Crisis contagion in the world trade network
CCoquid´e et al.
RESEARCH
Crisis contagion in the world trade network
C´elestin Coquid´e , Jos´e Lages and Dima L Shepelyansky * Correspondence:[email protected] Institut UTINAM, UMR 6213,CNRS, Universit´e BourgogneFranche-Comt´e, Besan¸con, FranceFull list of author information isavailable at the end of the article
Abstract
We present a model of worldwide crisis contagion based on the Google matrixanalysis of the world trade network obtained from the UN Comtrade database.The fraction of bankrupted countries exhibits an on-off phase transition governedby a bankruptcy threshold κ related to the trade balance of the countries. For κ > κ c , the contagion is circumscribed to less than 10% of the countries,whereas, for κ < κ c , the crisis is global with about 90% of the countries going tobankruptcy. We measure the total cost of the crisis during the contagion process.In addition to providing contagion scenarios, our model allows to probe thestructural trading dependencies between countries. For different networksextracted from the world trade exchanges of the last two decades, the globalcrisis comes from the Western world. In particular, the source of the global crisisis systematically the Old Continent and The Americas (mainly US and Mexico).Besides the economy of Australia, those of Asian countries, such as China, India,Indonesia, Malaysia and Thailand, are the last to fall during the contagion. Also,the four BRIC are among the most robust countries to the world trade crisis. Keywords:
Complex networks; world trade; contagion crisis; Google matrix;PageRank; phase transition
Introduction
The financial crisis of 2007-2008 highlighted the enormous effect of contagion overworld bank networks (see e.g. [1, 2, 3]). Similar contagion effects appear also inthe world trade which is especially vulnerable to energy crisis mainly related tothe trade of petroleum and gas (see e.g. [4, 5]). In this work, we model the crisiscontagion in the world trade using the UN Comtrade database [6]. We use theGoogle matrix analysis [7, 8, 9] of the world trade network (WTN) developed in[10, 11]. In comparison with the usual import-export analysis based on the countingof trade volumes directly exchanged between countries, the advantage of the Googlematrix analysis is that the long range interactions between the network nodes,i.e., the countries, are taken into account. Otherwise stated, this analysis capturesthe fact that even two countries which are not direct trade partners can possiblyhave their economies correlated through the cascade of trade exchanges betweena chain of intermediary countries. The power of the specific Google matrix relatedalgorithms, such as the PageRank algorithm, is well illustrated by the success of theGoogle search engine [7, 8], and also by their possible applications to a rich varietyof directed networks (see [9] for a review). The detailed UN Comtrade database,collected for about 50 years, allows to perform a thorough modeling of the crisiscontagion in the WTN. In the following, we use the contagion model inspired bythe analysis of the crisis in the Bitcoin transactions network presented in [12, 13]. a r X i v : . [ q -f i n . T R ] F e b oquid´e et al. Page 2 of 26
We note that various research groups studied the statistical properties of the worldtrade network (see e.g. [14, 15, 16, 17, 18, 19, 20]) but the contagion process has notbeen analyzed so far. We think that our study will attract research interest to thisnontrivial and complex process. Such an analysis can be also extended to networksof interconnected banks (see e.g. [21]) where the contagion process is of primaryimportance.
Datasets
Using the UN Comtrade database [6], we construct the multiproduct World TradeNetwork (WTN) for the years 2004, 2008, 2012 and 2016. Each year is characterizedby a money matrix, M pcc (cid:48) , giving the export flow of product p , expressed in USD,from country c (cid:48) to country c . The data concern a set C of N c = 227 countries andterritories, and a set P of N p = 61 principal type of products. The list of theseproducts, which belongs to the Standard International Trade Classification (Rev.1), is given in [11]. The 2016 WTN is represented in Fig. 1. The set C comprises N c = 227 sovereign states and territories which are listed, with their associatedISO 3166-1 alpha-2 code, in the Abbreviations section. Among territories, mostof them belong to a sovereign state, some are disputed territories, such as WesternSahara, and Antarctica is a international condominium. The UN Comtrade databaseinventories commodities flows, not only between sovereign states, but also from andto these territories. The present study complies with the UN Comtrade terms ofuse. Model
In this section, we recall the construction process of the google matrix G associatedto the WTN, and the PageRank-CheiRank trade balance (PCTB) [11, 22, 23]. Weintroduce also a model of crisis contagion in the WTN. Multiproduct World Trade Network
For a given year, the multiproduct WTN is characterized by N c N p nodes, each onerepresenting a couple of country and product ( cp ). We assign a weight M pcc (cid:48) to thedirected link from node c (cid:48) p to cp . We define V cp = (cid:80) c (cid:48) M pcc (cid:48) as the total volume ofproduct p imported by the country c , and V ∗ cp = (cid:80) c (cid:48) M pc (cid:48) c as the total volume ofproduct p exported from the country c . Google matrix of the World Trade Network
The Google matrix G is constructed as G = αS + (1 − α ) ve T (1)where S is a stochastic matrix, the elements of which are S cp,c (cid:48) p (cid:48) = (cid:40) δ pp (cid:48) M pcc (cid:48) /V ∗ c (cid:48) p if V ∗ c (cid:48) p (cid:54) = 01 /N if V ∗ c (cid:48) p = 0 . (2)Here, α ∈ [0 . ,
1[ is the damping factor, v is a preferential probability vector,and e T = (1 , , . . . ,
1) is a row vector. The Google matrix G (1) describes the oquid´e et al. Page 3 of 26
MZ SOKM SCMUTZMW MGUGKE KHIDCXCCMY BNSGMMTHNPBT VNLAMNBD PHCN PWKR PGMPGUTL JPKP MHKI TVNR FJVUFMSBSABH OMQAAEDJYEERSD ET KGPKMVIO LKIN AU NC NZNFPTGIFOMAEH ESIS IE KWIQ AZ TJUZTMAMGE KZAFIRSESKSICZNODK HUATHRDEFR CHLUGBDZADBENL LBSYEGCYJOPSILBYEEFILVLT RUTRUAMDGRTNSMLYMTVAIT PLRSBAMEALMKBGROZWLSBV SZSH ZMNA BWGS ZAMLLR BFSLGN CI NE TDNG BICD RWCGAOHM GAAQ GQ CFCMTGBJSTGHCV GMMRGWSNNICUBZGT CRSV KYHNMX FKARUYPY BRBBDMMSGDVCAGAIKNLCTTBOPA COECPE CLJM AWDOVEUS TC VGBS HT BMCA SRGYPMGL
Figure 1 2016 World trade network.
Two countries A and B are related by a directed link, thedirection of which is given by its curvature. If A points to B following the bent path in theclockwise direction (A (cid:95)
B) then A exports to B, otherwise, i.e. (A (cid:94)
B), B exports to A. Thewidth of the link is proportional to the exportation volume in the WTN from the source countryto the target country. The colors of country nodes range from red (blue) for a country going tobankruptcy at stage τ = 0 ( τ = τ ∞ ) in the case of a bankruptcy threshold κ = 0 . and for thefollowing crisis scenario: once a country goes to bankruptcy, it is prevented to import productswith the exception of petroleum and gas (see details in the Contagion model section). Onlytransactions above USD are shown. Most of the Polynesian islands have been removed, hereand in the following figures, to improve visibility. transition probabilities of a random surfer which, with a probability α , followsthe architecture of the multiproduct WTN encoded in the stochastic matrix S ,and, with a probability (1 − α ), jumps to any node of the WTN according to thepreferential probability vector v . Below, we use either α = 0 . α = 0 .
85. Thissecond value is the one used in the seminal paper of Brin and Page devoted to thePageRank algorithm [7]. The PageRank vector P characterizes the steady state ofthe Markovian process described by the Google matrix G (1), i.e., G P = P . The cp component of the PageRank vector P , i.e., P cp , gives the fraction of time therandom surfer spent on the node cp during its infinite journey in the WTN.Following [11, 22], the final WTN Google matrix is obtained after two contructionsteps. We use a first preferential probability vector v , the components of which are v cp = V cp / ( N c V c ) where V c = (cid:80) p V cp is the total volume of commodities importedby the country c . This choice of the preferential probability vector ensures equityfor the random jumps between countries. This preferential probability vector v allows to compute the PageRank vector P associated to the Google matrix G .As a second step, we use the PageRank vector P to define a new preferentialprobability vector v , the components of which are v cp = P p /N c where P p = (cid:80) c (cid:48) P c (cid:48) p gives the ability of a product p to be imported. The final Google matrix G (1) isconstructed using the latter defined preferential probability vector v . The PageRankvector component P cp naturally characterizes the ability of a country c to import aproduct p [11, 22].It is interesting to consider the complex network built by inverting the directedlinks of the WTN. The Google matrix G ∗ associated to this inverted network is oquid´e et al. Page 4 of 26 obtained from the stochastic matrix S ∗ , the elements of which are S cp,c (cid:48) p (cid:48) = (cid:40) δ pp (cid:48) M pc (cid:48) c /V c (cid:48) p if V c (cid:48) p (cid:54) = 01 /N if V c (cid:48) p = 0 , (3)and from the preferential probability vectors v ∗ and v ∗ , the components of whichare v ∗ cp = V ∗ cp / ( N c V ∗ c ) and v ∗ cp = P ∗ p /N c , where V ∗ c = (cid:80) p V ∗ cp is the total exportvolume of the country c and P ∗ p = (cid:80) c (cid:48) P ∗ c (cid:48) p gives the ability of the product p to beexported. Here, P ∗ and P ∗ are the CheiRank vectors defined such as G ∗ P ∗ = P ∗ and G ∗ P ∗ = P ∗ . The CheiRank vector component P ∗ cp naturally characterizes theability of a country c to export a product p [11, 22, 23].In addition to the PageRank vector P and the CheiRank vector P ∗ , we can definethe ImportRank vector I and the ExportRank vector E , the components of whichare I cp = V cp /V and E cp = V ∗ cp /V where V is the total volume exchanged throughthe WTN. The ImportRank and ExportRank constitute crude accounting measuresof the capabilities of a country c to import or export a given product p . It has beenshown [11, 23] that the rankings by PageRank and CheiRank provide a more finermeasure of these capabilities since it takes account of the all the direct ( c (cid:48) p → cp )and indirect ( c (cid:48) p → c p → c p → · · · → cp ) economical exchanges of any commodity p between any pair of countries c (cid:48) and c . The PageRank and CheiRank algorithmsexpress the economical importance of a ( cp )-pair, i.e., a country-product pair, insidethe complex network constituted by the international trade. PageRank-CheiRank trade balance
As the PageRank and CheiRank algorithms measure the capabilities of a country toimport or to export products, we can define the PageRank-CheiRank trade balance(PCTB) of a given country c as B c = P ∗ c − P c P ∗ c + P c (4)where P c = (cid:80) p P cp is the country c PageRank component and P ∗ c = (cid:80) p P ∗ cp thecountry c CheiRank component. The PCTB is bounded, B c ∈ [ − , B c is positive, the more the country c is a more efficient exporter than importer in theWTN. Consequently, the country c economic health should be correlated with thevalue of B c .Analogously, the usual normalized import-export trade balance can be definedusing the ImportRank and the ExportRank asˆ B c = E c − I c E c + I c (5)where E c = (cid:80) p E cp is the country c total export volume (divided by V ) and where I c = (cid:80) p I cp is the country c total import volume (divided by V ). Contagion model
Countries with large negative PCTB naturally have to restrain their imports of nonvital goods. This restriction can be de facto , as not enough liquidity are available for oquid´e et al.
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WTN money matrix M input : Bankruptcy threshold κ output: Countries went in bankruptcy B τ at the crisis stage ττ = 0 , B − = ∅ repeat B τ = ∅ Using M , compute G , G ∗ , P and P ∗ for c ∈ C − (cid:83) τ − i =0 B i doif B c ≤ − κ then B τ = B τ + c if model A thenforeach c (cid:48) p ∈ C × ˜ P| M pcc (cid:48) (cid:54) = 0 do M pcc (cid:48) = 0 endendif model B thenforeach c (cid:48) p ∈ C × P| M pcc (cid:48) (cid:54) = 0 do M pcc (cid:48) = 0 endendendend τ = τ + 1 until B τ = ∅ ; τ ∞ = τ Algorithm 1:
Crisis contagion in the WTN.these countries, or can be imposed by a supranational organization in order to holdback a possible crisis contagion (e.g. the European Union for countries belongingto the Eurozone). Thus, let us assume that every country c with B c ≤ − κ goes tobankruptcy. Here, κ ≥ τ = 0, using the Google matrix G = G defined by (1), wecompute the PCTB B c for each country c . We obtain a set of countries B = { c ∈ C | B c ≤ − κ } which go to bankruptcy at the crisis stage τ = 0 and whichremain in this state in the following crisis stages τ >
0. Let us assume that allthe bankrupted countries are prevented to import products at the following stages, τ ≥
1. We will consider two cases: the import ban concerns all the products withthe exception of petroleum and gas (model A) or the import ban concerns all theproducts (model B). At the stage τ = 1, the world trade network is modified settingto zero the money matrix elements corresponding to the banned trade exchanges,i.e., M pcc (cid:48) = 0 , ∀ c (cid:48) ∈ C , ∀ c ∈ B , (cid:40) ∀ p ∈ ˜ P (model A) ∀ p ∈ P (model B) (6)where ˜ P = P − { petroleum , gas } is the set of all the exchanged commodities in theWTN with the exception of petroleum and gas. The Google matrix G is constructedusing the above modified money matrix M (6). We compute again the PCTB foreach country, and establish the set of countries, B = { c ∈ C − B | B c ≤ − κ } , whichgo to bankruptcy at the stage τ = 1 and will remain in this state at later stages τ >
1, according to model A or model B. The crisis contagion stops at the contagionstep τ ∞ for which no more countries go to bankruptcy. The WTN crisis contagionmodel is described by the Algorithm 1. This contagion model has already been usedto analyze the crisis contagion in the bitcoin transaction network [13]. oquid´e et al. Page 6 of 26
Let us define the proportion η ( τ, κ ) of the world countries in bankruptcy at thecrisis stage τ for the bankruptcy threshold κ . Here, η = 0 if no countries are inbankruptcy, and η = 1 if all the N c countries and territories are in bankruptcy. Fora given bankruptcy threshold κ , let us also define the cost of the crisis up to theend of the contagion stage τC ( κ, τ ) = (cid:88) c ∈ τ (cid:83) τ (cid:48) =0 B τ (cid:48) (cid:88) p ∈ ˜ P (model A)or p ∈ P (model B) V cp . (7)The value of C ∞ ( κ ) = C ( κ, τ ∞ ) gives the total cost of the crisis, i.e., it gives thetotal volume of all the non accomplished commercial exchanges due to the successivebankruptcy of countries during the crisis contagion. Results
Phase transition of the crisis contagion
The crisis contagion in the 2016 WTN is observed in Fig. 2 where the fraction η ofcountries which go to bankruptcy is displayed as a function of the crisis contagionstage τ and of the bankruptcy threshold κ . We clearly see a transition from aregime of contained contagion for κ > κ c to a regime of global contagion for κ < κ c .The Brin & Page original damping factor value, i.e., α = 0 .
85, leads to a lessfrank transition (Fig. 2, second row) than the α = 0 . κ c (Fig. 2, first row). We note also that the criticalbankruptcy threshold is, for α = 0 . κ c (cid:39) .
15 (model A) and κ c (cid:39) .
175 (modelB), and, for α = 0 . κ c (cid:39) .
18 (model A) and κ c (cid:39) .
24 (model B). For a givenbankruptcy threshold κ , the more α is low, the more the contagion is able to spreadall over the WTN. This explain that for α = 0 . κ c is lower than for α = 0 .
85. The model A ismore realistic than the model B since a country in bankruptcy still needs to importvital commodities, as petroleum and gas, in order to support its industry which inreturn will provide commodities to export. For low κ , the model B leads to a moreglobal contagion crisis ( η (cid:39)
1) than the model A since the latter model indirectlyprotects countries which are petroleum and/or gas exporters.Let us use the ImportRank and the ExportRank, and consequently, the normalizedimport-export trade balance ˆ B c (5), to monitor the crisis contagion. In this case,the third row of Fig. 2 shows the fraction η of countries in bankruptcy as a functionof the bankruptcy threshold κ . We observe that for any κ , more than a third of thecountries go to bankruptcy already at the τ = 0 crisis stage. Moreover, there is nocrisis containment for κ > κ c , since for a bankruptcy threshold κ just above thecritical value κ c half of the world countries and territories are in bankruptcy alreadyat the stage τ = 0 of the contagion. This ImportRank-ExportRank description isless suitable than the PageRank-CheiRank description to follow the crisis contagionsince the transition around κ c (cid:39) . η goes from 0 . . η goes from 0 .
15 to 0 . oquid´e et al. Page 7 of 26 η κ τ = τ = τ = τ = τ = τ∞ η κ τ = τ = τ = τ = τ =8 τ ∞ η κ τ = τ = τ = τ = τ = τ∞ η κ τ = τ = τ =3 τ =5 τ =8 τ∞ τ κ τ κ τ κ τ κ ηη ηη η κ τ = τ = τ = τ = τ∞ η κ τ = τ = τ = τ = τ∞ τ κ τ κ Figure 2 Fraction of bankrupted countries for the 2016 WTN.
Fraction η of countries went tobankruptcy up to the τ th stage of the crisis contagion as a function of the bankruptcy threshold κ . For the first column, once a country goes to bankruptcy, it is prevented to import productswith the exception of petroleum and gas (model A). For the second column, once a country goesto bankruptcy, it is prevented to import any product (model B). The first (second) rowcorresponds to a damping factor α = 0 . ( α = 0 . ). The evolution of the fraction of bankruptedcountries is monitored by the PCTB (4) (first and second rows) and by theImportRank-ExportRank balance (5) (third row). The insets show the corresponding fraction η ofbankrupted countries in the ( τ, κ ) plane. Dark red corresponds to the case where all the countrieswent to bankruptcy ( η = 1 ), and dark blue to the case where all the countries are safe ( η = 0 ).oquid´e et al. Page 8 of 26 η κ τ = τ = τ = τ = τ = τ ∞ η κ τ = τ = τ = τ = τ = τ∞ η κ τ = τ = τ = τ = τ = τ ∞ η κ = τ τ = τ = τ = τ = τ∞ Figure 3 Fraction of bankrupted countries for the WTN of 2004, 2008, 2012, and 2016.
Fraction η of countries went to bankruptcy up to the τ th stage of crisis contagion as a function ofthe bankruptcy threshold κ . The crisis contagion has been computed for the WTN of 2004 (topleft), 2008 (top right), 2012 (bottom left), and 2016 (bottom right). Once a country goes tobankruptcy, it is prevented to import products with the exception of petroleum and gas (modelA). The damping factor is α = 0 . . row, left column). For a given country c , the ImportRank-ExportRank descriptionis based only on the relative balance between the total export and import volumes.Contrarily to the PageRank-CheiRank description, it does not take into accountthe relative centrality of the country c in the WTN. Otherwise stated, it does nottake account of the possible strong indirect economical relations between countries.In the following, we analyze the crisis contagion in the WTN for different yearsusing the PageRank-CheiRank trade balance B c with the model A and with α = 0 . κ > κ c ), forwhich the crisis only spreads over a small fraction (less than 10%) of the countries, toa regime of global crisis contagion ( κ < κ c ), for which the crisis spreads over about90% of the countries. For these years, the transition occurs at about the same criticalbankruptcy threshold κ c (cid:39) . − . , κ c ]region. This is due to the interplay between the WTN rewiring occurring at the oquid´e et al. Page 9 of 26 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4 C ∞ / V κ / κ c -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4 0 4 8 12 16 0 0.5 1 1.5 2 2.5 3 3.5 4 τ ∞ κ / κ c Figure 4 (Left) Total number τ ∞ of crisis contagion stages as a function of the bankruptcythreshold κ and (right) total crisis cost C ∞ as a function of the bankruptcy threshold κ . Thetotal cost C ∞ is defined according to the formula (7), i.e., C ∞ ( κ ) = C ( κ, τ ∞ ) . We use the WTNfor years 2004 (black), 2008 (red), 2012 (blue), and 2016 (green). Solid lines correspond to themodel A: once a country goes to bankruptcy, it is prevented to import products with theexception of petroleum and gas. Dashed lines correspond to the model B: once a country goes tobankruptcy, it is prevented to import any product. The lines allow to adapt an eye between thedots which represent the numerically computed values. The total amount of the World Tradetransactions is V = 9 . × USD in 2004, . × USD in 2008, . × USD in2012, and . × USD in 2016. The damping factor is α = 0 . . successive stages of the crisis contagion and the relative protection of the mainpetroleum and gas exporters since even countries which went to bankruptcy canimport these commodities from these suppliers. Such irregular profile is absent forthe less realistic model B which exhibits an even more sharper phase transition thanthe model A (see Additional file 1 - Fig. A1). As an example, for the 2016 WTN,with the model A, Russia never goes to bankruptcy in the bankruptcy thresholdinterval κ < .
04, it goes to bankruptcy in the κ ∈ [0 . , .
11] interval, it stayssafe for κ ∈ [0 . , . κ = 0 .
14, then it stays safe for κ > .
14. The intervals for which Russia goes to bankruptcy are concomitant withthe bumps and the peaks observed for the 2016 WTN in the region κ < κ c (Fig. 3).In the model A, the fall of Russia is responsible for a almost complete WTN crisis.As seen in Fig. 3, in the κ ∈ [0 , κ c ] region, about 90% of the countries go tobankruptcy. The countries which remain safe at a bankruptcy threshold κ = 0 . κ = 0 .
1, the list of remainingsafe countries is short, and even countries with a strong component of petroleumand gas in their export volume go to bankruptcy. oquid´e et al.
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The total number of crisis contagion stages, τ ∞ , as a function of the bankruptcythreshold κ , also exhibits a phase transition (see Fig. 4 left) from a regime ( κ < κ c )for which τ ∞ rapidly increases with κ , from τ ∞ (cid:39) κ = 0 up to τ ∞ (cid:39) − κ = κ c , and a regime ( κ > κ c ) for which the crisis contagion stops after fewstages, τ ∞ (cid:46)
5. In the latter regime, it is not infrequent that the contagion evenstops after τ ∞ = 1 or 2 stages. We clearly observe that, for all the considered years,we obtain the same curve τ ∞ vs. κ/κ c whether we use the model A or the model B.The phase transition is also clearly seen in the evolution of the total crisis cost C ∞ (7) as a function of the bankruptcy threshold κ (Fig. 4 right). For κ < κ c , thecost of the crisis is about 80-90% of the total USD volume V exchanged between thecountries in the WTN. By contrast, for κ > κ c , the total cost of the crisis is less than5% of V . Such a graph could help any supranational agency to limit the cost of acrisis induced by the application of austerity policies to indebted countries. Indeed,the calculus of the PCBT (4) allows to select a bankruptcy threshold limiting thecrisis cost below a given value. Eg, for κ (cid:38) . κ c , the cost of a crisis is less than thehundredth of the total volume exchanged. We also observe that the curves for all theconsidered years, whether we use the model A or the model B, fall into practicallythe same curve. Differences between different years are visible for κ (cid:38) . κ c . In thisregion, going from κ = 3 . κ c to κ (cid:39) . κ c , the stairway structure of the curves is dueto the successive sudden bankruptcies of countries at specific bankruptcy thresholds κ . These bankruptcies are dependent of the details of the WTN structure for theconsidered years. Let us also note that, in the region κ < κ c , the total cost of thecrisis is about 80-90% of the total USD volume V exchanged, the remaining 10-20%of the volume V still flows through the WTN since exports to the remaining 10%of the countries are still allowed even for countries in bankruptcy.In the following, we analyze with details the role of the countries in the crisiscontagion. Geographical distibution of the PageRank-CheiRank trade balance
In Fig. 5, we present the PCTB for each country. As an example, let us considerthat the bankruptcy threshold is κ = 0 .
1. Hence, countries with B c < − κ = − . B c (cid:46) oquid´e et al. Page 11 of 26 c oun t r i e s B c c oun t r i e s B c c oun t r i e s B c c oun t r i e s B c Figure 5 Geographical distribution of the PageRank-CheiRank trade balance B c at thecontagion stage τ = 0 with the bankruptcy threshold κ = 0 . . For each year, at the contagionstage τ = 0 , the countries colored in red (blue) have the most negative (positive) balance B c > − κ . Color categories are obtained using the Jenks natural breaks classification method [24].Countries going to bankruptcy at contagion steps τ = 0 are colored in magenta. The dampingfactor is α = 0 . . region, the northern South America, Bolivia, and Paraguay. Also, we note thatNorth American countries, e.g., US (excepting for 2012), West European countries,e.g., France, UK, Ireland, Switzerland, and East European countries have B c (cid:46) . (cid:46) B c (cid:46) . κ max at whicha country goes to bankruptcy at least at the final stage of the crisis contagion.Otherwise stated, for a κ max value associated to a given country c , this country donot go to bankruptcy for κ > κ max, i.e., B c is always greater than − κ at any stage ofthe contagion process for any κ > κ max. Fig. 6 shows the geographical distributionof κ max. For the model A, we observe that Russia is the less affected country bythe crisis for the years 2004, 2008, and 2012 (for Russia, κ max (cid:39) − .
2, i.e., at anycrisis stage τ , B c > − κ, ∀ κ > − . κ max (cid:46) . B c > − κ, ∀ κ (cid:38) . κ max <
0, this means that, for the years 2004, 2008,and 2012, Russia occupies a peculiar protected position in the WTN.For each considered years in Fig. 6, we observe a peak in the country distributionat κ max just below κ c (cid:39) .
175 (2004), 0 .
15 (2008), 0 .
14 (2012), 0 .
15 (2016). Sucha country distribution can be used to precisely determine the critical bankruptcythreshold κ c . oquid´e et al. Page 12 of 26 c oun t r i e s κ c oun t r i e s κ c oun t r i e s κ c oun t r i e s κ
134 149169131 maxmax maxmax
Figure 6 Geographical distribution of the maximum bankruptcy threshold κ max at which acountry goes to bankruptcy at any step of the contagion. Countries with the highest (lowest) κ max are colored in red (blue). Color categories are obtained using the Jenks natural breaksclassification method [24]. Here, once a country goes to bankruptcy, it is prevented to importproducts with the exception of petroleum and gas (model A). The damping factor is α = 0 . . Atthis scale, small sized islands are not visible. For information, the blue colored countries for the2016 WTN are the following islands BV, IO, CC, HM, YT, AN, PN and GS (not visible in theworld map). The most vulnerable countries (with κ max (cid:38) .
2) are Central and South Ameri-can countries (2004, 2008, 2016), including Mexico (2004, 2008, 2016), Guatemala(2004, 2008, 2016), El Salvador (2004, 2016), Honduras (2004), Costa Rica (2004),Dominican Republic (2004, 2008), Venezuela (2008, 2012), Guyana (2016), and Suri-name (2016), Sub Saharan countries, including Mali (2004, 2008, 2012, 2016), Burk-ina Faso (2004, 2008, 2012, 2016), Togo (2004), Benin (2004, 2016), Niger (2004,2016), RDC (2004, 2008, 2012, 2016), Liberia (2008), Ghana (2008, 2012, 2016),Nigeria (2008), Sudan (2008, 2012), Uganda (2008, 2012, 2016), Rwanda (2008,2012, 2016), Tanzania (2008, 2012, 2016), Zambia (2008, 2012, 2016), Zimbabwe(2008, 2012, 2016), Malawi (2008, 2012), Senegal (2012, 2016), Egypt (2012), Re-public of Congo (2012), Angola (2012), Burundi (2012, 2016), Kenya (2012, 2016),Mozambique (2012, 2016), Bostwana (2012, 2016), Nigeria (2016), Ethiopia (2016),Algeria (2016), and Namibia (2016), Middle East countries, including Syria (2004),Iraq (2004), Georgia (2004), Egypt (2012), Israel (2016), Jordan (2016), and SaudiArabia (2016), few European countries, Slovenia (2008), Bosnia-Herzegovina (2008)and Serbia (2008), and Asian countries, including Pakistan (2004, 2008, 2016),Afghanistan (2008, 2016), and Philippines (2016), and Papua New Guineas (2012).As a summary, for the considered year, the most fragile countries in the WTN areprimarily many Sub Saharan countries, Central and South American countries, andsome Middle East and Asian countries. oquid´e et al.
Page 13 of 26 c oun t r i e s Fraction of blocked products 0102030 10.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c oun t r i e s Fraction of blocked products35+ Figure 7 Fraction of products which can not be exported by countries by lack of importers.
The color is function of the fraction of products which can not be exported by countries.Countries in blue can still export most of their products. Countries in red can almost no moreexport any of their products. Color categories are obtained using the Jenks natural breaksclassification method [24]. The computed data concern the 2008 and 2016 WTNs with κ = 0 . at τ ∞ and α = 0 . . Once a country goes to bankruptcy, it is prevented to import products with theexception of petroleum and gas (model A). The fact that bankrupted countries are prevented to import products implies that,during the contagion process, more and more products can not be exchanged. Asan example, we show in Fig. 7, for a bankruptcy threshold κ = 0 .
1, the fraction ofproducts which, at the end of the contagion, can not be exported by countries bylack of importers. For the 2008 WTN (Fig. 7, left column), we observe that most ofthe countries of the Western world have less than 17% of their exports blocked dueto the crisis contagion. This means that at the end of the contagion process, thesecountries have at least one importer for almost each of their product. The samesituation is found for some former USSR countries or satellites, such as Ukraine,Belarus, Moldova, Bulgaria, and Kazakhstan, some Middle Eastern countries, suchas Turkey, and Asian countries, such as China, India, Thailand, Taiwan, SouthKorea, Japan, Singapore, and Indonesia. Although Russia do not go to bankruptcyduring the crisis contagion at κ = 0 .
1, nevertheless more than 87% of its exportshave been indirectly prevented by the crisis contagion. Russia remains safe in the2008 WTN crisis contagion thanks to petroleum and gas exports which correspondto 60% of the total Russian exports and which can be imported by any country inthe model A. For the 2016 WTN (Fig. 7, right column), the crisis is more severeas most of the countries have more than 90% of their exports prevented. OnlyUK, Poland, South Africa, and New Zealand have less than 30% of their exportedproducts blocked.
Crisis contagion networks
Let us define a network of causality where a country c points to a country c (cid:48) , ifthe country c goes to bankruptcy at the crisis contagion stage τ and the country c (cid:48) goes to bankruptcy at the next stage τ + 1. Otherwise stated, the bankruptcyof the country c (cid:48) follows right away the bankruptcy of the country c . In Fig. 8,we show the network of crisis contagion causality for 2004, 2008, 2012, and 2016WTNs and for a bankruptcy threshold κ = 0 .
1. A country is colored according tothe crisis contagion stage at which it goes to bankruptcy, from red for τ = 0 to bluefor τ = τ ∞ . The direction of the links is given by the bending of the links, i.e., if oquid´e et al. Page 14 of 26
NISVHNGT LCVCBBAWEC AIJMCO MSDMANBO ARCLPE GDCRPA TTVE MRKNAGBZ INPKOM AF AUPG NC TVNF FJSB VU NZUZTMIR JPCNKG KPKZ KRTJ MN PWBNMYMV LK SGMM CXIO CC ID TL FMVN MHKITH GUMPPHKH NRLABTQABHAE BDNPBIRW KECD SCCGAOSH TZKMZMMWZWNA MG MUYTFK LSBWBV MZGS SZZABRUYPY SD YEERSN ETCV GQBJTGGH CFCMNEBFMLGMGW NGGNSL UGGAHMAQ SOSTSRGY LR CI PLLTDEDK RUBYSKSICA HUCHIS NOGLPM LVSE EEFIFO ATHR UARSLU MDFR CZ AMTRROES AZALAD SM BAMEVA MK GEIT BGBEGBNLIEMX CU TCHTDO EHKYBS VG SAJO KWPSEGBMUM PT DZ TNUS GRGI LYMA IQMT CYLBSYIL
KG BTPKIN BD VNTHNP MMMV LAAF KR JPCNKZTJ MNUZ KPTLMYSGIDCX PH PWCCLK KHIO MP SBGU NRFMAU TVNCKI FJNFVU NZKEET YE SCRUSD GEMD AMUAEGUGBIRW IQSYJOLBPSILCYTR TMSA OMIRAEQAKWBHAZNEHMAQGHBFBJTGCISL MLGNGMGWEHLRSNMR STGA CGCDCMNG CFLYGQ TDVE VCLCANAW AIVGDO PMGLUSCA BMHNNICUCRKYPABSCOPEJMHTEC BOCLTCMX BZGTSV BVSHFK GSBRUY MGKMNA BW MZ TZ MUYTAO MWARPYGYSRTTGDBBAG SZLSZAZWZMCVMSKNDM ESMA GBGIIS PTIEFO LUCHFRNLAD TNBEDZ BACZSIATSESKHRDKNO BYROFILTLVEEGRBGMKALMERSHUPLDEITVAMT
AR BR GSPYFK BVUY LRGNSN MLGMCVJMCOECPE BO DMVCLCCLAWCUMX HN KYSVGT PANICRBZ MSGYANVE TTGD SRBBUSCABSHT SB FJNRAU NCFM KIVUNF NZMYCCTH CXSGKH PGMPGUJPPWIO BDLK MMMVTM PKINTJMU KG CNVN KRID PHBN KPTC AG MRPM ISAIKNDO GLVGBM ROBGGRLVMKPLRSEEFILTMESESKHUBA KZ MNAF LAUZ NPIE GBMA FRESAD OMAMAZRU GEBYAT UAAL MDTRHRCHSMVAITSIDEDKCZNOPTFO LUBENL TNDZNEGI MTLY EGCYLBILPSJOSYNA SDCGST CDNGCM BIAOGQSH BJHMAQGHBFTGCI BHIQ AEYESA IRYTMG SCKMDJBW MWZMZWRWMZUGZALS TZERETKE
PYBO GSFKUYARCL BRDOANAWEC HTTCPE BMPA COBSKYJMCU VEVGAIKNMSAGGDTT PMUS GLCASVHNGTMX BZNICR LCGYVCDM SRBB LUGBNLFRBECVIS NP MMPK BTKG BDIN KPMN JPKRVNMYTH PH PWSG CNBYEELTLV UAFI RU AFTJKZUZPLSEATCZDEFOIE NODK AZIRBHSCQA TMOMAEDJYESAKWGEAMIQKMSOERSD YTETTZUGKELBILPSSYJORWMWBIZMMDCYEGTRGRBGMKALME RORSHUBASMVALYTDSISKMTCHTNITHR AUPGTL GUMPLALKIO KHMG BNCC CXMVMU ID TVKIMHVUNCNFFM NRSB NZFJBV NA SZLSBWZAZWMZGWSLGNSNMRGMEH STBJAQSH HMTGNEGHCI ADMLGILR BFDZESPTMA NG CFAOGQGA CDCGCM
Figure 8 Crisis contagion network for years 2004, 2008, 2012, and 2016.
The colors of countrynodes range from red, for countries going to bankruptcy at stage τ = 0 , to blue, for countriesgoing to bankruptcy at stage τ = τ ∞ . White nodes corresponds to countries which never go tobankruptcy. The direction of the link between two countries A and B is given by its curvature. If Apoints to B following the bent path in the counterclockwise direction (A (cid:94) B) then A went tobankruptcy at the stage just before B, otherwise, i.e. (A (cid:95)
B), B went to bankruptcy at the stagejust before A. The color of the link is the color of the node source. The width of the link isproportional to the export volume from the target country to the source country in the unmodifiedWTN . Here, once a country goes to bankruptcy, it is prevented to import products with theexception of petroleum and gas (model A). The bankruptcy threshold is κ = 0 . and the dampingfactor is α = 0 . .oquid´e et al. Page 15 of 26 the country c points to the country c (cid:48) following a bent path in the counterclockwisedirection, c (cid:94) c (cid:48) , then the country c (cid:48) goes to bankruptcy right after the country c . In the other hand, if the path direction from c to c (cid:48) is clockwise, c (cid:95) c (cid:48) , thenthe country c (cid:48) goes to bankruptcy right before the country c . The width of the linkfrom country c to country c (cid:48) is proportional to the prevented export volume fromcountry c (cid:48) to country c , i.e., M cc (cid:48) = (cid:80) p ∈ ˜ P M pcc (cid:48) . The links are colored according tothe color of the source. Consequently, the patients zero of the crisis are the reddishcountries and the very first banned trade exchanges are the reddish links. For allthe considered years, the seeds of the crisis are mainly countries from Sub Sahara,Middle East, Central America, and Eastern Europe. We can observe only very fewAsian countries as seeds of the crisis. The directions of the very first banned tradeexchanges are meaningful. For the 2004 WTN, bunches of them come from Africa,Middle East, and Central America to Europe. Another bunches come from EasternEurope and Central America to North America. Thus the fall of the US, whichoccurs at the second stage of the crisis contagion, stems mainly from the failures ofMexico, and Central American countries and East European countries. Once fallen,US drives to bankruptcy Western European countries and ignite the crisis in Asiawhere Japan and South Korea go to bankruptcy at the third stage of the contagion.The failure of these latter countries then induce the failure of China and Australia.A similar contagion scheme occurs in the 2008 WTN. For the 2012 WTN, the US goto bankruptcy at the third stage of the crisis contagion after the failure of Mexico,South Korea, Singapore, and France. Singapore and South Korea propagate thecrisis to Japan. For the 2016 WTN, the US also fall at the third contagion stagebeing impacted by the previous failure of Singapore, Great Britain, and France.Then, the failure of the US directly impacts China, South Korea, and Japan. Ananimation shows the contagion dynamics for the 2016 WTN (see Additional file 6— Evolution of the crisis contagion in the 2016 WTN).Let us focus on the greatest volume trade exchanges between countries. Fig. 9shows the hierarchy of the crisis contagion causality for imports greater than 10 USD (as a complement, Fig. A2 shows the same data but geographically dis-tributed).For the 2004 WTN, among the big exporters, Mexico and Israel contribute tothe fall of the US. From the fall of the US, one of the main paths of contagioncan be followed, US → JP → (Asian countries and Australia). The bankruptcyof all the European countries are due to the conjugated effect of the fall at stage τ = 1 of the US and of part of the main European economies. Bankruptcies ofthe European countries, and also of South Korea, contribute then to the fall of theAsian big exporters (such as China) and of Australia, which, as stressed before, arethe last countries to go to bankruptcy. Let us note that France and Great Britainalso contribute to the failure of Japan.For the 2008 WTN, we obtain a similar scenario excepting that Venezuela andSaudi Arabia, in addition to Mexico, lead the US to the bankruptcy. Also, EasternEurope countries, Poland and Slovakia, are the seeds of the contagion in Europe. Wecan observe that the BRIC, i.e., Brazil, India, and China, are among the countrieswhich are the last affected by the contagion. This remark is also true for 2012 and2016 WTNs. Russia, which is a petroleum and gas exporter, is never affected bythe contagion excepting for the 2016 WTN. oquid´e et al. Page 16 of 26 E u r o p e A s i a A m e r i c a s A m e r i c a s A s i a E u r o p e Americas A m e r i c a s E u r o p e A s i a A s i a E u r o p e Figure 9 Crisis contagion network with a hierarchical layout for years 2004, 2008, 2012, and2016.
Only imports greater than USD are represented. Countries going to bankruptcy at thesame contagion step τ are aligned in the same row. From bottom (red country nodes) to top(blue country nodes), the rows are associated to contagion step from τ = 0 to τ = 3 for 2004, τ = 4 for 2008, τ = 5 for 2012 and 2016. The width of the link going from a country c whichgoes to bankruptcy at the stage τ to a country c (cid:48) which goes to bankruptcy at the stage τ (cid:48) = τ + 1 is proportional to the volume usually imported by country c from c (cid:48) . Here, once acountry goes to bankruptcy, it is prevented to import products with the exception of petroleumand gas (model A). Colored zones gather countries from the same continent (green for Europeancountries, blue for American countries, and pink for Asian countries). The bankruptcy threshold is κ = 0 . and the damping factor is α = 0 . . For the 2012 WTN, at the crisis stage τ = 0 and τ = 1, there are several sourcesof contagion: Central America with Mexico, Panama and Costa Rica, Europe with oquid´e et al. Page 17 of 26
Austria, France, and Slovakia, Middle East with Iraq and Turkey, and Asia withSingapore and South Korea. The crisis is already present, at the very first stages ofthe contagion, in all the continents excepting Oceania. At stage τ = 2, the US aremainly affected by the previous fall of some Central American countries, some Asiancountries and France. Then the US contributes to propagate the crisis to the restof the world. The crisis in Asia is also brought by the fall of Japan induced by thebankruptcy of Singapore and South Korea, and in a somewhat lesser importance bythe bankruptcy of France. The bankruptcy of the rest of European countries followsmainly the fall of France, and Austria, and follows secondarily the fall of Turkey,South Korea, Singapore, and Mexico.For the 2016 WTN, the European countries ignite the crisis with Slovakia as seedof the contagion. The crisis propagates to North American countries and then toAsia and the rest of the world. We note that in addition to European countriesSingapore is also in bankruptcy at the early stages of the contagion and contributesto the fall of the US. Here, Russia is the last country going to bankruptcy. Conclusion and discussion
The Google matrix analysis of the world trade network allows to probe the di-rect and indirect trade exchange dependencies between countries. Unlike the sim-ple accounting view obtained from the usual import-export balance, relying onthe total volumes of exchanged commodities between countries (5), the PageRank-CheiRank trade balance (PCTB) (4) allows to take account of the long range inter-dependencies between world economies. The WTN crisis contagion model is buildupon the iterative measure of the PCTB for each country. Once a country have aPCTB below a threshold − κ , it is declared in a bankruptcy state in which it can nomore import commodities excepting some vital one for the industry, i.e., petroleumand gas. This state corresponds either to the fact that a country with a very nega-tive trade balance have not enough liquidity to import non essential commodities,or to the decision of a supranational economic authority trying to contain a cri-sis by placing an unhealthy national economy in bankruptcy. The bankruptcies ofeconomies with PCTB less than − κ induce a rewiring of the world trade networkwhich possibly weaken other economies. In the phase corresponding to a bankruptcythreshold κ > κ c , the crisis contagion is rapid and contained since it affects onlyless than 10% of the world countries and induces a total cost of less than 5% ofthe total USD volume exchanged in the WTN. This total cost of the crisis dropsexponentially with the increase of κ . In the phase corresponding to a bankruptcythreshold κ < κ c , the cascade of bankruptcies can not be contained and the crisisis global, affecting about 90% of the world countries. The bankruptcy threshold κ is the order parameter of the phase transition. In the global crisis phase ( κ < κ c ),at the first stage ( τ = 0) of the contagion, myriads of countries with low exchangedvolume (ie, low import and export volumes) go to bankruptcy. These countries be-long mainly to Sub Saharan Africa, Central and South America, Middle East, andEastern Europe. In the next stage of the crisis contagion, the conjugated effect ofthe bankruptcies of these countries contribute to the fall of big exporters, such asthe US or Western European countries. As an example, for 2004, 2012, and 2016WTNs, the bankruptcy of France at the contagion stage τ = 1 is solely due to the oquid´e et al. Page 18 of 26 failure of many low exchanged volume countries, which, here, individually importfrom France a volume of commodities less than 10 USD. Otherwise stated, Francefailure is caused by the failure of many small importers. Great Britain is a simi-lar case for the 2004, 2008, and 2016 WTNs. Among the big exporters (ie, witha exchanged volume greater than 10 USD), European and American countriesare the sources of the crisis contagion. The gates from which crisis enters Asia areusually Japan, Korea, and Singapore. Generally, Asian countries go to bankruptcyat the end of the crisis contagion, with China, India, Indonesia, Malaysia and Thai-land, being, with Australia, usually the last economies to fall. We also observe thatfailures of the four BRIC occur during the last stages of the crisis contagion.As a future development of the presented WTN crisis contagion analysis, it wouldbe interesting to study the cascades of country bankruptcies induced by a sharp in-crease of the price of a given commodity. Indeed, within our model, such an increaseof the price of petroleum and/or gas would highlight the structural vulnerability ofthe countries to an energy crisis contagion.
Availability of data and materials
The raw data is available from the UN Comtrade database [6]. Additional output data and/or plots of datagenerated are available upon request.
Abbreviations
WTN: World trade network; PCTB: PageRank-CheiRank trade balance; DRC: Democratic Republic of the Congo;BRIC: Brazil, Russia, India, China; BRICS: Brazil, Russia, India, China, South Africa.ISO 3166-1 alpha-2 code for countries:AF: Afghanistan; AL: Albania; DZ: Algeria; AS: American Samoa; AD: Andorra; AO: Angola; AI: Anguilla; AQ:Antarctica; AG: Antigua and Barbuda; AR: Argentina; AM: Armenia; AW: Aruba; AU: Australia; AT: Austria; AZ:Azerbaijan; BS: The Bahamas; BH: Bahrain; BD: Bangladesh; BB: Barbados; BY: Belarus; BE: Belgium; BZ:Belize; BJ: Benin; BM: Bermuda; BT: Bhutan; BO: Bolivia; BA: Bosnia and Herzegovina; BW: Botswana; BV:Bouvet Island; IO: British Indian Ocean Territory; VG: British Virgin Islands; BR: Brazil; BN: Brunei; BG: Bulgaria;BF: Burkina Faso; BI: Burundi; KH: Cambodia; CM: Cameroon; CA: Canada; CV: Cape Verde; KY: CaymanIslands; CF: Central African Republic; TD: Chad; CL: Chile; CN: China; CX: Christmas Island; CC: Cocos (Keeling)Islands; CO: Colombia; KM: Comoros; CG: Republic of the Congo; CK: Cook Islands; CR: Costa Rica; CI: IvoryCoast; HR: Croatia; CU: Cuba; CY: Cyprus; CZ: Czech Republic; KP: North Korea; CD: Democratic Republic of theCongo; DK: Denmark; DJ: Djibouti; DM: Dominica; DO: Dominican Republic; EC: Ecuador; EG: Egypt; SV: ElSalvador; GQ: Equatorial Guinea; ER: Eritrea; EE: Estonia; ET: Ethiopia; FO: Faroe Islands; FK: Falkland Islands;FJ: Fiji; FI: Finland; FR: France; PF: French Polynesia; FM: Micronesia; GA: Gabon; GM: The Gambia; GE:Georgia; DE: Germany; GH: Ghana; GI: Gibraltar; GR: Greece; GL: Greenland; GD: Grenada; GU: Guam; GT:Guatemala; GN: Guinea; GW: Guinea-Bissau; GY: Guyana; HT: Haiti; HM: Heard Island and McDonald Islands; VA:Vatican; HN: Honduras; HU: Hungary; IS: Iceland; IN: India; ID: Indonesia; IR: Iran; IQ: Iraq; IE: Ireland; IL: Israel;IT: Italy; JM: Jamaica; JP: Japan Ryukyu Island; JO: Jordan; KZ: Kazakhstan; KE: Kenya; KI: Kiribati; KW:Kuwait; KG: Kyrgyzstan; LA: Laos; LV: Latvia; LB: Lebanon; LS: Lesotho; LR: Liberia; LY: Libya; LT: Lithuania;LU: Luxembourg; MG: Madagascar; MW: Malawi; MY: Malaysia; MV: Maldives; ML: Mali; MT: Malta; MH:Marshall Islands; MR: Mauritania; MU: Mauritius; YT: Mayotte; MX: Mexico; MN: Mongolia; ME: Montenegro;MS: Montserrat; MA: Morocco; MZ: Mozambique; MM: Myanmar; MP: Northern Mariana Islands; NA: Namibia;NR: Nauru; NP: Nepal; AN: Netherlands Antilles; NL: Netherlands; NC: New Caledonia; NZ: New Zealand; NI:Nicaragua; NE: Niger; NG: Nigeria; NU: Niue; NF: Norfolk Islands; NO: Norway; PS: State of Palestine; OM:Oman; PK: Pakistan; PW: Palau; PA: Panama; PG: Papua New Guinea; PY: Paraguay; PE: Peru; PH: Philippines;PN: Pitcairn; PL: Poland; PT: Portugal; QA: Qatar; KR: South Korea; MD: Moldova; RO: Romania; RU: Russia;RW: Rwanda; SH: Saint Helena; KN: Saint Kitts and Nevis; LC: Saint Lucia; PM: Saint Pierre and Miquelon; VC:Saint Vincent and the Grenadines; WS: Samoa; SM: San Marino; ST: Sao Tome and Principe; SA: Saudi Arabia;SN: Senegal; RS: Serbia; SC: Seychelles; SL: Sierra Leone; SG: Singapore; SK: Slovakia; SI: Slovenia; SB: SolomonIslands; SO: Somalia; ZA: South Africa; GS: South Georgia and the South Sandwich Islands; ES: Spain; LK: SriLanka; SD: Sudan; SR: Suriname; SZ: Swaziland; SE: Sweden; CH: Switzerland; SY: Syria; TJ: Tajikistan; MK:Macedonia; TH: Thailand; TL: Timor-Leste; TG: Togo; TK: Tokelau; TO: Tonga; TT: Trinidad and Tobago; TN:Tunisia; TR: Turkey; TM: Turkmenistan; TC: Turks and Caicos Islands; TV: Tuvalu; UG: Uganda; UA: Ukraine; AE:United Arab Emirates; GB: United Kingdom; TZ: Tanzania; UM: United States Minor Outlying Islands; UY:Uruguay; US: United States; UZ: Uzbekistan; VU: Vanuatu; VE: Venezuela; VN: Vietnam; WF: Wallis and Futuna;EH: Western Sahara; YE: Yemen; ZM: Zambia; ZW: Zimbabwe.
Author details Institut UTINAM, UMR 6213, CNRS, Universit´e Bourgogne Franche-Comt´e, Besan¸con, France. Laboratoire dePhysique Th´eorique, IRSAMC, Universit´e de Toulouse, CNRS, UPS, Toulouse, France. oquid´e et al.
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The authors declare that they have no competing interests.
Author’s contributions
The authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
We thank Leonardo Ermann for useful discussions. We thank UN Comtrade for providing to us a friendly access totheir database.
Funding
Programme Investissements d’Avenir ANR-15-IDEX-0003, ISITE-BFC (GNETWORKS project); BourgogneFranche-Comt´e region (APEX project); ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT France(project THETRACOM). oquid´e et al.
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Additional Files
Additional file 1 — Fraction of bankrupted countries for the WTN of 2004, 2008, 2012, and 2016 (model B) η κ τ = τ = τ = τ = τ = τ∞ η κ τ = τ = τ = τ = τ = τ ∞ η κ τ = τ = τ = τ = τ = τ ∞ η κ τ = τ = τ = τ = τ = τ ∞ Figure A1 Fraction of bankrupted countries for the WTN of 2004, 2008, 2012, and 2016.
Fraction η of countries went to bankruptcy up to the τ th stage of crisis contagion as a function ofthe bankruptcy threshold κ . The crisis contagion has been computed for the WTN of 2004 (topleft), 2008 (top right), 2012 (bottom left), and 2016 (bottom right). Once a country goes tobankruptcy, it is prevented to import any product (model B). The damping factor is α = 0 . .oquid´e et al. Page 21 of 26
Additional file 2 — List of the 38 countries remaining safe at τ ∞ for κ = 0 . in 2004 (model A) Table A1
List of the 38 countries remaining safe at τ ∞ for κ = 0 . in 2004 (model A). The CC andSC columns give the ISO 3166-2 codes of the country and of its Sovereign country, respectively. TheLT column gives the land type of the country, i.e. either Mainland (M) or Island (I). The three lastcolumns give the percentages of gas and petroleum exported by the country. Countries are sortedprimarily by the proportion of gas and petroleum in their exports (last column) and secondarily by theexport total amount (fifth column).Country ExportationCC SC LT Name Total( USD) Gas( (cid:104) ) Petroleum( (cid:104) ) Gas &petroleum( (cid:104) ) NG NG
M Nigeria 34582.35 49.45 910.87 960.32
SA SA
M Saudi Arabia 110667.14 34.88 787.04 821.92
RU RU
M Russia 225850.70 51.36 447.20 498.56
NR NR
I Nauru 17.58 28.38 425.38 453.76
PN UK
I Pitcairn Islands 20.02 0.00 448.97 448.97
TL TL
I East Timor 155.31 21.89 274.94 296.83
SC SC
I Seychelles 520.47 0.02 194.06 194.08
GU US
I Guam 68.83 1.60 167.09 168.69
KE KE
M Kenya 3507.94 0.65 139.40 140.05
BM UK
I Bermuda 189.97 0.00 74.22 74.22
MH MH
I Marshall Islands 651.83 0.00 57.53 57.53
KY UK
I Cayman Islands 674.49 0.00 46.41 46.41
VA VA
M Vatican 3.19 0.00 18.58 18.58
MP US
I Northern Mariana Is-lands 18.97 0.00 10.81 10.81
SH UK
I Saint Helena, Ascensionand Tristan da Cunha 15.52 0.00 9.99 9.99
TC UK
I Turks and Caicos Islands 30.78 9.52 0.01 9.53
AS US
I American Samoa 22.63 0.00 8.62 8.62
FK UK
I Falkland Islands 136.20 0.00 4.53 4.53
IO UK
I British Indian OceanTerritory 3.23 0.00 3.07 3.07
TV TV
I Tuvalu 2.12 1.95 0.01 1.95
TK NZ
I Tokelau 20.41 0.00 1.48 1.48
SM SM
M San Marino 53.24 0.00 0.92 0.92
SB SB
I Solomon Islands 193.42 0.00 0.84 0.84
GL DK
M Greenland 538.59 0.00 0.26 0.26
PW PW
I Palau 22.80 0.00 0.10 0.10
CK CK
I Cook Islands 17.01 0.00 0.04 0.04
BT BT
M Bhutan 57.88 0.00 0.01 0.01
UM US
I United States MinorOutlying Islands 33.55 0.00 0.00 0.00
CX AU
I Christmas Island 15.54 0.00 0.00 0.00
PM FR
I Saint Pierre andMiquelon 7.80 0.00 0.00 0.00
CC AU
I Cocos (Keeling) Islands 6.05 0.00 0.00 0.00
NF AU
I Norfolk Island 4.01 0.00 0.00 0.00
GS UK
I South Georgia and theSouth Sandwich Islands 3.50 0.00 0.00 0.00 EH M Western Sahara 2.11 0.00 0.00 0.00 AQ M Antarctica 2.05 0.00 0.00 0.00
NU NU
I Niue 1.88 0.00 0.00 0.00
HM AU
I Heard and McDonald Is-lands 0.95 0.00 0.00 0.00
BV NO
I Bouvet Island 0.25 0.00 0.00 0.00oquid´e et al.
Page 22 of 26
Additional file 3 — List of the 38 countries remaining safe at τ ∞ for κ = 0 . in 2008 (model A) Table A2
List of the 38 countries remaining safe at τ ∞ for κ = 0 . in 2008 (model A). The CC andSC columns give the ISO 3166-2 codes of the country and of its Sovereign country, respectively. TheLT column gives the land type of the country, i.e. either Mainland (M) or Island (I). The three lastcolumns give the percentages of gas and petroleum exported by the country. Countries are sortedprimarily by the proportion of gas and petroleum in their exports (last column) and secondarily by theexport total amount (fifth column).Country ExportationCC SC LT Name Total( USD) Gas( (cid:104) ) Petroleum( (cid:104) ) Gas &petroleum( (cid:104) ) TL TL
I East Timor 169.40 816.28 0.28 816.56
RU RU
M Russia 570605.26 44.12 550.37 594.49
BV NO
I Bouvet Island 41.07 0.00 357.29 357.29
CK CK
I Cook Islands 33.11 0.00 239.31 239.31
BM UK
I Bermuda 1849.83 3.54 104.77 108.30
UM US
I United States MinorOutlying Islands 24.16 0.00 68.68 68.68
SZ SZL
M Eswatini 1058.15 0.00 61.57 61.57
GS UK
I South Georgia and theSouth Sandwich Islands 1.62 0.00 21.96 21.96
AD AD
M Andorra 175.92 7.70 9.86 17.56
NF AU
I Norfolk Island 4.04 0.00 15.57 15.57 EH M Western Sahara 11.31 0.00 14.72 14.72
TV TV
I Tuvalu 4.02 0.00 11.10 11.10
SH UK
I Saint Helena, Ascensionand Tristan da Cunha 43.99 0.00 3.38 3.38
NU NU
I Niue 9.67 0.00 2.77 2.77
GU US
I Guam 77.22 0.00 2.33 2.33 AQ M Antarctica 2.55 0.00 1.75 1.75
GL DK
M Greenland 753.16 0.00 1.38 1.38
WS WS
I Samoa 89.95 0.00 1.28 1.28
WF FR
I Wallis and Futuna 18.64 0.00 0.98 0.98
VU VU
I Vanuatu 568.61 0.00 0.67 0.67
LS LS
M Lesotho 873.69 0.27 0.39 0.66
KI KI
I Kiribati 14.02 0.00 0.47 0.47
NR NR
I Nauru 126.51 0.00 0.33 0.33
FM FM
I Federated States of Mi-cronesia 28.57 0.00 0.13 0.13
AS US
I American Samoa 70.09 0.00 0.08 0.08
BT BT
M Bhutan 688.82 0.00 0.03 0.03
IO UK
I British Indian OceanTerritory 8.25 0.00 0.02 0.02
SB SB
I Solomon Islands 383.50 0.00 0.01 0.01
PW PW
I Palau 29.04 0.00 0.01 0.01
FK UK
I Falkland Islands 196.72 0.00 0.00 0.00
GW GW
M Guinea-Bissau 135.22 0.00 0.00 0.00
CX AU
I Christmas Island 52.61 0.00 0.00 0.00
CC AU
I Cocos (Keeling) Islands 29.68 0.00 0.00 0.00
PM FR
I Saint Pierre andMiquelon 17.42 0.00 0.00 0.00
MP US
I Northern Mariana Is-lands 12.64 0.00 0.00 0.00
PN UK
I Pitcairn Islands 9.38 0.00 0.00 0.00
VA VA
M Vatican 2.52 0.00 0.00 0.00
HM AU
I Heard and McDonald Is-lands 0.53 0.00 0.00 0.00oquid´e et al.
Page 23 of 26
Additional file 4 — List of the 32 countries remaining safe at τ ∞ for κ = 0 . in 2012 (model A) Table A3
List of the 32 countries remaining safe at τ ∞ for κ = 0 . in 2012 (model A). The CC andSC columns give the ISO 3166-2 codes of the country and of its Sovereign country, respectively. TheLT column gives the land type of the country, i.e. either Mainland (M) or Island (I). The three lastcolumns give the percentages of gas and petroleum exported by the country. Countries are sortedprimarily by the proportion of gas and petroleum in their exports (last column) and secondarily by theexport total amount (fifth column).Country ExportationCC SC LT Name Total( USD) Gas( (cid:104) ) Petroleum( (cid:104) ) Gas &petroleum( (cid:104) ) RU RU
M Russia 640181.69 82.16 565.46 647.63
TC UK
I Turks and Caicos Islands 94.84 0.00 585.21 585.21
GU US
I Guam 146.08 7.94 519.57 527.51
AS US
I American Samoa 91.34 6.58 379.91 386.49
BV NO
I Bouvet Island 55.78 0.00 288.63 288.63 AQ M Antarctica 150.45 0.00 243.27 243.27
KY UK
I Cayman Islands 622.35 0.00 19.92 19.92
HM AU
I Heard and McDonald Is-lands 245.23 0.00 11.25 11.25
VA VA
M Vatican 7.71 0.00 1.45 1.45
NU NU
I Niue 3.59 0.00 1.17 1.17
SH UK
I Saint Helena, Ascensionand Tristan da Cunha 19.12 0.00 0.99 0.99
MP US
I Northern Mariana Is-lands 3.65 0.01 0.86 0.86
NR NR
I Nauru 100.46 0.00 0.58 0.58
GS UK
I South Georgia and theSouth Sandwich Islands 4.03 0.00 0.46 0.46
SB SB
I Solomon Islands 572.38 0.00 0.40 0.40
YT FR
I Mayotte 28.29 0.00 0.16 0.16
AI UK
I Anguilla 10.29 0.00 0.15 0.15
CK CK
I Cook Islands 51.41 0.00 0.14 0.14
FO DK
I Faroe Islands 1001.96 0.00 0.10 0.10
CX AU
I Christmas Island 38.71 0.00 0.05 0.05
IO UK
I British Indian OceanTerritory 32.05 0.00 0.05 0.05
UM US
I United States MinorOutlying Islands 20.42 0.00 0.03 0.03
TK NZ
I Tokelau 43.62 0.00 0.03 0.03
VU VU
I Vanuatu 454.29 0.00 0.01 0.01
FK UK
I Falkland Islands 210.46 0.01 0.00 0.01
SM SM
M San Marino 128.83 0.00 0.00 0.00
ER ER
M Eritrea 47.89 0.00 0.00 0.00
PM FR
I Saint Pierre andMiquelon 7.12 0.00 0.00 0.00
CC AU
I Cocos (Keeling) Islands 7.05 0.00 0.00 0.00
PN UK
I Pitcairn Islands 6.71 0.00 0.00 0.00
NF AU
I Norfolk Island 3.85 0.00 0.00 0.00
WF FR
I Wallis and Futuna 1.30 0.00 0.00 0.00oquid´e et al.
Page 24 of 26
Additional file 5 — List of the 11 countries remaining safe at τ ∞ for κ = 0 . in 2016 (model A) Table A4
List of the 11 countries remaining safe at τ ∞ for κ = 0 . in 2016 (model A). The CC andSC columns give the ISO 3166-2 codes of the country and of its Sovereign country, respectively. TheLT column gives the land type of the country, i.e. either Mainland (M) or Island (I). The three lastcolumns give the percentages of gas and petroleum exported by the country. Countries are sortedprimarily by the proportion of gas and petroleum in their exports (last column) and secondarily by theexport total amount (fifth column).Country ExportationCC SC LT Name Total( USD) Gas( (cid:104) ) Petroleum( (cid:104) ) Gas &petroleum( (cid:104) ) GS UK
I South Georgia and theSouth Sandwich Islands 0.34 0.00 83.01 83.01 AQ M Antarctica 10.28 0.00 11.67 11.67
NU NU
I Niue 2.30 0.00 10.50 10.50
FK UK
I Falkland Islands 257.30 0.00 5.87 5.87
CC AU
I Cocos (Keeling) Islands 4.59 0.00 0.29 0.29 EH M Western Sahara 8.92 0.00 0.01 0.01
BV NO
I Bouvet Island 0.86 0.00 0.00 0.00
IO UK
I British Indian OceanTerritory 20.16 0.00 0.00 0.00
SH UK
I Saint Helena, Ascensionand Tristan da Cunha 26.64 0.00 0.00 0.00
PN UK
I Pitcairn Islands 1.29 0.00 0.00 0.00
HM AU
I Heard and McDonald Is-lands 0.12 0.00 0.00 0.00oquid´e et al.
Page 25 of 26
Additional file 6 — Evolution of the crisis contagion in the 2016 WTNSee also the video at http://perso.utinam.cnrs.fr/~lages/datasets/WTNcrisis/ oquid´e et al.
Page 26 of 26
Additional file 7 — Crisis contagion network for years 2004, 2008, 2012, and 2016
FRIE BE CHGB NL ATSEDKDECA TRESVEMX ILUS IT CZ KRSG AUCNMYTH ID JP
SKDK SEPLDE ATBE FINL CZCHPT ES ITFRIE GB ZATR SA INOMAE SG PHKRCNVNMY JP AUIDTHPAMX CA US VE BR
GB NLES BEFRIE TRROHUCZPLSKMX CA US CLPACR BR INAEIQSA KZ TH KRIDMY VN JPSG AUCNSEDKDE ATITCH
TRAEOM IN VNTH CN KRGB NOIE NLBE CZDE DK ROPL RUSK BYSEATUS ZABR FRCO CA CH ITES AUJPIDMY SG
Figure A2 Crisis contagion network for years 2004, 2008, 2012, and 2016.
Same as Figure 8 butonly exports greater than10