Entropy of international trades

The organization of international trades is highly complex under the collective efforts towards economic profits of participating countries given inhomogeneous resources for production. Considering the trade flux as the probability of exporting a product from a country to another, we evaluate the entropy of the world trades in the period 1950–2000. The trade entropy has increased with time, and we show that it is mainly due to the extension of trade partnership. For a given number of trade partners, the mean trade entropy is about 60% of the maximum possible entropy, independent of time, which can be regarded as a characteristic of the trade fluxes' heterogeneity and is shown to be derived from the scaling and functional behaviors of the universal trade-flux distribution. The correlation and time evolution of the individual countries' gross-domestic products and the number of trade partners show that most countries achieved their economic growth partly by extending their trade relationship.

Figure 1

Time evolution of the trade entropy and uniformity. (a) The global trade entropy $S$ in year $t$. Inset: the global trade uniformity $U\left(t\right)$. (b) The local trade entropy $s_i\left(t\right)$ of country $i$ in year $t$. Inset: the number of trading partner countries ${\ell }_i\left(t\right)$. Its mean and the standard deviation over countries having at least one trade partner are shown. (c) The local trade uniformity $u_i\left(t\right)=s_i\left(t\right)/{log}_2{\ell }_i\left(t\right)$. Its mean and standard deviation over countries having at least one trade partner are shown. Also, the uniformity in random trade networks preserving $L\left(t\right)$ and $F\left(t\right)$ is shown; links (trade partnership) are rewired in both “random(1)” and “random(2)” networks. The “random(1)” networks have link weights (trade fluxes) randomized as well. (d) The average of the local trade entropy ${\langle s\rangle }_{\ell }\left(t\right)$ for the countries having $\ell$ trade partners is plotted as a function of ${log}_2\ell$ for selected years. The line with slope 0.65 fits the data aggregated for all years.

Figure 2

Distribution of the global and local trade fluxes. (a) Cumulative distribution of $f_{ij}\left(t\right)$ for selected years. $\langle f\rangle \left(t\right)$ is the average of $f_{ij}\left(t\right)$ in each year $t$. The solid line fits the slow decaying part of $C_t\left(f\right)$ in year $t=2000$. Inset: data collapse of $C_t\left(f\right)L{\left(t\right)}^{z_f\nu }$ as a function of $x/L{\left(t\right)}^{z_f}$ with $x=f/\langle f\rangle \left(t\right)$ for different $t$'s. (b) Cumulative distribution of $p_{ij}\left(t\right)$ for the countries having $\ell$ trade partners with $\ell$ lying in the most probable range ${\ell }^{\left(1\right)}\left(t\right)\le \ell <{\ell }^{\left(2\right)}\left(t\right)$ as shown in the legend. $\langle p\rangle \left(t\right)$ is the average of $p_{ij}\left(t\right)$ over those countries. The solid line fits the slow decaying part of $C_t\left(p\right)$ in year $t=2000$. Inset: data collapse of $C_t\left(p\right)\ell {\left(t\right)}^{z_p\eta }$ as a function of $x/\ell {\left(t\right)}^{z_p}$ with $x=p/\langle p\rangle \left(t\right)$ and $\ell \left(t\right)=\sqrt{{\ell }^{\left(1\right)}\left(t\right){\ell }^{\left(2\right)}\left(t\right)}$.

Figure 3

Correlation and evolution of the standardized GDP and number of trade partners. (a) The average standardized GDP $\langle z^{(lng)}\rangle$ of the countries having a given standardized number of trading partners $z^{\left(\ell \right)}$. The correlation coefficient between $\left\{z_i^{(lng)}\left(t\right)\right\}$ and $\left\{z_i^{\left(\ell \right)}\left(t\right)\right\}$ is 0.82. (b) The average standardized growth rate $\langle z^{\left(v\right)}\rangle$ versus $z^{\left(\ell \right)}$. The correlation coefficient between $\left\{z_i^{\left(v\right)}\left(t\right)\right\}$ and $\left\{z_i^{\left(\ell \right)}\left(t\right)\right\}$ is 0.025 with $P$ value 0.03. (c) $z^{(lng)}$ and $z^{\left(\ell \right)}$ for rich, intermediate, poor countries, being top 25%, middle 50%, and bottom 25% in their standardized GDP, respectively, as a function of age $\tau$. (d) The displacement of selected countries between 1950 and 2000 in the plane $(z^{\left(\ell \right)},z^{(lng)})$.

Figure 4

Time evolution of various quantities in the random networks. The presented data for random(1) and random(2) networks are obtained by averaging over 1000 realizations of the corresponding random networks. The error bars represent the standard deviations over countries averaged over realizations of random networks. (a) The total number of links $L\left(t\right)={\sum }_{i,j,F_{ij}\left(t\right)>0}1$. (b) The number of countries having at least one partner country $N\left(t\right)={\sum }_{i,{\ell }_i\left(t\right)>0}1$. (c) The global trade entropy $S\left(t\right)$ in year $t$. (d) The global trade uniformity $U\left(t\right)=S\left(t\right)/{log}_{2}L$. (e) The number of trading partner countries ${\ell }_i\left(t\right)$. Its mean and standard deviation over countries having at least one trade partner are shown. (f) The local trade entropy $s_i\left(t\right)$. Its mean and standard deviation over countries having at least one trade partner are shown. (g) The average of the local trade entropy ${\langle s\rangle }_{\ell }\left(t\right)$ for the countries having $\ell$ trade partners is plotted as a function of ${log}_2\ell$ obtained from the data aggregated for all years in each type of network. Different slopes of the lines fitting the data in the real and the random networks are shown. In random(2) networks, two fitting lines are shown.

Figure 5

Statistical significance of the time invariance of $\langle u\rangle \left(t\right)$. Plots of the fraction of the pairs $(t_1,t_2)$ for which the $P$ value for the hypothesis $\langle u\rangle \left(t_1\right)=\langle u\rangle \left(t_2\right)$ is not larger than $x$. The same plots for the real trade networks and for the random networks are shown. The same plot for different samples obtained from a given normal distribution is also shown, denoted by “identical”.

Figure 6

Statistical significance of the time invariance of $c\left(t\right)$. (a) The average of the local trade entropy ${\langle s\rangle }_{\ell }\left(t\right)$ versus the number of trade partners $\ell$ in selected years for random(1) networks. (b) The same plot as (a) for random(2) networks. (c) The coefficient $c\left(t\right)$ in the relation ${\langle s\rangle }_{\ell }\left(t\right)\simeq c\left(t\right){log}_2\ell \left(t\right)$ for the real and the random networks. (d) Plots of the fraction of the pairs $(t_1,t_2)$ for which the $P$ value for the hypothesis $c\left(t_1\right)=c\left(t_2\right)$ is not larger than $x$.

Figure 7

The exponents $\nu$ and $\eta$ for the power-law behaviors of the cumulative distributions of the trade flux $C_t\left(f\right)\sim f^{-\nu }$ and $C_t\left(p\right)\sim p^{-\eta }$ each year $t$. These exponents are obtained by using the maximum likelihood estimation and the Kolmogorov-Smirnov statistic [19].

Figure 8

Distributions of local trade fluxes for random(2) networks in selected years. Inset: data collapse of $C_t\left(p\right)\ell {\left(t\right)}^{z_p\nu }$ as a function of $x/\ell {\left(t\right)}^{z_p}$ with $x=p/\langle p\rangle \left(t\right)$.