# The interconnected wealth of nations: Shock propagation on global trade-investment multiplex networks

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Beyond country vulnerabilities, our model allows us to measure the potential risk that each country poses for the international economic system as a whole. The global impact of a shock in a given country across the GTI multiplex can be quantified by defining its systemic impact ({{mathcal{S}}}_{i}(alpha ,beta )), as the total economic value that is affected by a shock originated in country i with parameters (α, β). The systemic impact is expected to depend crucially on the propagation of the shock from financial to trade layer, and vice versa. These spillover effects between layers can be addressed by considering separately the impacts on trade and investment. One can define the systemic impact on trade, ({{mathcal{S}}}_{i}^{T}(alpha ,beta )), and investment ({{mathcal{S}}}_{i}^{I}(alpha ,beta )), as the affected value of traded goods and financial securities expressed as a fraction of the global value of traded goods WT and financial securities WI, respectively, that is

$$begin{array}{lllll}{{mathcal{S}}}_{i}^{T}(alpha ,beta ) & = & frac{{sum }_{j},langle Delta {X}_{j}(alpha ,beta ,i)rangle }{{W}_{T}} & = & frac{{sum }_{j},langle Delta {M}_{j}(alpha ,beta ,irangle }{{W}_{T}}, {{mathcal{S}}}_{i}^{I}(alpha ,beta ) & = & frac{{sum }_{j},langle Delta {L}_{j}(alpha ,beta ,i)rangle }{{W}_{I}} & = & frac{{sum }_{j},langle Delta {A}_{j}(alpha ,beta ,i)rangle }{{W}_{I}}.end{array}$$

(4)

Note that the second equality holds because of Eq. (1).

Figure 2 shows the systemic impact on trade, ({{mathcal{S}}}_{i}^{T}(alpha ,beta )), and investment, ({{mathcal{S}}}_{i}^{I}(alpha ,beta )), of a shock originated only in the financial layer (α = 0, Fig. 2a), or trade layer (β = 0, Fig. 2b) of the GTI multiplex networks, with the United States as epicenter country. As expected, the larger the initial distress, represented by parameters (α, β), the larger the systemic impact on the rest of the world. Even if the initial shock only involves one layer, the economic distress spreads from the financial to the trade layer, and viceversa. Interestingly, the different magnitudes of systemic impact reported in Fig. 2 can be quantitatively explained by our model, as we will see in the next section.

Different countries exhibit different magnitudes of systemic impact on trade or investment, that can be taken as a measure of their relevance for the stability of the GTI multiplex network. The systemic impact of a country i, indeed, is expected to depend on the economic value of the initial shock ({mathcal{I}}_{i}), determined simply as ({{mathcal{I}}}_{i}=(alpha {M}_{i}+beta {A}_{i})/({W}_{T}+{W}_{I})). Figure 3a shows the systemic impact on trade ({{mathcal{S}}}_{i}^{T}), and investment ({{mathcal{S}}}_{i}^{I}), as a function of the value of the initial shock ({mathcal{I}}_{i}), characterized by (alpha =beta =-,0.2), for countries belonging to the G20 group. Surprisingly, we found that the systemic impacts of these countries on global trade ((ell =T)) or investment ((ell =I)) are well fitted by linear regressions, whose coefficients ({gamma }_{ell }(alpha ,beta )) represent scale factors for the initial shock. This implies that, at least for big economies, the systemic impact of a country i can be described simply as ({{mathcal{S}}}_{i}^{ell }(alpha ,beta )simeq {gamma }_{ell }(alpha ,beta ){mathcal{I}}_{i}), where ({gamma }_{ell }(alpha ,beta )) encodes the sensitivity of the GTI network to the shock. The larger the coefficients ({gamma }_{ell }(alpha ,beta )), the larger the shock amplification. Notice that these coefficients depend on the initial shock but are country-independent. Even if the elasticity relations (2), determining the internal contagion within countries, are linear, the pass-through coefficients are quite heterogeneous across countries (see SM), and the inter-country propagation phase modeled by the spreading dynamics introduces highly non-linear effects.

Furthermore, it is interesting to consider the regression residuals of different countries. For each country i, one can define the deviations of the systemic impact of each country from the expected value obtained by the fitting function, as

$${{mathcal{D}}}_{i}^{ell }(alpha ,beta )={gamma }_{ell }(alpha ,beta ){mathcal{I}}_{i}-{{mathcal{S}}}_{i}^{ell }(alpha ,beta ).$$

(5)

The trade (financial) deviation ({{mathcal{D}}}_{i}^{T}) (({{mathcal{D}}}_{i}^{I})) of a country i can be positive, if its systemic impact on trade (investment) is smaller than the fitted value, or negative, if ({{mathcal{S}}}_{i}^{T}) (({{mathcal{S}}}_{i}^{I})) is larger than what expected by considering the magnitude of the initial shock ({mathcal{I}}_{i}). Figure 3b shows the trade and financial deviations, ({{mathcal{D}}}_{i}^{T}) and ({{mathcal{D}}}_{i}^{I}), respectively, of the systemic impact of each country i belonging to the G20 group. These deviations are affected by both the statistical error on the systemic impact and the uncertainty of the fitting function, and thus few countries show statistically significant values of ({D}_{i}^{ell }(alpha ,beta )). However, one can see that countries having a significant, positive deviation on trade, generally show a significant, negative deviation on investment, and viceversa. China and Germany, for instance, have a larger systemic impact on trade and a smaller impact on investment than expected, while the United Kingdom and Japan show a considerably smaller impact on trade and a larger impact on investment. Even though ({D}_{i}^{ell }(alpha ,beta )) are expected to depend on the magnitude of the initial shock, these countries presenting significant values of the deviations have qualitatively similar behavior regardless the value of (α, β), as shown in the SM. It is worth to note that it is not possible to verify the linear scaling between initial shock and systemic impact, and consequently its deviations, for small economies, due to large uncertainties over the impact of these countries.