World impact of kernel European Union 9 countries from Google matrix analysis of the world trade network

We consider the trade exchange between \(N_c=186\) (185 countries + KEU9) world countries and \(N_p=10\) products given by 1 digit from the the Standard International Trade Classification (SITC) Rev. 1, and for years 2012, 2014, 2016, 2018 taken from UN COMTRADE (2020). These 10 products contain all smaller subdivided specific products which number goes up to\(\sim 10^4\). The list of these 10 products is given in Table 1. The list of world countries is available at Ermann and Shepelyansky (2011, 2015). Following the approach developed in Ermann and Shepelyansky (2011, 2015) we obtain \(N_p\) money matrices \(M^p_{c,c^\prime }\) which give product p transfer (in USD) from country \(c'\) to country c. The Google matrices G for the direct trade flow and \(G^*\) for the inverted trade flow have the size of \(N=N_c N_p=1860\) nodes. They are constructed by normalization of all column of outgoing weighted links to unity. There is also the part with a damping factor \(\alpha =0.5\) describing random trade-surfer jumps to all nodes with a certain personalized vector taking into account the weight of each product in the global trade volume. The construction procedure of G and \(G^*\)is described in detail in Ermann and Shepelyansky (2015), Coquidé et al. (2019). The general properties and various examples of Google matrices of various networks are given in Brin and Page (1998), Langville and Meyer (2006), Ermann et al. (2015).

The stationary probability distribution of Markov transitions described by the Google matrix G is given by the PageRank vector P with maximal eigenvalue \(\lambda =1\): \(GP=\lambda P =P\) (Brin and Page 1998; Langville and Meyer 2006). For the inverted flow described by \(G^*\) matrix we have similarly the CheiRank vector \(P^*\), being the eigenvector of \(G^* P^* = P^*\). The importance and detailed statistical analysis of the CheiRank vector were demonstrated in Chepelianskii (2010) (see also Ermann and Shepelyansky 2011, 2015; Zhirov et al. 2010). We define PageRankK and CheiRank \(K^*\) indexes by monotonic ordering of probabilities of PageRank vector P and of CheiRank vector \(P^*\) as \(P(K)\ge P(K+1)\) and \(P^*(K^*)\ge P^*(K^*+1)\) with \(K,K^*=1,\ldots ,N\). By taking a sum over all products p we obtain the PageRank and CheiRank probabilities of a given country as \(P_c =\sum _p P_{cp}\) and \({P^*}_c =\sum _p {P^*}_{cp}\) (and in a similar way product probabilities \(P_p, {P^*}_p\)) (Ermann and Shepelyansky 2015; Coquidé et al. 2019). From these probabilities we obtain the related indexes \(K_c, {K^*}_c\). In a similar way we define from import and export trade volume the probabilities \(\hat{P}_p\), \(\hat{P}^*_p\), \(\hat{P}_c\), \(\hat{P}^*_c\), \(\hat{P}_{pc}\), \(\hat{P}^*_{pc}\) and corresponding indexes \(\hat{K}_p\), \(\hat{K}^*_p\), \(\hat{K}_c\), \(\hat{K}^*_c\), \(\hat{K}\), \(\hat{K}^*\) (the import and export probabilities are normalized to unity via the total import and export volumes, see details in Ermann and Shepelyansky (2015), Coquidé et al. (2019)). We note that qualitatively PageRank probability is proportional to the volume of ingoing trade flow and CheiRank respectively to outgoing flow. Thus, approximately we can say that the high import gives a high PageRankP and a high export a high CheiRank \(P^*\) probabilities.

We also use the REGOMAX algorithm described in detail in Frahm (2016), Frahm et al. (2016). This algorithm allows to compute efficiently areduced Google matrix \(G_R\) of size \(N_r \times N_r\) that accounts all transitions of direct and indirect pathways happening in the full Google matrix G between \(N_r\) nodes of interest. For the selected \(N_r\) nodes their PageRank probabilities are the same as for the global network with N nodes (up to a constant multiplicative factor which takes into account that the sum of PageRank probabilities over \(N_r\) nodes is unity). The matrix \(G_R\) can be presented as as a sum of three matrix components that clearly distinguish direct and indirect interactions: \(G_{\text{R}} = G_{rr} + G_{\mathrm{pr}} + G_{\mathrm{qr}}\) (Frahm et al. 2016). Thus \(G_{rr}\) is produced by the direct links between selected \(N_r\) nodes in the global network of \(N \gg N_r\) nodes. The component \(G_{pr}\) is rather close to the matrix in which each column is given by the PageRank vector \(P_r\) (up to a constant multiplier). Due to that \(G_{pr}\) does not give much information about direct and indirect links between selected \(N_r\) nodes. The most interesting and nontrivial role is played by the component \(G_{qr}\), which accumulates the contribution of all indirect links between selected \(N_r\) nodes appearing due to multiple pathways via the global network of N nodes. The exact formulas for these three components of  \(G_R\) are given in (Frahm 2016; Frahm et al. 2016).

Following Ermann and Shepelyansky (2015), Coquidé et al. (2019), we use the trade balance of a given country with PageRank and CheiRank probabilities defined as\(B_c = (P^*_c - P_c)/(P^*_c + P_c)\). In a similar way we have from ImportRank and ExportRank probabilities as \(\hat{B}_c= ({\hat{P}^*}_c - \hat{P}_c)/({\hat{P}^*}_c + \hat{P}_c)\). The sensitivity of trade balance \(B_c\) to the price of energy or machinery can be obtained from the change of corresponding money volume flow related to SITC Rev.1 code \(p=3\) (mineral fuels) or \(p=7\) (machinery) by multiplying it by \((1+\delta )\), then computing all rank probabilities and the derivative \(dB_c/d\delta\).

The efficiency of the above Google matrix methods has been demonstrated not only for the WTN but also for variety of other directed networks including Wikipedia networks (Zhirov et al. 2010; Zant et al. 2018; Demidov et al. 2020) and biological networks of protein–protein interactions (Lages et al. 2018; Zinovyev et al. 2020; Frahm and Shepelyansky 2020).

We start the presentation of obtained results from showing the distribution of world countries on the plane of CheiRank-PageRank indexes \((K,K^*)\) given in Fig. 1 (left panel). Here, for a better visibility, we show only countries with \(K,K^* \le 60\), each country is marked by a circle with its flag. For a comparison we also present in Fig. 1 (right panel) the distribution of countries on the plane of ExportRank-ImportRank \(\hat{K}\), \(\hat{K}^*\) (in both panels, for compactness, we keep index K which in fact corresponds to \(K_c\) index of a country obtained by a summation over all products). The top 20 countries with their indexes are given in Table 2.

The main feature of Fig. 1 and Table 2 is that KEU9 takes the top leading position in PageRank and CheiRank indexes \(K, K^*\) in 2018 (this leadership is also present in other studied years 2012, 2014, 206 as it is shown in Additional file 1: Fig. A1). This result is significantly different from the Import-Export volume ranking where in 2018 China is leading in export and USA in import. We argue that the Google matrix analysis via PageRank and CheiRank treats in a deeper way the multiplicity of trade relations between world countries compared to the standard Import-Export approach which takes into account only one step trade links.

Another important feature of Google matrix analysis is a significant improvement of positions of certain countries compared to their usual Import-Export ranking (see Fig. 1, Table 2). Thus Russia moves to the fourth CheiRank position \(K^*=6\) compared to its ExportRank \(\hat{K}^* =7\). Also India has strong CheiRank-PageRank position \(K^*=7, K=5\) compared to Export–ImportRanks \(\hat{K}^* =11\), \(\hat{K}= 7\). Also there is a significant reduction of positions of Switzerland from \(\hat{K}^* =12\), \(\hat{K}= 11\) to \(K^*=18, K=14\). In our opinion these results demonstrate a significant hidden power or weakness of trade relations of certain countries due to the multiplicity and variety of their trade relations which are not visible in a standard Export–Import approach.

We also considered the WTN rank positions for other unions of countries instead of KEU9 including countries of Shengen; European Economic Area; post Brexit EU of 27 countries (EU27 without UK). For all these cases we obtained the same PageRank and CheiRank positions as in Table 2for USA, China and the above union replacing KEU9. The main message of the results of this part is the world top leading position of KEU9 in CheiRank and PageRank trade that gives confirmation of the strength and importance of KEU9 countries discussed in Saint-Etienne (2018). The detailed analysis of EU27 case will be presented elsewhere.

We present the world map of trade balance \(B_c\) of countries obtained from CheiRank-PageRank and ExportRank-ImportRank probabilities in Fig. 2 for year 2018 (other years 2012, 2014, 2016 are given in Additional file 1: Fig. A2; the distributions of import and export of countries for all years are shown in Additional file 1: Figs. A3, A4).

The comparison of two ways of balance computation shows that Export–Import approach does not capture the influence of Russia and China on the world trade exchange. In contrast the CheiRank-PageRank approach directly highlights the multistep network influence of Russia and China on the world trade flows and their balance. We also see a strong positive CheiRank-PageRank balance for Japan. In both approaches the balance of US is close slightly negative. There is a relative increase of KEU9 balance in CheiRank-PageRank description compared to the standard Export–Import one. We attribute this to the fact that CheiRank-PageRank description takes into account the multiplicity of trade links which better describes a broad variety of KEU9 trade exchange.

As described above we determine the sensitivity of trade balance of countries \(d B_c/d \delta _s\) to specific products using the sensitivity definition from CheiRank-PageRank and Export–Import probabilities. The sensitivity results for \(s=3\) (mineral fuels) are given in Fig. 3. The CheiRank-PageRank approach shows that the most profitable countries with the highest values of \(d B_c/d \delta _3\) are Saudi Arabia and Russia (Kazakhstan also has high sensitivity). This is rather natural since these countries are the highest petroleum producers. The strongly negative impact is well visible for Australia, China and countries of Latin America. USA and KEU9 sensitivities being close to zero.

In contrast the sensitivity from Export–Import approach gives of the top position Algeria (followed by Brunei). Among countries with strongly negative sensitivities we have India, Pakistan and China while Australia is slightly positive. In this Export–Import approach USA is slightly positive and KEU9 is slightly negative.

This shows a significant difference between the usual Export–Import analysis and the Google matrix approach. We argue that the latter approach takes into account the multiplicity of trade links and flows thus highlighting in a better way the multistep trade relations between countries.

The sensitivities of countries to the product \(s=7\) (machinery) is shown in Fig. 4. Here both approaches give the most positive countries being Japan, S.Korea and China. In the Export–Import approach KEU9 has a bit higher positive sensitivity compared to the CheiRank-PageRank method. Thus we have for both methods of CheiRank-PageRank and Export–Import: KEU9 \(dB_c/d\delta _7 = 0.015\), \(d\hat{B}_c/d\delta _7 = 0.043\); slightly negative values for USA \(dB_c/d\delta _7 = -0.019\), \(d\hat{B}_c/d\delta _7 = -0.027\); Russia has strongly negative values \(dB_c/d\delta _7 = -0.145\), \(d\hat{B}_c/d\delta _7 = -0.169\).

In the above Figs. 3, 4 we presented results for year 2018. The same type of data for years 2012, 2014, 2016 are given in Additional file 1: Figs. A5, A6.

Above we considered the sensitivity of trade balance to a global price variation of a given product applicable to the whole world with a homogeneous price increase of product for all countries. It is also interesting to consider the sensitivity of country trade balance when the product price is changed only by one country. In this way we obtain the sensitivity \(d B_c/ d \delta _{cs}\) of countries to a product of a given country. This specific sensitivity is shown in Fig. 5 in respect to price variation of \(s=7\) (machinery) from KEU9, USA, and China in year 2018. The Export–Import approach gives strongly positive sensitivity for the country which increased price of machinery (respectively KEU9, USA, China). All other countries have sensitivity close to zero or negative. The result from CheiRank-PageRank analysis is different. For KEU9 machinery price increase the positive sensitivity is obtained for Czechia, Slovakia, Hungary (with the sensitivity values 0.028, 0.017, 0.015 respectively). For USA case the positive sensitivity is obtained for Mexico, Canada with repective values 0.045, 0.014. For China case the positive sensitivity is obtained for Korea, Philippines, Malaysia (with sensitivity respective values 0.031, 0.023, 0.014). This increase is related to strong network links between these countries well captured by the Google matrix analysis.

In Fig. 6 we show the sensitivity \(d B_c/ d \delta _{cs}\) from both approaches for \(s=3\) (mineral fuels) of Russia. Again as for \(s=7\) we see that the Export–Import approach gives the strong positive sensitivity only for Russia. In contrast the CheiRank-PageRank approach shows that Uzbekistan, Kazakhstan, Ukraine (with values \(d B_c/ d \delta _{cs} = 0.032,0.021,0.012\) respectively) also gain the positive sensitivity in the case of price increase of \(s=3\) of Russia. This also confirms the strength of Google matrix analysis which captures multiple trade links between countries.

It is interesting to analyze the sensitivity of a country trade balance \(d B_c/d \sigma _{c'}\) to a labor cost variation in a given country. This analysis is done by increasing the price of all products of a given country by a factor \(1+\sigma _{c'}\)followed by a renormalization of sum all column elements to unity. Such an approach has been developed and studied in Kandiah et al. (2015) for the world economic activities from World Trade Organization data. Here, at the difference of price shock of one product, the price increase affects all product flows from a given country corresponding to a global increase of labor cost in a given country. Of course, the price increase is considered to be very small corresponding to the linear response regime. The labor cost sensitivity\(d B_c/d \sigma _{c'}\) is computed numerically in the same manner as the product sensitivity \(d B_c/d \delta _{c'}\) discussed above.

As discussed in Kandiah et al. (2015) the most strong labor cost sensitivity\(d B_c/d \delta _{c'}\) is naturally obtained for the country itself with \(c=c'\). Therefore, below in Fig. 7 we present the diagonal results for \(d B_c/d \delta _{c'}\) at \(c \ne c'\) by a separate magenta color while all other countries sensitivity are characterized by color bar.

For KEU9 the strongest sensitivity values \(d B_c/d \delta _{c'}\) are obtained for countries: Czechia, Tunisia, Morocco with positive values 0.027, 0.014, 0.013 and S.Korea, Canada, Russia with negative values \(-0.074,-0.072,-0.062\). For USA case these are: Canada, Mexico, Venezuela with positive values 0.086, 0.063, 0.008 and S.Korea, Japan, United Kingdom with negative values \(-0.045,-0.044,-0.044\); for China we find: Korea, Congo, Philippines with positive values 0.033, 0.015, 0.014 and Mexico, Canada, Poland with negative values \(-0.090, -0.076, -0.074\). For Russia we obtain: Uzbekistan, Belarus, Kazakhstan with positive values 0.028, 0.026, 0.020 and Norway, Sweden, United Kingdom with negative values \(-0.019,-0.016,-0.013\).

This shows that the Google matrix analysis allows to determine on pure mathematical grounds the mutual trade dependences of world countries.

We use the REGOMAX algorithm described above to obtain the reduced Google matrix of trade flows between certain selected countries. We choose the case of 4 countries which has a strong world trade influence KEU9, USA, China, Russia with 10 trade products of Table 1. In this way the size of \(G_R\) (Import or PageRank direct flow direction) and \({G^*}_R\) (Export or CheiRank inverted flow direction) is equal to 40. For clarity we show only 4 directed links corresponding to the most strong matrix elements from a given node (only non-diagonal terms are shown). The obtained networks are shown in Figs. 8 and 9 respectively.

The network structure shown in Fig. 8 from \(G_R\) shows that the main importing nodes are machinery (\(s=7\)) of KEU9 and USA. In a similar way the main exporting nodes of \({G^*}_R\) are again machinery product of KEU9, China and USA (Fig. 9). This clearly show the importance of machinery product for the world trade.