Figure 4

Empirical four laws for the temporal network structure 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 in-degree and out-degree cumulative distribution functions (CDFs). Figure (a) shows the empirical inter-firm trading network in Japan in 2011 and 2020, where the exponent \(-1.30\) can be observed. A fitting function \(C e^{-a k} k^{-b}\) for 2020, with parameters \((a,b,C)=(3.80\times 10^{-4},1.30,1.83)\) 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 \(C=1.15\), is also plotted as “Data (rescaled)”. Figures (c) and (d) correspond to the lifetime distribution of firms in Japan. The lifetime in Fig. (c) is estimated by the earliest of the foundation or establishment date, to the bankruptcy or closing date of a company as observed between 2011 and 2020. The rates 2.3%, 5.3%, and 1.2% are according to the disappearance rates in the time span [0,30], [40,140], and \([140,\infty )\), respectively. Figure (d) relates to the results of simulated networks of 10 samples with \(p_{pa}=0.02\). The decay constant is consistent with the data. Figures (e) and (f) represent the mean degree, plotted as a function of the firm age. Figure (e) is the real data for Japan in 2020, showing an average increase of 1.8% per year. The dotted line corresponds to the fit of an exponential function in [0, 100]. Figure (f) shows the simulation results for different \(p_m\). They are fitted in [0, 100] at \(p_m=0\) and [40, 100] at the other parameters. The average degree growth rate is an increasing function of \(p_m\). Figures (g) and (h) show the preferential attachment of the newcomers. Figure (g) is the real data of the cumulation of the preferential attachment probability for Japan in 2020, with an exponent of 2 (we fit the data in [10,200]), thus indicating that the probability of a newcomer attaching to a firm is proportional to that firm’s degree. Figure (h) plots the simulation results which are consistent with the empirical data.