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

We use the United Nations COMTRADE database for analysis of the multiproduct world trade network. With this data, considered for years 2012-2018, we determined the world trade impact of the Kernel of EU 9 countries (KEU9), being Austria, Belgium, France, Germany, Italy, Luxembourg, Netherlands, Portugal, Spain, considered as one united country. We apply the advanced Google matrix analysis for investigation of the influence of KEU9 and show that KEU9 takes the top trade network rank positions thus becoming the main player of the world trade being ahead of USA and China. Our network analysis provides additional mathematical grounds in favor of the recent proposal of KEU9 super-union which is based only on historical, political and economy basis.


Introduction
The economy of European Union (EU) is considered as the second world largest economy after United States (US) [2] even if there are also other opinions placing China (CN) on the first position [3]. At present EU includes 27 member states and about 447 million population [4]. While the global EU economy and population are really huge the political action of member states [5] is not always coherent pushing in some cases in different directions. Due to this reason there is a proposal, pushed forward by Christian Saint-Etienne, to consider the Kernel EU 9 (KEU9) states (or countries), which are tightly linked by historical, political and economic relations, as a strongly united kernel group of EU that would allow to perform coherent actions of these KEU9 states [1] (follow also discussion of this proposal at [6]). These 9 kernel states include Austria (AT), Belgium (BE), France (FR), Germany (DE), Italy (IT), Luxembourg (LU), Netherlands (NL), Portugal (PT), Spain (ES) [1] with the total population of about 305 millions [4].
A variety of arguments in favor of possible coherent political and economic actions of KEU9 group is presented and analyzed in [1]. However, this analysis is not based on detailed mathematical grounds pushing forward arguments of historical, political and economic heuristic reasons. Here, we put forward the mathematical foundations for this KEU9 proposal presenting the mathematical and statistical analysis of the world trade database of UN COMTRADE [7]. This database presents an exceptional variety of data on trade exchange between all world UN registered countries on a scale of more than 50 years with more than 10 4 of trade commodities (products). The transactions are expressed in their dollar (USD) values of a given year. The World Trade Organization (WTO) Statistical Review 2018 [8] demonstrates the vital importance of the international trade between countries for their development and progress. Also the whole world economy is deeply influenced by the world trade [9]. Hence, this database is well appropriate for verification of how strong and important is the world influence of KEU9 group on the world economy. Thus here we use the UN COMTRADE database [7] for mathematical and statistical analysis of heuristic arguments presented in favor of KEU9 in [1].
The trade transfer between countries represents the multiproduct World Trade Network (WTN). The modern methods of Google matrix approach [10][11][12] are well suited for the analysis of transactions on the WTN. The detailed description of Google matrix applications to WTN are described in [13][14][15][16]. Here we apply these methods considering KEU9 countries as one country thus excluding trade transfers between them and keeping only ingoing and outgoing trade flows to this group from other countries.
We point that various research groups investigated the statistical properties of WTN (see e.g. [17][18][19][20][21][22][23]). However, as discussed in [13][14][15] the Google matrix approach has significant advantages for analysis of weighted directed trade networks since it takes into account multiple iterative transactions and thus provides a new and more detailed analysis of trade influence propagation compared to the usual approach based on export and import flows.

Google matrix construction of WTN
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 [7]. These 10 products contain all smaller subdivided specific products which number goes up to ∼ 10 4 . The list of these 10 products is given in Table 1. The list of world countries is available at [13,14]. Following the approach developed in [13,14] we obtain N p money matrices M p c,c 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 α = 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 [14,15]. The general properties and various examples of Google matrices of various networks are given in [10][11][12].
The stationary probability distribution of Markov transitions described by the Google matrix G is given by the PageRank vector P with maximal eigenvalue λ = 1: GP = λP = P [10,11]. 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 [24] (see also [13,14,25]). We define PageRank K and CheiRank K * indexes by monotonic ordering of probabilities of PageRank vector P and of CheiRank vector P * as P (K) ≥ P (K + 1) and P * (K * ) ≥ P * (K * + 1) with K, K * = 1, . . . , N . By taking a sum over all products p we obtain the PageRank and CheiRank probabilities of a given country as P c = p P cp and P * c = p P * cp (and in a similar way product probabilities P p , P * p ) [14,15]). 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 probabilitiesP p ,P * p ,P c ,P * c ,P pc ,P * pc and corresponding indexesK p ,K * p ,K c ,K * c ,K,K * (the import and export probabilities are normalized to unity via the total import and export volumes, see details in [14,15]). 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 PageRank P and a high export a high CheiRank P * probabilities.

Reduced Google matrix
We also use the REGOMAX algorithm described in detail in [26,27]. This algorithm allows to compute efficiently a reduced Google matrix G R of size N r × 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: . Thus G rr is produced by the direct links between selected N r nodes in the global network of N 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 [26,27].

Sensitivity of trade balance
Following [14,15], 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 asB c = (P * c −P c )/(P * c +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 + δ), then computing all rank probabilities and the derivative dB c /dδ.

CheiRank and PageRank of countries
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 * ≤ 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-ImportRankK,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 Supporting Information (SupInf) Fig.S1). 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 ExportRankK * = 7. Also India has strong CheiRank-PageRank position K * = 7, K = 5 compared to Export-ImportRankŝ K * = 11,K = 7. Also there is a significant reduction of positions of Switzerland fromK * = 12, 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.
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 [1].

Trade balance of countries
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 SupInf FigS2; the distributions of import and export of countries for all years are shown in Figs.S3,S4). 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.
We note that by definition we have the balance bounds −1 ≤ B c ≤ 1. The actual obtained bounds are (−0.94, 0.73) and (−0.25, 0.31) for Export-Import and CheiRank-PageRank descriptions (see Fig. 2). We attribute a reduction of bounds for the latter case to a multiplicity of network links that reduce fluctuations in trade exchange.
Sensitivity of trade balance to specific products As described above we determine the sensitivity of trade balance of countries dB c /dδ 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 dB c /dδ 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 /dδ 7 = 0.015, dB c /dδ 7 = 0.043; slightly negative values for USA dB c /dδ 7 = −0.019, dB c /dδ 7 = −0.027; Russia has strongly negative values dB c /dδ 7 = −0.145, dB c /dδ 7 = −0.169.
In 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 dB c /dδ cs of countries to a product of a given country. This specific sensitivity is shown in Fig. 5  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 dB c /dδ 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 dB c /dδ 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.

Sensitivity of trade balance to labor cost
It is interesting to analyze the sensitivity of a country trade balance dB c /dσ 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 + σ c followed by a renormalization of sum all column elements to unity. Such an approach has been developed and studied in [33] 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 dB c /dσ c is computed numerically in the same manner as the product sensitivity dB c /dδ c discussed above.
As discussed in [33] the most strong labor cost sensitivity dB c /dδ c is naturally obtained for the country itself with c = c . Therefore, below in Fig. 7

Network structure of trade from reduced Google matrix
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 Fig. 8 and Fig. 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.

Discussion
Above we considered the trade influence of kernel EU 9 countries (KEU9) considered as a one united state following the proposal pushed forward by Christian Saint-Etienne in [1]. The analysis is done on the bases of multiproduct trade data provided by UN COMTRADE [7]. Our results are based on the advanced Google matrix analysis of multiproduct world trade network flows for years 2012-2018 between all world countries registered at UN. They clearly show that KEU9 takes the world leading position in PageRank and CheiRank probabilities being ahead of USA and China. This mathematical network analysis demonstrates that KEU9 becomes the main player in the international trade. This provides additional mathematical foundation for the historical, economical and political arguments presented in [1] in the favor of coherent strong impact of KEU9 (if united) on the world development.
We also show that the Google matrix analysis allows to obtained significantly deeper information about world trade comparing to the Import-Export analysis usually used in economy studies.  Food and live animals 1 Beverages and tobacco 2 Crude materials,inedible,except fuels 3 Mineral fuels etc 4 Animal and vegetable oils and fats 5 Chemicals and related products,n.e.s. 6 Basic manufactures 7 Machinery,transport equipment 8 Miscellaneous manufactured articles 9 Goods not classified elsewhere         Figure 9. Same in Fig. 8