Influence of Petroleum and Gas Trade on EU Economies from The

EPJ manuscript No. (will be inserted by the editor)

Inﬂuence of petroleum and gas trade on EU economies from the reduced Google matrix analysis of UN COMTRADE data

CélestinCoquidé1, Leonardo Ermann2, JoséLages1 and D.L.Shepelyansky3 1 Institut UTINAM, OSU THETA, Universitéde Bourgogne Franche-Comté, CNRS, Besan¸con,France 2 Departamento de F´ısicaTeórica,GIyA, CNEA, Av. Libertador 8250, (C1429BNP) Buenos Aires, Argentina. 3 Laboratoire de Physique Théorique,IRSAMC, Universitéde Toulouse, CNRS, UPS, 31062 Toulouse, France

Dated: February 5, 2019

Abstract. Using the United Nations COMTRADE database [1] we apply the reduced Google matrix (RE- GOMAX) algorithm to analyze the multiproduct world trade in years 2004-2016. Our approach allows to determine the trade balance sensitivity of a group of countries to a specific product price increase from a specific exporting country taking into account all direct and indirect trade pathways via all world countries exchanging 61 UN COMTRADE identified trade products. On the basis of this approach we present the influence of trade in petroleum and gas products from Russia, USA, Saudi Arabia and Norway determin- ing the sensitivity of each EU country. We show that the REGOMAX approach provides a new and more detailed analysis of trade influence propagation comparing to the usual approach based on export and import flows.

PACS. XX.XX.XX No PACS code given

1 Introduction and poor countries have equal consideration) and also the PageRank and CheiRank algorithms take into account the The statistical data of UN COMTRADE [1] and the World whole chain of transactions incorporating the importance Trade Organization (WTO) Statistical Review 2018 [2] of specific network nodes. This is drastically different from demonstrate the vital importance of the international trade the simple bilateral transactions of import and export. between world countries for their development and pro- gress. Also the whole world economy deeply depends on Usually in directed networks, like WWW or Wikipedia, the world trade [3]. At present the UN COMTRADE data- the PageRank vector of the Google matrix plays the dominant role since its components are on average proportional base contains data for Nc = 294 UN countries with up to 4 to the number of ingoing links. For the WTN the ingoing Np ≈ 10 trade products. Thus the whole matrix of trade monetary flows reaches a large size N = N N ∼ 106. In flows are related to import. However, the outgoing flows, p c related to export, are also important for trade. Thus we fact for each year the commercial exchange between coun- ∗ tries represents the directed network with transactions of also use the Google matrix G , constructed from the in- various commodities (products) expressed in their US dol- verted transaction flows, with its PageRank eigenvector, lar (USD) values of the given year. called CheiRank vector [10,11]. The components of this It is clear that the recent research developments in the vector are on average proportional to the number of out- field of complex networks (see e.g. [4]) should find use- going links in the original WTN. The construction rules of

arXiv:1903.01820v1 [q-fin.ST] 5 Mar 2019 ∗ ful applications for analysis of this multiproduct World G and G for the case of multiproduct WTN are described Trade Network (WTN). In [5,6] it was proposed to use in detail in [6]. the methods of the Google matrix G, PageRank and Chei- In many cases it is important to know the effective in- Rank algorithms for analysis of the WTN. The PageRank teractions of trade transactions for a specific region (i.e., algorithm had been invented by Brin and Page [7] for the for selected nodes of the global network) on which one ranking of nodes of the World Wide Web (WWW) be- wants to focus the analysis. This requires to know not ing at the foundation grounds of the Google search engine only direct links between nodes but also the indirect (or [7,8]. The applications of these methods to a variety of hidden) links which connect the selected nodes via the real directed networks are described in [9]. In contrast to remaining part of the global network. Recently the re- the usual economy approach based on bilateral import and duced Google matrix (REGOMAX) algorithm has been export flows, the Google matrix analysis treats all world invented in [12] and tested with various directed networks countries on equal grounds (since all columns with outgo- of Wikipedia [13,14] and protein-protein interactions [15] ing country flows of G are normalized to unity so that rich showing its efficiency. This algorithm originates from the 2 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies

0 scattering theory of nuclear and mesoscopic physics and with country indexes c, c = 1,...,Nc and product index the field of quantum chaos. In this work, using the COM- p = 1,...,Np. For future notation we also define TRADE data, we apply the REGOMAX algorithm to an- p X p ∗p X p alyze the influence on European Union (EU) countries of Vc = Mc,c0 ,Vc = Mc0,c. (2) petroleum and gas trade from Russia (RU), USA (US), c0 c0 Saudi Arabia (SA) and Norway (NO). With this approach p ∗p which are the volume of imports (Vc ) and exports (Vc ) we are able to measure the sensitivity of EU countries to for a given country c and a given product p. The global the supply of petroleum and gas from one of these four import and export volumes are given by V = P V p and countries taking into account the global WTN, i.e., tak- c p c ∗ P ∗p ˆ ing into account all direct and indirect transactions of 61 Vc = p Vc . Thus the ImportRank (P ) and Export- major products with the rest of the world. Rank (Pˆ∗) vector probabilities are given by the normal- We note that there is a variety of papers with network ized import and export volumes methods applied to financial and trade networks (see e.g. Pˆ = V p/V , Pˆ∗ = V ∗p/V , (3) [16,17,18,19,20,21]). However, the applications of the Pa- i c i c geRank algorithm to the WTN is rarely used (see e.g. one where i = p + (c − 1)Np ∈ {1,...,N = NcNp} is the of the first cases in [22]) but the outgoing flows with the index associated to the country c – product p couple, and CheiRank analysis were not considered apart from [5,6]. P p P p the total trade volume is V = p,c,c0 Mc,c0 = p,c Vc = The analysis of hubs and authorities is performed in [23] P ∗p p,c Vc . but in our opinion this approach has lower performance The list of 61 products and 227 countries are given in comparing to the Google matrix methods. Thus for the [6]. bitcoin transaction network the Google matrix methods demonstrate the existence of oligarchy type structure [24]. Till present the matrix methods are rather rarely used in 2.1 Google matrix construction for the WTN the field of transactions even if it was shown that the Ran- dom Matrix Theory finds useful applications for financial The Google matrices G for the direct trade flow and G∗ and credit risk analysis [25,26]. The methods of statistical for the inverted trade flow have the size N = NcNp = mechanics also demonstrated their efficiency for analysis 227 × 61 = 13847 and are constructed as it is described in of market economies [27]. However, the flows considered [6]. By the definition the sum of elements in each column in [25,26] are non-directional while the WTN typically de- is equal to unity. The Google matrices have the form scribes directed flows. Due to these reasons we hope that G = αS + (1 − α)v , the REGOMAX algorithm will find further useful applica- ij ij i (4) G∗ = αS∗ + (1 − α)v∗, tion for the treatment of trade and financial transactions. ij ij i ∗ The paper is constructed as follows: in Section 2, we where α ∈]0, 1] is the damping factor, and vi and vi construct the Google matrix for the World Trade Network are components of positive column vectors called person- P P ∗ and introduce the REGOMAX method. In Section 3, we alization vectors with i vi = i vi = 1 [8]. In this present the network structure of petroleum and gas trade work we fix α = 0.5, its variation in the range [0.5, 0.9] in EU exhibiting direct and indirect effects of petroleum does not significantly affect the results. The PageRank P and gas trade between EU economies and non EU major and CheiRank P ∗ vectors have each an eigenvalue λ = actors as Russia, Saudi Arabia and USA. We also inves- 1 since GP = P and G∗P ∗ = P ∗. According to the tigate the EU countries trade balance sensitivity to Rus- Perron-Frobenius theorem the components {Pi}i=1,...,N ∗ sian, Saudi Arabian, and US petroleum and to Russian and {Pi }i=1,...,N are positive and give probabilities to and Norwegian gas over the time period 2004-2016. find a random surfer (seller) traveling on the network of N nodes. The PageRank K and CheiRank K∗ indexes are defined from the decreasing ordering of probabilities of Pa- geRank vector P and of CheiRank vector P ∗ as P (K) ≥ P (K + 1) and P ∗(K∗) ≥ P ∗(K∗ + 1) with K,K∗ = 2 Methods 1,...,N. A similar definition of ranks from import and export trade volume can be also done via probabilities ˆ ˆ∗ ˆ ˆ∗ ˆ ˆ∗ ˆ Pp, Pp , Pc, Pc , Ppc, Ppc and corresponding indexes Kp, We collected the multiproduct (multicommodities) trade Kˆ ∗, Kˆ , Kˆ ∗, Kˆ , Kˆ ∗. data from UN COMTRADE database [1] for N = 227 p c c c The matrices S and S∗ are built from money matrices countries, Np = 61 products given by 2 digits from the p M 0 as Standard International Trade Classification (SITC) Rev. c,c 1, and for years 2004, 2008, 2012, 2016. Following the ap- ( p ∗p ∗p0 Mc,c0 δpp0 /Vc0 if Vc0 6= 0 proach developed in [6], for a given year, we build Np Sii0 = p ∗p0 1/N if V 0 = 0 money matrices Mc,c0 of the WTN defined as c (5) ( p p p0 M 0 δpp0 /V 0 if V 0 6= 0 p product p transfer (in USD) ∗ c ,c c c Sii0 = 0 Mc,c0 = 0 (1) p from country c to c 1/N if Vc0 = 0 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies 3

0 0 where c, c = 1,...,Nc; p, p = 1,...,Np; i = p+(c−1)Np; The interesting role is played by Gqr, which takes into ac- 0 0 0 0 i = p + (c − 1)Np; and therefore i, i = 1,...,N. count all indirect links between selected nodes appearing Following [6] we defined the personalized vectors in (4) due to multiple pathways via the N global network nodes via the relative import and export product volume per (see [12,13]). The matrix Gqr = Gqrd +Gqrnd has diagonal country (Gqrd) and non-diagonal (Gqrnd) parts where Gqrnd de- scribes indirect interactions between nodes. The explicit V p V ∗p formulas with the mathematical and numerical computa- v = c , v∗ = c , (6) i P p0 i P ∗p0 tion methods of all three matrix components of G are Nc 0 Vc Nc 0 Vc R p p given in [12,13]. We discuss the properties of these matrix components below for the multiproduct WTN. using the definitions (2) and the relation i = p+(c−1)Np. In this way we obtain the first iteration for PageRank P and CheiRank P ∗ vectors keeping the democracy in countries and proportionality of products to their trade vol- 2.3 WTN datasets ume. Then in the second iteration we use the personalized vectors from the results of the first iteration With the REGOMAX approach we consider 27 EU countries dated by 2008 and presented in Table1 and Table2; ∗ 0 Pp 0∗ Pp countries are marked by their country code ISO 3166-1 vi = , vi = . (7) Nc Nc alpha-2 [28]. The Table of 61 products is given in [6]. In Table1 in addition to 27 EU countries (marked by Here we use the tracing over product or countries getting blue) we also take 10 best non-EU petroleum (SITC Rev.1 P P respectively Pc = p Ppc = p P (p + (c − 1)Np) and code p = 33 for petroleum and petroleum products) ex- ∗ P ∗ P ∗ Pc = p Ppc = p P (p + (c − 1)Np) with their corre- porters in 2016 (marked by red) showing their PageRank, ∗ sponding Kc and Kc indexes. Also after tracing over coun- CheiRank, ImportRank and ExportRank in 2016. Here P P the PageRank and CheiRank are given by the local order tries we obtain Pp = c Ppc = c P (p + (c − 1)Np) and ∗ P ∗ P ∗ of P and P ∗ with fixed p = 33 with highest probabilities Pp = c Ppc = c P (p + (c − 1)Np) with their corre- pc pc ∗ ∗ sponding product indexes Kp and Kp (Pp, P p are used in at index being 1, 2, ... (probability in decreasing order). In (7)). This second iteration is used for further construction the same way ImportRank and ExportRank are obtained ∗ ˆ ˆ∗ of G and G matrices with which we work in the following. from Ppc and Ppc at fixed p = 33. For petroleum we see in Table1 that in 2016 the top position is taken by Russia in CheiRank and ExportRank 2.2 Reduced Google matrix for the WTN while USA is the first in PageRank and ImportRank. We also see that for CheiRank not only the trade volume counts but also the broad trade network of a given coun- The REGOMAX algorithm, invented in [12], is described try. Thus Saudi Arabia (SA) is 2nd in ExportRank but it in detail in [13]. Here we give the main elements of this has only 6th position in CheiRank since its trade is mainly method keeping the notations of [13]. oriented towards US. Another example is Singapore (SG) The reduced Google matrix G is constructed for a R which goes from 4th position in ImportRank to 2nd posi- selected subset of N nodes. The construction is based on r tion in PageRank demonstrating the importance of broad concepts of scattering theory used in different fields inclu- trade relations of SG. Among EU countries the first place ding mesoscopic and nuclear physics, and quantum chaos. in all 4 ranks is taken by Netherlands (NL) due to its It captures, in a N -by-N Perron-Frobenius matrix, the r r broad commercial maritime connections. full contribution of direct and indirect interactions hap- For gas in Table2 we have similar observations. Al- pening in the full G matrix between N selected nodes r though France (FR), Italy (IT) and UK (GB) occupy the of interest. In addition the PageRank probabilities of the first ImportRank places for EU countries, i.e., they are the N nodes are the same as for the global network with N r top EU importer by volume trade of gas, NL and Belgium nodes, up to a constant factor taking into account that the (BE) supersede them in PageRank top positions, indicat- sum of PageRank probabilities over N nodes is unity. The r ing that NL and BE import gas from more diverse and (i, j)-element of G can be interpreted as the probability R important sources than FR, IT and GB. Also Qatar (QA) for a random surfer starting at node j to arrive in node is first in ExportRank but is only at the 4th position in i using direct and indirect interactions. Indirect interac- CheiRank due to its rather specific trade orientation. tions refer to pathways composed in part of nodes different from the Nr ones of interest. The intermediate computa- tion steps of GR offer a decomposition of GR into matrices that clearly distinguish direct from indirect interactions: 2.4 Sensitivity of trade balance GR = Grr + Gpr + Gqr [13]. Here Grr is given by the direct As in [6], we determine the trade balance of a given coun- links between selected Nr nodes in the global G matrix with N nodes, G is usually rather close to the matrix try with PageRank and CheiRank probabilities as Bc = pr ∗ ∗ in which each column is given by the PageRank vector (Pc − Pc)/(Pc + Pc) and in a similar way via ImportRank ˆ ˆ∗ ˆ ˆ∗ Pr. Due to that Gpr does not provide much information and ExportRank probabilities as Bc = (Pc − Pc)/(Pc + about direct and indirect links between selected nodes. Pˆc). The sensitivity of trade balance Bc to the price of 4 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies

Table 1. List of 27 EU countries (in blue) and 10 best non- Table 2. List of 27 EU countries (in blue) and 10 best non- EU exporters regarding to ExportRank for SITC Rev.1 code EU exporters regarding to ExportRank for SITC Rev.1 code p = 33 (petroleum and petroleum products, in red), sorted p = 34 (gas, natural and manufactured, in red), sorted by Pa- by PageRank, CheiRank, ImportRank and ExportRank order geRank, CheiRank, ImportRank and ExportRank order from from UN COMTRADE 2016. UN COMTRADE 2016. PageRank CheiRank ImportRank ExportRank PageRank CheiRank ImportRank ExportRank 1US RUUS RU 1NLUSFR QA 2SGUSNLSA 2BECAITNO 3NLAEINUS 3FR RUGB RU 4ININSGAE 4ITQAUSUS 5FRSGDENL 5GBNODE AU 6DESAITCA 6ES AUBEDZ 7ESNLFRIQ 7HUNLESMY 8GBBEGBSG 8USGBNLBE 9ITGRBEKW 9DEDZAECA 10BENGESNG 10PTAECAAE 11CAITCAIN 11BGBEIDID 12AEDESEGB 12SKDECZNL 13NGCAPLBE 13PLITSKGB 14PLIQNGDE 14SIFRPTDE 15SIKWAEIT 15CA SEHUFR 16CZGBGRES 16 ROIDPLES 17 ATESFIFR 17IDDKMY AT 18SEFR ATGR 18GRMY IESK 19HUFIPTSE 19 RUGRGRCZ 20PTSE LVFI 20MYPLSEIT 21 ROPTMT LT 21CZESBGPL 22BG ROCZDK 22SE AT LTSE 23SKDKDKPL 23 AUPT ROHU 24GRBG LTPT 24IEHU LVDK 25MT LT RO RO 25AEIESI RO 26SAPLIEBG 26 LTSK AUSI 27 RUHUHUSK 27NO LT RUPT 28 LT ATSK AT 28 AT RODKGR 29IESKSA LV 29DKCZEE LT 30CY LVBGMT 30CYSINOLU 31DKMTSIHU 31EE LVLU LV 32FICZ RUCZ 32MTBGFIFI 33 LVSIEEEE 33 LVFI ATMT 34LUCYLUSI 34LULUCYIE 35IQEECYIE 35FIMTMTEE 36EEIEIQCY 36QAEEQABG 37KWLUKWLU 37DZCYDZCY

Examples of GR and its 3 matrix components are shown petroleum or gas can be obtained by the change of the in Fig.1 for 27 EU countries with code p = 33 (27 nodes) corresponding money volume ﬂow related to code 33 or plus petroleum of Russia, i.e., a total of 28 nodes for GR 34 by multiplying it by (1 + δ), computing all rank prob- (from the global network with N = 13847 nodes). The abilities and then the derivative dBc/dδ. same GR matrix but for gas from Russia is shown in Fig.2.

This approach was used in [6]. However, in this way We discuss the properties of these GR matrix compo- there we had the effect of global price change of petroleum nents shown in Figs.1,2 in the next Section. Here we only or gas for all countries. Here, we want to determine the note that for selected countries this GR matrix captures sensitivity of country balance to a flow of petroleum from only trade in petroleum (or gas). This can be interesting a specific country (e.g. RU, US, or SA). Thus we first in itself but in this way we cannot obtain the country bal- compute all 4 matrix components of the reduced Google ance and its sensitivity. Thus we follow another approach. matrix GR, Gpr, Grr, Gqr and then we recompute these We take 27 EU countries with all their products (that matrices with the price modification factor (1+δ) applied gives us 27 × 61 = 1647 nodes) and we add to these nodes only for the trade of a given EU country with Russia (or the node of RU-petroleum. In this way we obtain GR ma- with US, or SA). trix with the size of Nr = 1647 + 1 = 1648 nodes (from C. Coquidé et al.: Influence of petroleum and gas trade on EU economies 5

NL NL NL NL 0.12 0.07 FR FR FR 0.35 FR 0.45 DE DE DE DE ES ES ES ES GB GB GB GB 0.4 0.06 IT IT 0.1 IT 0.3 IT BE BE BE BE PL PL PL PL 0.35

SI SI SI SI 0.25 0.05 CZ CZ CZ CZ 0.08 AT AT AT 0.3 AT

SE SE SE SE HU HU HU 0.2 HU 0.04 PT PT PT 0.25 PT

RO RO RO RO 0.06

BG BG BG BG 0.2 SK 0.15 SK 0.03 SK SK GR GR GR GR

MT MT MT MT 0.15 0.04 RU RU RU RU

LT 0.1 LT 0.02 LT LT

IE IE IE IE 0.1 CY CY CY CY

DK DK DK DK 0.02 0.05 0.01 FI FI FI FI 0.05 LV LV LV LV

LU LU LU LU

EE EE EE EE 0 0 0 0 FI FI IT IT IE IE SI SI LT LT FI FI IT IT LV LV PT PT PL PL IE IE SI SI LU LU CZ AT CZ AT NL FR NL FR EE EE SE SE ES ES CY CY SK SK BE BE DE DE MT RU MT RU DK DK HU HU RO BG GR RO BG GR GB GB LT LT LV LV PT PT PL PL LU LU CZ AT CZ AT NL FR NL FR EE EE SE SE ES ES CY CY SK SK BE BE DE DE MT RU MT RU DK DK HU HU RO BG GR RO BG GR GB GB

NL NL NL NL

FR FR FR FR

DE DE DE DE 0.3 0.35 0.05 ES ES ES ES

GB GB 0.02 GB GB

IT IT IT IT BE BE BE 0.3 BE PL 0.25 PL PL PL 0.04 SI SI SI SI

CZ CZ CZ CZ 0.015 0.25 AT AT AT AT

SE 0.2 SE SE SE

HU HU HU HU 0.03 PT PT PT 0.2 PT RO RO RO RO BG 0.15 BG BG BG 0.01 SK SK SK SK

GR GR GR 0.15 GR 0.02 MT MT MT MT RU 0.1 RU RU RU LT LT LT 0.1 LT IE IE IE IE 0.005 CY CY CY CY 0.01 DK 0.05 DK DK DK FI FI FI 0.05 FI

LV LV LV LV

LU LU LU LU

EE EE EE EE 0 0 0 0 FI FI IT IT IE IE SI SI FI FI IT IT IE IE SI SI LT LT LV LV PT PT PL PL LU LU CZ AT CZ AT NL FR NL FR EE EE SE SE ES ES LT LT CY CY SK SK BE BE DE DE LV LV PT PT PL PL MT RU MT RU DK DK HU HU LU LU CZ AT CZ AT NL FR NL FR EE EE SE SE RO BG GR RO BG GR ES ES GB GB CY CY SK SK BE BE DE DE MT RU MT RU DK DK HU HU RO BG GR RO BG GR GB GB

Fig. 1. Left four panels: reduced Google matrix GR (top left) and its matrix components Gpr (top right), Grr (bottom left) and Gqrnd (bottom right) for the petroleum product (code p = 33) exchanged among the 27 EU countries and Russia in 2016. ∗ Right four panels: the same as on the left but for reduced Google matrix GR and its three matrix components in the same order as on the left. Here, the EU countries and RU are ordered as in the PageRank column of Table1.

the total size of G being N = 13847). In this GR matrix We characterize the weight WR, Wpr, Wrr, Wqr of GR we have all direct and indirect links of all products of 27 and its 3 matrix components Gpr, Grr, Gqr by the sum EU countries with petroleum of RU. In this GR matrix of all its elements divided by the matrix size Nr (Wqrnd we can change the petroleum price using the multiplier for Gqrnd). By definition we have WR = 1. It is usual (1 + δ) for links from RU petroleum to other nodes with for Wikipedia networks that the weight Wpr ≈ 0.95 (see the renormalization of all matrix elements in this column e.g. [13,14]) is rather close to unity since Gpr is approx- to unity. Then we obtain the probabilities Ppc for all 27+1 imately composed from identical columns of PageRank countries. The same procedure is done for the CheiRank vector, while the remaining weight of about 0.05 is ap- ∗ GR matrix getting Ppc and then the balance sensitivity proximately equally distributed between Wrr and Wqr. dBc/dδ of country (including all its products) to Russian We find that for the WTN the situation is different. We petroleum. The same procedure is used to obtain the sen- have Wpr = 0.651568, Wrr = 0.30849, Wqr = 0.039942 sitivity to Russian gas (or US or other country gas). The and Wqrnd = 0.036512 so that the weight of Wpr is sig- sensitivity computed in this way gives us the real sensiti- nificantly reduced and the weight of Wrr is significantly vity of country balance taking into account all direct and larger than the weight of Wqr. We attribute this to the indirect links present in the WTN. fact that the global S matrix of WTN contains many links (about 2000 links per node for matrix elements with amplitude being larger than ∼ 10−4) in contrast with 3 Results the very sparsed Wikipedia S matrix. Hence, for WTN the importance of direct links is significantly higher. For ∗ GR and its 3 matrix components we obtain the following Here we present the results for EU trade obtained with ∗ ∗ ∗ weights Wpr = 0.6051, Wrr = 0.34379, Wqr = 0.05111 and the reduced Google matrix algorithm. ∗ Wqrnd = 0.047 which are similar to the GR case.

In Fig.1 (left 4 panels) we show GR matrix with its 3 ∗ 3.1 Examples of reduced Google matrices GR and GR matrix components for petroleum product (code p = 33) trade of 27 EU countries with Russia. For GR and Grr the ∗ In Fig.1 we show the reduced Google matrices GR and GR dominant matrix elements correspond to trade flow from and their 3 matrix components for EU petroleum trade Republic of Ireland (IE) to UK (GB). Indeed, since UK with Russia 2016. The matrix size of selected nodes is and IE both have territories on island of Ireland the trade Nr = 27 + 1 = 28, the direct and indirect link con- flow between two countries is very high. The next by the tributions from other network nodes Ns = N − Nr = amplitude is the trade flow from Denmark (DK) to Swe- 13847 − 28 = 13819 are taken into account by the REGO- den (SE) both in GR and Grr. Among the indirect links in MAX algorithm. The nodes are ordered by the PageRank Gqr we find as the strongest the flow from Portugal (PT) index of countries given in Table1. to Spain (ES) and from Romania (RO) to Bulgaria (BG) 6 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies

NL NL NL NL 0.1 0.5 BE BE 0.2 BE BE

FR FR FR 0.5 FR

IT IT IT IT

GB GB GB GB

ES ES ES ES 0.08 HU 0.4 HU HU HU DE DE DE 0.4 DE 0.15 PT PT PT PT

BG BG BG BG

SK SK SK SK

PL PL PL PL 0.06 0.3 SI SI SI 0.3 SI RO RO RO RO GR GR 0.1 GR GR RU RU RU RU

CZ CZ CZ CZ 0.2 0.04 SE SE SE 0.2 SE IE IE IE IE

LT LT LT LT

AT AT AT AT

DK DK 0.05 DK DK CY 0.1 CY CY 0.1 CY 0.02 EE EE EE EE

MT MT MT MT

LV LV LV LV

LU LU LU LU

FI FI FI FI 0 0 0 0 FI FI FI FI IT IT IT IT IE IE IE IE SI SI SI SI LT LT LT LT LV LV LV LV PT PL PT PL PT PL PT PL AT LU AT LU AT LU AT LU NL FR CZ NL FR CZ NL FR CZ NL FR CZ EE EE EE EE SE SE SE SE ES ES ES ES CY CY CY CY BE BE BE BE DE SK DE SK DE SK DE SK MT MT MT MT DK DK DK DK RU RU RU RU HU HU HU HU RO GR RO GR RO GR RO GR GB BG GB BG GB BG GB BG NL NL 0.14 NL NL 0.09 BE BE BE BE

FR 0.45 FR FR 0.45 FR IT IT IT IT 0.08 GB GB 0.12 GB GB

ES 0.4 ES ES 0.4 ES HU HU HU HU 0.07 DE DE DE DE 0.35 0.35 PT PT 0.1 PT PT BG BG BG BG 0.06 SK 0.3 SK SK 0.3 SK PL PL PL PL 0.08 SI SI SI SI 0.05 RO 0.25 RO RO 0.25 RO GR GR GR GR

RU RU RU RU 0.06 0.04 CZ 0.2 CZ CZ 0.2 CZ SE SE SE SE IE IE IE IE 0.03 LT 0.15 LT LT 0.15 LT 0.04 AT AT AT AT

DK DK DK DK 0.02 CY 0.1 CY CY 0.1 CY

EE EE EE EE 0.02 MT MT MT MT 0.05 0.05 0.01 LV LV LV LV

LU LU LU LU

FI FI FI FI 0 0 0 0 FI FI FI FI IT IT IT IT IE IE IE IE SI SI SI SI LT LT LT LT LV LV LV LV PT PL PT PL PT PL PT PL AT LU AT LU AT LU AT LU NL FR NL FR NL FR NL FR CZ CZ CZ CZ EE EE EE EE SE SE SE SE ES ES ES ES CY CY CY CY BE BE BE BE DE SK DE SK DE SK DE SK MT MT MT MT DK DK DK DK RU RU RU RU HU HU HU HU RO GR RO GR RO GR RO GR GB GB GB GB BG BG BG BG

Fig. 2. Left four panels: reduced Google matrix GR (top left) and its matrix components Gpr (top right), Grr (bottom left) and Gqrnd (bottom right) for the gas product (code p = 34) exchanged among the 27 EU countries and Russia in 2016. Right ∗ four panels: the same as on the left but for reduced Google matrix GR and its three matrix components in the same order as on the left. Here, the EU countries and RU are ordered as in the PageRank column of Table2.

and Cyprus (CY) to Italy (IT). However, the amplitude GR matrix gives the strongest gas import flows which are, of these transitions is relatively small. In the matrix com- by decreasing importance, from CY to IT, IE to GB, DK ponent Gpr the dominant transitions points to top Page- to SE, BE to FR, LT to PL, ES to PT, ... The Gpr matrix Rank countries NL, FR, DE, ES, GB, IT. In all the matrix component shows that the main importer for gas in EU is components the contribution of petroleum from RU is not NL. Indeed import flows toward NL are at least about one very pronounced. We see the similar features for the pe- order of magnitude more important than toward the other troleum trade from US and SA shown in Figs. A1 and A2 EU states and in particular FR which is nonetheless the of Appendix. These results show that the contribution of top importer according to ImportRank (see Table2). For petroleum trade is masked by the active trade between the case of gas trade between EU countries and RU, we EU countries with other products. note that the maximum matrix elements in Gqr have the ∗ same magnitude than the maximum matrix elements of The reduced Google matrix GR and its 3 matrix components are presented in Fig.1 (right 4 panels). Here, we the other matrix components. In particular, hidden indi- keep in mind that the flow directions have been inverted rect import flows from LU and PL toward Hungary (HU) to compute CheiRank probabilities. Thus to obtain the are clearly visible from Gqr. In Fig.2 (right 4 panels), ∗ highest petroleum exports from Russia we have to focus from GR, the strongest gas export flows emanate mainly on the largest matrix elements on the RU horizontal line. from RU toward (by decreasing importance) Latvia (LV), EE, FI, BG, RO, ... Besides this Russian gas export, the Contrarily to the GR case, here RU exports of petroleum clearly dominate the G∗ matrix and its 3 matrix com- second and third most important gas export flows are in R ∗ ponents; this is mainly due to the fact that RU is the fact from GB to IE and from BE to LU. From Gpr we see petroleum top exporter (see CheiRank and ExportRank that gas export flows from NL, which is the top EU gas ∗ exporter according to CheiRank (see Table2), although in Table1). From GR we observe that the strongest petroleum flows from RU point (in decreasing importance) weaker than the ones from RU are nonetheless of the same to Latvia (LV), Lithuania (LT), Finland (FI), BG, Poland order of magnitude. Among EU countries and Russia, NL (PL), Estonia (EE), ... which are countries peripheral to and RU compete for the best gas supplier. Although the ∗ RU. We also note non negligible petroleum flows from NL weight of Gqr is weaker than the weight of the other ma- to Belgium (BE), and from BE to Luxembourg (LU). trix components, hidden indirect gas export flows can be ∗ ∗ seen in Gqr from, SE and RU, to DE. Fig.2 presents reduced Google matrices GR and GR for EU gas trade with Russia. The weights of the GR ∗ and GR matrix components WR = 1, Wpr = 0.634069, 3.2 Network structure of petroleum and gas EU trade Wrr = 0.308960, Wqr = 0.056971 (Wqrnd = 0.051085) ∗ ∗ ∗ ∗ ∗ and WR = 1, Wpr = 0.611761, Wrr = 0.322066, Wqr = From GR and G matrices shown in Fig.1, we are able ∗ R 0.066173 (Wqrnd = 0.058111), are similar to those of EU to extract the network structure of the petroleum trade petroleum trade with Russia. In Fig.2 (left 4 panels), the between EU countries and RU. Fig.3 left panel shows C. Coquidé et al.: Influence of petroleum and gas trade on EU economies 7

Fig. 3. Network of petroleum import and network of petroleum export between EU countries and Russia in 2016. The EU and ∗ Russia petroleum reduced network is built from GR for import (left panel) and from GR for export (right panel). The network construction rule is the following: for each country c, we determine the 4 best petroleum importers from (exporters to) country ∗ c according to GR (GR). The directed links illustrate the petroleum flows. the petroleum import trade network between EU and RU. EU petroleum exporters are NL, BE, DE, IT, GR. From The top 6 EU economies by nominal GDP (i.e. DE, GB, both of the EU+SA petroleum trade networks shown in FR, IT, ES, NL in 2016 [29]) are the main petroleum im- Fig.4 we observe a situation different from the EU+RU porters, NL and FR being the more central. The perfor- case (Fig.3) as not only NL but also DE, IT, GB, con- mances of these economies are consequently correlated to stitute each one a hub for petroleum exchanges. Although their abilities to efficiently import petroleum. The four SA is the top petroleum exporter worldwide, RU is the main direct and/or indirect EU gates for RU petroleum main supplier for EU, this is the reason why trade net- are DE, FR, NL, IT. We note closed loop petroleum ex- works with SA allows also to unveil secondary petroleum change between (almost) neighboring countries, e.g. DE- exchange hubs. AT, CZ-SK, DE-PL, AT-HU, AT-SK, PT-ES, ES-IT, SE- FI. Fig.3 right panel shows the petroleum export trade network between EU and RU. We clearly retrieve the fact 3.3 Sensitivity of EU to petroleum price that RU is the first petroleum supplier of EU and that NL is the top EU exporter of petroleum (see CheiRank and ExportRank in Table1). From both of the petroleum Above we have considered the reduced Google matrices ∗ trade networks shown in Fig.3 we observe that NL con- GR and GR with related networks only for petroleum or stitutes the main European hub for petroleum exchanges. gas flows of 27 EU countries plus Russia (or SA, US). Secondary petroleum exporters are GR, IT, BE, GB, SE, However, this approach does not capture the global influ- and DE. ence of petroleum or gas trade on the all products trade balance of a given EU country. Therefore we extend our We also construct the reduced Google matrices GR and G∗ associated to petroleum import and export be- REGOMAX analysis taking into account the matrix size R ∗ tween EU countries and Saudi Arabia (SA). Fig.4 shows Nr = 1648 for GR and GR (see Section 2.4). As the main the petroleum trade network between EU countries and characteristic we analyze the sensitivity of country global SA. The EU+SA petroleum import trade network (Fig.4 trade balance in respect to small petroleum price increase left panel) is similar to the one obtain for EU+RU (see (from unit price 1 to price 1 + δ) expressed by the deriva- Fig.3 left panel). This illustrate the robustness of the EU tive dBc/dδ. As described in section 2.4 we express the ∗ intramarket in regards to petroleum import. The main country c balance Bc via CheiRank Pc and PageRank Pc ˆ∗ entrances in EU for SA petroleum are FR, IT, ES, and probabilities and also via ExportRank Pc and Import- NL. Fig.4 right panel shows that besides SA the main Rank Pˆc probabilities. 8 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies

Fig. 4. Network of petroleum import and network of petroleum export between EU countries and Saudi Arabia in 2016. The ∗ EU and Arabia petroleum reduced network is built from GR for import (left panel) and from GR for export (right panel). The network construction rule is the following: for each country c, we determine the 4 best petroleum importers from (exporters to) ∗ country c according to GR (GR). The directed links illustrate the petroleum ﬂows.

In Fig.5 we present the sensitivity dBc/dδ, shown by count all direct and hidden links between selected nodes color, to petroleum trade with RU, US and SA on the EU of the WTN. Due to these reasons below we focus mainly political map for year 2016. The sensitivity to petroleum on results obtained with the REGOMAX analysis. from Russia is shown in Fig.5 top left panel. We see that The sensitivity of EU to petroleum price from SA and the strongest negative effect is produced on NL which is US are shown in the bottom panels of Fig.5. For SA the on at the top PageRank position (see Table1) due to its most sensitive countries are Spain (ES) and NL while for strong maritime relations which bring a lot of petroleum to US the most sensitive is NL. However, for EU the sensi- NL and then redistributed to other EU countries. The next tivity to petroleum of SA is by a factor 3-4 smaller than most sensitive EU countries are Italy (IT), Greece (GR), for those from RU. Thus the sensitivity of Germany to Bulgaria (BG), Poland (PL), Lithuania (LT) and Latvia Russian petroleum is by a factor 5 stronger than of SA (LV). We note that here the sensitivity dBc/dδ is defined petroleum. In contrast the maximum EU sensitivity to via CheiRank and PageRank probabilities taking into ac- US petroleum is by a factor 2 stronger than to those of count the multiplicity of WTN links. The result is very dif- RU. The sensitivity of Germany is comparable for US and ferent (see Fig.5 top right panel) if the sensitivity dBˆc/dδ RU. Let us note that GR is not affected and even benefit is defined by ExportRank and ImportRank probabilities, from SA petroleum price increase. The same for FI bene- which are usually used in economy for the trade analysis. fiting from RU petroleum price increase. Also we observe This crude Export-Import analysis gives the most strong a rigid component of Eastern EU countries from Sweden negative sensitivity for Latvia (LV). The next is Lithuania to Greece and from Baltic countries to Germany which are (LT) which as LV keeps close trade relations with RU be- almost insensitive to US petroleum (Fig.5 bottom right ing ex-USSR republics. Moreover the Export-Import anal- panel). ysis gives a rigid component of Western EU countries al- The time evolution of EU sensitivity to petroleum from most not sensitive to RU petroleum. The drastic global RU, SA, and US is shown in Fig.6 for years 2004, 2008, difference between REGOMAX analysis and the simple 2012 (for year 2016 see previous Fig.5). For RU petro- standard Import-Export analysis is that the first consid- leum the most sensitive country are Netherlands (NL), ers the multilateral cascade of direct or indirect trades Italy (IT), Cyprus (CY) in 2004, NL and IT in 2008 and between two countries and the second only considers the NL in 2012 and 2016. Also the maximal negative sensitiv- direct bilateral trade between two countries. We consider ity is changing from −0.0016 in 2004 to −0.0029 in 2008, that the REGOMAX algorithm provides much more de- −0.0037 in 2012 and −0.0017 in 2016. From these maximal tailed and realistic information on sensitivity to petroleum sensitivities and also from the distribution of sensitivities price compared to the usual Export-Import consideration. among EU countries, we observe an overall increase of the We attribute this advantage of REGOMAX analysis to its balance trade sensitivity to RU petroleum until 2012, then deep mathematical properties that allows to take into ac- we remark that EU trade sensitivities in 2016 decreases be- C. Coquidé et al.: Influence of petroleum and gas trade on EU economies 9

by petroleum price increases. This strong indirect feature is absolutely not captured by the standard Import-Export analysis picture (see e.g. Fig.5 top right panel). In global we see that EU countries are more sensitive to US petroleum that is by a factor 2-3 stronger comparing to those of RU. We relate this to the fact that US is the world top PageRank country so that it has more global world inﬂuence on other countries.

3.4 Sensitivity of EU to gas price

^ We present in Fig.7 EU trade balance sensitivity to gas from RU and Norway (NO) in 2016. EU sensitivities to RU gas is one order of magnitude weaker for gas than petroleum. The price increase of the RU gas mainly affects Italy (IT) while other Western EU countries being relatively not sensitive. Again RU neighboring countries are the most sensitives to RU gas import. The most sensitive EU economies to Norwegian gas are DE economy (and to a lesser extent GB and BE economies) which would be affected by NO gas price increase and SE economy which would benefit from it. The positive balance trade sensitivity for SE is certainly due to the entanglement of NO-SE economies. The others economies are insensitive to NO gas (see peak of fourteen EU countries with balance trade around 0 in Fig.7 right panel). Fig. 5. EU countries balance derivative dBc/dδ induced by Figs.8 and9 show from 2004 to 2016 time evolution an increase of petroleum price from Russia, Saudi Arabia, and of EU economies trade balance sensitivity to RU and NO United States in 2016. For each EU country c, we compute the gas. In Fig.8 we observe that during this time period the balance derivative dBc/dδ induced by an infinitesimal change Western EU bloc from PT to DE remained insensitive to of petroleum price to EU country c from Russia (top left), from RU gas with the exception of IT economy which became Saudi Arabia (bottom left), from United States (bottom right). the most affected since 2012 (also FR economy were tem- ˆ The balance derivative dBc/dδ, computed using ImportRank porarily sensitive to RU gas around 2004). In Fig.9 we ˆ ˆ∗ Pc and ExportRank Pc , induced by an increase of petroleum observe that during the same period EU east end coun- price from Russia to EU country c is shown in top right panel. tries are insensitive to NO gas. The most affected countries by NO gas volume import and/or price increase are FR, BE, DE in 2004, BE in 2008, BE, NL, DE in 2012 and ing comparable back to those in 2004. We attribute this DE in 2016. SE economy always benefit from volume in- to a significant drop of petroleum price happened in the crease of NO gas excepting in 2008, at that time SE was world after the financial crisis of 2007-2008. A similar ten- relatively affected and GB was benefiting from NO gas. dency is visible for SA and US petroleum sensitivity. For SA petroleum the most sensitive countries are NL, IT and Greece (GR) in 2004, NL, IT in 2008, NL, Spain 4 Discussion (ES) in 2012 and 2016 with the maximal negative sensitivity changing from −0.0006 in 2004, −0.0008 in 2008, In this work we developed the reduced Google matrix (RE- −0.001 in 2012 and −0.0005 in 2016. As in the RU case, GOMAX) analysis of the multiproduct world trade net- trade balance sensitivities of EU countries to SA petro- work with a specific accent to sensitivity of EU country leum increases until 2012 and then decreases in 2016 to trade balance to petroleum and gas prices from Russia, attain values comparable to year 2004. USA, Saudi Arabia and Norway. In particular we observe For US petroleum the most sensitive countries are NL that, during the 2004-2016 time period, most of the EU and ES in 2004 and NL in 2008, 2012 and 2016 with the countries are sensitive to price increase of Russian petro- maximal negative sensitivity changing from −0.0052 in leum and petroleum products. The situation is different 2004, −0.0127 in 2008, −0.0122 in 2012 and −0.0037 in for Saudi Arabia and US petroleum price influence for 2016. which east and central EU countries are relatively less af- Globally, the ancient USSR satellites and more glob- fected. The Netherlands, which is the best EU petroleum ally central EU economies are less affected by the increase importer and exporter, is during this time period the most of petroleum from US or SA. We also observe that due to affected EU country by the price increase of either Rus- NL central position in petroleum import and export for sia, Saudi Arabia, or USA. The influence of Russian gas EU, the performance of NL economy is the most affected is mostly exerted to Eastern EU countries among which 10 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies

Fig. 6. EU countries balance derivative dBc/dδ induced by an increase of petroleum price from Russia (left column), Saudi Arabia (middle column), and United States (right column), for 2004 (top row), 2008 (middle row), and 2012 (bottom row). ancient USSR satellites, Western EU countries being in- ing the whole 2004-2016 time period, France in 2004, Bel- sensitive with the exception of Italy. Although Norway is gium in 2004 and 2012, and The Netherlands in 2012, but the second gas supplier for EU, the Norway price increase benefiting to Sweden (excepting around 2008). influences only few EU countries, affecting Germany dur- C. Coquidé et al.: Influence of petroleum and gas trade on EU economies 11

Fig. 7. EU countries balance derivative dBc/dδ induced by an increase of gas price from Russia (left panel) and Norway (right panel) in 2016.

Fig. 9. EU countries balance derivative dBc/dδ induced by an increase of gas price from Norway for 2004 (top left), 2008 (top right), 2012 (bottom left) and 2016 (bottom right).

ther investigation of such indirect inﬂuence will play an important role in petroleum or gas crisis contamination propagation in EU trade.

Acknowledgments

We thank the representatives of UN COMTRADE [1] for providing us with the friendly access to this database. This work was supported in part by the Pogramme In- vestissements d’Avenir ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT (project THETRACOM); it was granted access to the HPC resources of CALMIP (Toulouse) under the allocation 2017-P0110. This work Fig. 8. EU countries balance derivative dBc/dδ induced by was also supported in part by the Programme Investisse- an increase of gas price from Russia for 2004 (top left), 2008 ments d’Avenir ANR-15-IDEX-0003, ISITE-BFC (project (top right), 2012 (bottom left) and 2016 (bottom right). GNETWORKS) and by Bourgogne Franche-Comt´eregion (project APEX).

We show that comparing to the usual export-import consideration this REGOMAX approach takes into ac- Appendix count the cascade of chain influence propagation via all nontrivial pathways of trade relations between countries. Here we present some additional figures of the reduced Due to this feature this approach is more powerful com- Google matrix analysis of EU trade. Fig. A1 shows the re- ∗ pared to only nearby link analysis considered in the import- duced Google matrices GR and GR for petroleum product export approach. Thus the REGOMAX method allows to associated to 27 EU countries and Saudi Arabia. Fig. A2 ∗ recover indirect influence of petroleum or gas price from shows the reduced Google matrices GR and GR for petro- a specific country on EU trade. We argue that the fur- leum product associated to 27 EU countries and US. 12 C. Coquidé et al.: Influence of petroleum and gas trade on EU economies

NL NL NL 0.35 NL 0.1 FR 0.35 FR FR FR

DE DE DE DE

ES ES 0.07 ES ES

GB GB GB 0.3 GB

IT 0.3 IT IT IT 0.08 BE BE BE BE 0.06 PL PL PL PL

SI SI SI 0.25 SI 0.25 CZ CZ CZ CZ

AT AT 0.05 AT AT SE SE SE SE 0.06 0.2 HU 0.2 HU HU HU PT PT PT PT 0.04 RO RO RO RO

BG BG BG BG 0.15 SK 0.15 SK SK SK 0.04 GR GR 0.03 GR GR

MT MT MT MT

SA SA SA SA 0.1 LT 0.1 LT LT LT 0.02 IE IE IE IE CY CY CY CY 0.02 DK DK DK DK 0.05 0.05 FI FI 0.01 FI FI LV LV LV LV

LU LU LU LU

EE EE EE EE 0 0 0 0 FI FI FI FI IT IT IT IT IE IE IE IE SI SI SI SI LT LT LT LT PT PT PT PT PL PL PL PL LV LV LV LV CZ AT CZ AT CZ AT CZ AT LU LU LU LU NL FR NL FR NL FR NL FR SE SE SE SE EE EE EE EE ES ES ES ES SK SA SK SA SK SA SK SA CY CY CY CY BE BE BE BE DE DE DE DE MT MT MT MT HU HU HU HU DK DK DK DK RO BG GR RO BG GR RO BG GR RO BG GR GB GB GB GB

NL NL NL NL FR FR FR 0.3 FR DE DE DE DE 0.3 ES ES ES ES 0.025 GB GB 0.02 GB GB

IT IT IT IT

BE BE BE 0.25 BE

PL 0.25 PL PL PL SI SI SI SI 0.02 CZ CZ CZ CZ 0.015 AT AT AT 0.2 AT SE 0.2 SE SE SE

HU HU HU HU PT PT PT PT 0.015 RO RO RO 0.15 RO BG 0.15 BG BG BG 0.01 SK SK SK SK

GR GR GR GR

MT MT MT MT 0.01 0.1 SA 0.1 SA SA SA LT LT LT LT

IE IE IE IE 0.005 CY CY CY CY 0.005 DK 0.05 DK DK 0.05 DK FI FI FI FI

LV LV LV LV

LU LU LU LU

Fig. A1. Left four panels: reduced Google matrix GR (top left) and its matrix components Gpr (top right), Grr (bottom left) and Gqrnd (bottom right) for the petroleum product (code p = 33) exchanged among the 27 EU countries and Saudi Arabia ∗ in 2016. Right four panels: the same as on the left but for reduced Google matrix GR and its three matrix components in the same order as on the left. Here, the EU countries and SA are ordered as in the PageRank column of Table1.

US 0.35 US US US NL NL NL NL

FR FR 0.14 FR FR 0.18

DE DE DE DE 0.3 ES 0.3 ES ES ES 0.16 GB GB 0.12 GB GB IT IT IT IT

BE BE BE BE 0.25 0.14 PL 0.25 PL PL PL SI SI 0.1 SI SI CZ CZ CZ CZ 0.12 AT AT AT AT 0.2 SE 0.2 SE SE SE 0.08 HU HU HU HU 0.1 PT PT PT PT

RO RO RO RO 0.15 BG 0.15 BG BG BG 0.08 0.06 SK SK SK SK

GR GR GR GR

MT MT MT MT 0.06 0.1 0.1 LT LT 0.04 LT LT IE IE IE IE

CY CY CY CY 0.04

DK DK DK DK 0.05 0.05 FI FI 0.02 FI FI 0.02 LV LV LV LV

LU LU LU LU

EE EE EE EE 0 0 0 0 FI FI FI FI IT IT IT IT IE IE IE IE SI SI SI SI LT LT LT LT PT PT PT PT PL PL PL PL LV LV LV LV CZ AT CZ AT CZ AT CZ AT LU LU LU LU NL FR NL FR NL FR NL FR SE SE SE SE EE EE EE EE ES ES ES ES SK CY SK CY SK CY SK CY BE BE BE BE US DE US DE US DE US DE MT MT MT MT HU DK HU DK HU DK HU DK RO BG GR RO BG GR RO BG GR RO BG GR GB GB GB GB

US US US US

NL NL NL NL 0.3 0.035 FR FR FR FR 0.3 0.03 DE DE DE DE

ES ES ES ES GB GB GB GB 0.03 IT IT IT 0.25 IT 0.025 BE 0.25 BE BE BE

PL PL PL PL

SI SI SI SI 0.025 CZ CZ CZ 0.2 CZ AT 0.2 AT 0.02 AT AT

SE SE SE SE

HU HU HU HU 0.02 PT PT PT 0.15 PT RO 0.15 RO 0.015 RO RO BG BG BG BG 0.015 SK SK SK SK

GR GR GR GR 0.1 MT 0.1 MT 0.01 MT MT LT LT LT LT 0.01 IE IE IE IE

CY CY CY CY DK 0.05 DK 0.005 DK 0.05 DK FI FI FI FI 0.005

LV LV LV LV

LU LU LU LU

EE EE EE EE 0 0 0 0 FI FI FI FI IT IT IT IT IE IE IE IE SI SI SI SI LT LT LT LT PT PT PT PT PL PL PL PL LV LV LV LV CZ AT CZ AT CZ AT CZ AT LU LU LU LU NL FR NL FR NL FR NL FR SE SE SE SE EE EE EE EE ES ES ES ES SK SK SK SK CY CY CY CY BE BE BE BE US DE US DE US DE US DE MT MT MT MT HU HU HU HU DK DK DK DK RO BG GR RO BG GR RO BG GR RO BG GR GB GB GB GB

Fig. A2. Left four panels: reduced Google matrix GR (top left) and its matrix components Gpr (top right), Grr (bottom left) and Gqrnd (bottom right) for the petroleum product (code p = 33) exchanged among the 27 EU countries and USA in 2016. ∗ Right four panels: the same as on the left but for reduced Google matrix GR and its three matrix components in the same order as on the left. Here, the EU countries and US are ordered as in the PageRank column of Table1.

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