-blue background (left side) relates to plots for key quantities characterising the actual data set and corresponding fitting elements, whereas the right side corresponds to the simulation results, compared to the actual data. Figures (a) and (b) correspond to the cumulative distribution function (CDFs) of the firm sales S. Figure (a) shows the empirical data in Japan in 2011 and 2020, where the exponent \(-0.98\) can be observed. A fitting function \(C e^{-a k} k^{-b}\) for 2020, with parameters \((a,b,C)=(3.39\times 10^{-7},0.98,188)\) is also plotted. Figure (b) shows the results of simulated networks of 10 samples at \(p_{pa}=0.02\). A data fitting result, where C is adjusted to \(C=130\), is also plotted as “Data (rescaled)”. Figures (c) and (d) correspond to the scaling relation between degree and sales. Figure (c) is a plot of the median for the empirical inter-firm trading network in Japan in 2011 and 2020. The bars show the quantiles, with the dotted line representing a power-law scaling. Figure (d) relates to the results of simulated networks of 10 samples at \(p_{pa}=0.02\). Figures (e) and (f) represent the probability density of the log growth rate of firm sales for different sales scale levels. Figure (e) corresponds to the empirical data in Japan during 2011-2020. The fitting function is \(2.93e^{-6.91|x|^{0.525}}+3.63 e^{-15.4|x|}\), where x indicates the log growth rate. We plot the cases of \(S<10^3\), \(10^3\le S < 10^4\), \(10^4\le S < 10^5\), \(10^5 \le S\), and all S, where S represent the total sales. It is observed that firms with large sales have a narrow distribution. Figure (f) represents the simulation results at different scale levels with \((N_0,p_m,p_{pa})=(10^6,0.37,0.02)\). The fitting function of (e) is also plotted for reference. We show the cases of \(S<10^3\), \(10^3\le S < 10^4\), \(10^4\le S < 10^5\), \(10^5 \le S\), and \(10^3\le S\) as “All (\(S\ge 10^3\))”. We discard the firms of \(S<10^3\) in “All (\(S\ge 10^3\))” because the model has the limitation of not calculating the smallest firm sales.
