Indirect Influences in International Trade
aa r X i v : . [ c s . S I] N ov Indirect Influences in International Trade
Rafael D´ıaz and Laura G´omez
Abstract
We address the problem of gauging the influence exerted by a given country on the global trademarket from the viewpoint of complex networks. In particular, we apply the PWP method forcomputing indirect influences on the world trade network.
Global and local, private and public institutions increasingly rely on numerical indexes for their decisionmaking processes. A single index may be the difference between approval or rejection. These indexes,and even the mathematical models behind them, are often made public to enhance transparency, ac-countability, and establish standards. The usefulness of an index is measured by the successes andfailures of decisions based on it. Thus index creation, computation, selection, comparison, evaluation,and consolidation has become an industry of its own, and an integral part of institutionalized decisionmaking processes.In this work we address the problem of finding suitable indexes for ranking countries according totheir influences on the international trade market. In this case there is an obvious candidate: the in-fluence of a country on the international trade market is proportional to the amount (counted in USdollars) of its international trade, i.e. the total amount of imports plus the total amount of exports.This primordial index is sound and should not be overlooked. Nevertheless, we claim that it disregardssome subtle but important issues.Suppose we have a couple of countries A and B both with high levels of international trade, so thatthey both are highly ranked with the above index. Suppose in addition that A and B trade essentiallywith each other, i.e. trade of A and B with other countries is negligible in comparison with trade betweenthemselves. In this case our feeling is that A and B do not exert a strong influence on the internationaltrade market: a disruption on A ’s economy will surely impact B ’s economy, but will have a negligibleimpact on global trade. Countries A and B although highly interdependent are in fact quite isolatedfrom the rest of the world, and should not be ranked as highly influential countries in the internationaltrade market.To address this sort of issues we take a network approach [5, 7] to our index creation problem. InSection 2, we regard the international trade market as a weighted network with nodes representing coun-tries, edges representing trade between countries, and weights measuring the influence that a countryexerts on another country trough trade. Indeed, in Sections 4 and 5, we are going to introduce a coupleof different weights leading to a couple of rankings.Once we have established the network settings for approaching our index creation problem, we facethe problem of ranking nodes in a network by their influence. This demands that we paid attention tothe distinction between direct and indirect influences in a network, a distinction emphasized by Godetand his collaborators [10], who stressed the power of studying indirect influences for uncovering hidden1elations. Direct influences in a network arise from directed edges. Indirect influences in a network arisefrom chains of direct influences, i.e. from directed paths.Let us consider an extreme case that illustrates the importance of taking indirect influences intoaccount. Cuba and the United States (US) have a fairly weak amount of trade, indeed since 1960 theUS has placed a series of economic, financial, and commercial prohibitions to trade with Cuba. Thus afairly low influence of the US on the Cuban economy is to be expected. However, the US trades withCuba’s main trading partners, and so we may expect that the US exerts a stronger indirect influence onthe Cuban economy than what one may naively think. Simply put, a disruption of the US’s economywill likely impact the Spaniard economy, and trough Spain it will also be felt by the Cuban economy.The main challenge that we confront in this work is to give a quantitative account of the latter impact.After building our mathematical models in Sections 2 trough 6, we proceed in Sections 7 and 8 toimplement them using real world data. Out of United Nations member states we restrict our attentionto 177 countries, namely, those that have recent data available in the Economic Commission for LatinAmerica and the Caribbean (ECLAC) web site [8]. We collect the 2011 data of exports and importsbetween each pair of countries, as well as the GDP, and the exports and imports totals for each country.Using this data we build the ”international trade network,” where nodes are countries, and a pair ofnodes is connected by an edge if there were exports or imports between them in 2011. Suppose thereactually was trade between countries A and B , then we define a couple of weights on the edge from A to B as follows:1. Direct Influences on Trade.
This weight computes the proportion of the international trade of B (exports + imports) that involves country A. Direct Influences on Offer.
This weight computes the relative contribution of the trade between A and B to the offer of B (GDP + imports).To take indirect influences into account we start from a trade network, with one of the weights justintroduced, and apply one of the mathematical methods available for computing indirect influences incomplex networks. With any of these methods one goes beyond computing direct trade between countries A and B , and takes as well into account trade chains A = A , A , .... , A n = B such that country A i trades with country A i +1 .In this work we weight chains of trade using the PWP method introduced by D´ıaz [6]. To place thismethod whiting a general context and for the reader convenience we provide a in Section 3 a succinctdescription of four closely related methods for computing indirect influences, namely, Godet’s MICMAC,Google’s PageRank, Chung’s Heat Kernel, and the PWP method. For more details on the similaritiesand differences among these methods the reader may consult [6]. Our choice of the PWP method reston the fact that with it any chain of direct influences, of any length, generates indirect influences, andreciprocally, all indirect influences are generated in this fashion.Thus we obtain several rankings among nations using the methodology outlined above: for each ofthe above weights we get the direct and indirect influences rankings. We compare these rankings amongthemselves and with the GDP ranking for the American continent trade network in Section 7, and for theWorld trade network in Section 8. We show that there are remarkable differences between the variousrankings, and analyse them in economic terms. 2e remark once again that our economic data refers to year 2011, and is measured in US dollars.When we compare different indexes what is really at stake is to compare the rankings they induce. Inpractice what we do is to compare the corresponding normalized indexes. Numerical calculations in thiswork were made with Scilab’s module for computing indirect influences designed by Catumba [3]. Forthe economics terminology the reader may consult [12]. In this section we introduce basic definitions concerning the construction of international trade networksand their adjacency matrices, called matrices of direct influences in this work.
Definition 1.
A trade network G = ( V, E ) is a directed graph such that: • The set V of vertices is a family of countries. • There is an edge in E from vertex A to vertex B if and only if there is trade (exports or imports)between A and B .Note that trade networks are actually symmetric graphs. Nevertheless, we regard them as directedgraphs since we are soon going to introduced non-symmetric weights on them. For each edge e ∈ E , let s ( e ) be its source vertex, or equivalently, the country that exerts the influence, and let t ( e ) be its targetvertex, or equivalently, the country that is influenced. A weight w on a directed graph G assigns weight w ( e ) ∈ R to each edge of G , i.e. a weight on G is a map w : E −→ R .In Sections 4 and 5 we are going to introduce a couple of weights on trade networks.The bi-degree ( i ( A ) , o ( A )) ∈ R × R of a vertex A ∈ V in a weighted directed graph G is such that i ( A ) is the sum of the weights of edges with target A , and o ( A ) is the sum of the weights of edges withsource A , that is: i ( A ) = X e ∈ E, t ( e )= A w ( e ) and o ( A ) = X e ∈ E, s ( e )= A w ( e ) . We impose the alphabetic order on countries, so the set of countries V is identified with the set { , ..., n } , where n is the number of countries in our trade network. Definition 2.
The adjacency matrix or matrix of direct influences D of a trade network G is given for A, B ∈ V by: D AB = D ( A, B ) = X s ( e )= B, t ( e )= A w ( e ) . Thus D AB gives the direct dependence of A on B , or equivalently, the direct influence of B on A .The matrix of indirect influences T = T ( D ) is computed from the matrix of direct influences D usingone of the methods discussed in Section 3. In our applications, the matrix of indirect influences T iscomputed applying the PWP method. Definition 3.
Let G be a trade graph, D its matrix of direct influences, and T = T ( D ) its asso-ciated matrix of indirect influences. The indirect dependence d A , indirect influence f A , and indirectconnectedness c A of vertex A in G are given, respectively, by: d A = X B ∈ V T AB , f A = X B ∈ V T BA , and c A = d A + f A . The ordered pair ( d A , f A ) is the bi-degree of A in the network of indirect influences. The indirectconnectedness of vertices A and B is given by c AB = T AB + T BA emark 4. Direct dependencies, influences, and connectedness are computed in a similar way usingthe matrix of direct influences D instead of the matrix T of indirect influences.Following Godet a 2-dimensional representation of vertices bi-degrees can be displayed trough thedependence-influence plane which comes naturally divided in four sectors, see Figure 1. The horizontalaxis represents dependencies and the vertical axis represents influences. A country A is represented inthe dependence-influence plane by the ordered pair ( d A , f A ). The horizontal and vertical lines definingthe four sectors of the plane are located at the mean dependence d and at the mean influence f , givenrespectively, by d = 1 n X A ∈ V d A and f = 1 n X A ∈ V f A . Figure 1: Dependence-Influence Plane. • Sector 1:
Influential independent countries. • Sector 2:
Influential dependent countries. • Sector 3:
Low influence independent countries. • Sector 4:
Low influence dependent countries.
Although our focus is on the PWP method, for the reader convenience we introduce four methods forcomputing indirect influences, and describe how each method computes the matrix of indirect influences.For more on the similarities and differences among these methods the reader may consult [6].
With this method, introduced in 1992 by Godet [10], the matrix of indirect influences is T ( D ) = D k , where D is the matrix of direct influences, and k ∈ N is a parameter usually equal to 4 or 5. The relevantpaths in the network, with this method, are those of length k .4 .2 PageRank This method, registered in 1999 by Google [1, 2, 11], is quite well-known thanks to its application to websearching. With PageRank influences are normalized, and dependencies measure the relevance or relativeimportance of web pages. PageRank takes infinite powers of matrices, thus it gives greater importanceto infinite long paths. Let D be the matrix of direct influences, whose entries should be non negativereal numbers, and the sum of each column must be either 0 or 1. The matrix of indirect influences isgiven by T ( D ) = lim k →∞ [ pD + (1 − p ) E n ] k ,where D is the matrix obtained form D by replacing the entries of a zero column by n , E n is the matrixwith entries n , and the parameter p is usually 0 . B to page A if there is ahyperlink in B leading to A . The matrix of direct influences D is given by: D AB = o ( B ) if there is an edge from B to A, , where o ( B ) is out-degree of the vertex B .The ranking of page A is proportional to its indirect dependence d A . A higher rank means an earliershowing in a web search. This method was introduced in 2007 by Chung [4]. Let D be the matrix of direct influences, then thematrix of indirect influences is given by T ( D ) = e λ ( D − I ) = ∞ X k =0 ( D − I ) k λ k k ! , where I is the identity matrix with the size of D , and λ is a fixed positive real number. This method was introduced in 2009 by D´ıaz [6]. With PWP, as well as with MICMAC and Heat Kernel,one can compute indirect influences for any matrix of direct influences, even one with negative entries.Fix a positive real number λ (set to 1 in our applications) and let D be the matrix of direct influences.The matrix of indirect influences is given by: T ( D ) = e λD + e λ + , where for a matrix or a number x we set e x + = e x − ∞ X k =1 x k k ! . Direct Influences on Trade
We go back to the problem of defining direct influences on a trade network G . Definition 5.
The direct influence on trade C AB of country B on country A is given by: C AB = E ( B, A ) + I ( B, A ) E ( A ) + I ( A )where: • E ( B, A ) is the amount of exports from B to A , according to A ’s data. • I ( B, A ) is the amount of imports from B to A , according to A ’s data. • E ( A ) and I ( A ) are, respectively, the total exports and total imports of A .The direct influence on trade C AB measures the portion of A ’s international trade that involves B :out of each dollar the A trades with the world, it trades C AB dollars with B . A few remarks on thedefinition of direct influences on trade C AB : • Exports from country A to country B are often though as creating dependence of B on A . In ourapproach exports creates dependence in both directions: B needs some products coming from A (otherwise it would not buy), just as A needs the currency coming from B (otherwise it would notsell.) • Dependencies are normalized, indeed we have that d A = X B ∈ V C AB = X B ∈ V E ( B, A ) + I ( B, A ) E ( A ) + I ( A ) = E ( A ) + I ( A ) E ( A ) + I ( A ) = 1 . • A country trading significatively with many different partners will, in general, have low dependenceon any particular country.US and China are the first and the second world largest economies, respectively. Thus their mutualinfluences are a topic of global interest, see Figure 2 in which the diameter of vertices is roughly propor-tional to GDP. Definition 5 gives us a quantitative method for measuring these influences.According to US’s data its exports and its imports in 2011 (in thousands of US dollars) were1,479,730,169 and 2,262,585,634, respectively; while its exports to and its imports from China were103,878,414 and 417,302,859, respectively. So the influence of China on the US’s international trade is: C (US , China) = E (China , US) + I (China , US) E (US) + I (US) = 0 . . Thus a 13 .
9% of the US international trade involves China (either as a buyer or as a seller.)According to China’s data its exports and imports in 2011 were 1,898,388,435 and 1,743,394,866,respectively; while its exports to and its imports from the US were 325,010,987 and 123,124,009,respectively. So the influence of the US on China’s international trade is: C (China , US) = E (US , China) + I (US , China) E (China) + I (China) = 0 . . Thus a 12 .
3% of China’s international trade involves the US.So, with respect to trade, China has a higher direct influence on the US than viceversa, and influencesin both directions are quite high. 6igure 2: China-US bilateral influences.
The definition of direct influences given in the previous section measures international trade. In thissection we take a different approach that considers both international trade (trough exports and imports)and domestic trade (trough GDP).
Definition 6.
The direct influence D AB of country B on the total offer of country A is given by: D AB = E ( B, A ) + I ( B, A )GDP( A ) + I ( A )where: • E ( B, A ) is the amount of exports from B to A , according to A ’s data. • I ( B, A ) is the amount of imports from B to A , according to A ’s data. • GDP( A ) and I ( A ) are A ’s gross domestic product and imports total, respectively.The direct influence on offer D AB measures the relative size of the trade between A and B in com-parison with A ’s total offer GDP( A ) + I ( A ). In other words, D AB is the portion of products trade in A (both home grown and imported) that involve B (either because they were imported from or exportedto B ).A couple remarks on the definition of direct influences on offer D AB : • The total dependence d A of country A measures the portion of its offer that involves (troughimports or exports) the rest of the world. Indeed we have that d A = X B ∈ V D AB = X B ∈ V E ( B, A ) + I ( B, A ) GDP ( A ) + I ( A ) = E ( A ) + I ( A ) GDP ( A ) + I ( A ) . • The greater the GDP of a country, the lesser its dependence on other countries.7et us compute the China-US bilateral influences on offer. The GDP of China and US amounted in2011 to 7,321,935,025 and 14,991,300,000, respectively. Thus the influence of China on the US’s offeris given by: D (US , China) = E (China , US) + I (China , US)GDP(US) + I (US) = 0 . . Thus a 3 .
02% of the US’s offer involves China (either as a buyer or as a seller.)The influence of the US on China’s offer is in turn given by: D (China , US) = E (US , China) + I (US , China)GDP(China) + I (China) = 0 . . Thus a 4 .
94% of China’s offer involves the US.So, regarding offer, we find that the US have a higher influence on China than viceversa, reversingthe result obtained for direct influences on trade in Section 4.
To account for indirect influences on a trade network we apply the PWP method from Section 3 to thematrices C and D of Definitions 5 and 6 measuring direct influences on trade and offer. Thus we obtainthe matrices indirect influences on trade and offer given, respectively, by T ( C ) = e λC + e λ + and T ( D ) = e λD + e λ + . Let us consider in details an extreme case that illustrates the importance of taking indirect influencesinto account. We use data from the world trade network to be discussed in Section 8.For political reasons Cuba and the US have fairly low direct trade, and as a result the direct influenceof the US on Cuba in trade and offer are just 0 .
03 and 0 . .
6% of the Cuban international trade and offer, respectively. Comparing direct and indirectinfluences of the US on Cuba one obtains, after normalization, increments of 66% and 20% on trade andoffer, respectively. These increments in indirect vs direct influences are quite remarkable since, as weshall see in Section 8, the US has decreasing global indirect influences both on trade and on offer.The increments in indirect influences come from the fact that, beyond direct trade, the US alsoinfluences Cuba trough chains of direct trade, which in the simplest case take the form of triangulations.For example, in Figure 3 we show the triangle of direct influences US-Spain-Cuba: the US exerts adirect influence on Spain in trade and offer of 3 .
8% and 1 . .
8% and 1 . We consider 35 countries located on the American continent ranging from mighty US to tiny Dominica. The trade network for the American continent is a nearly complete graph with each country trading inaverage with 32.2 countries out of 34 possibilities. The fully connected countries are Argentina, Brazil,Canada, Colombia, Costa Rica, Cuba, Dominican Republic, Ecuador, Guatemala, Jamaica, Mexico,Peru, Salvador, and the US. The lesser connected countries are Bolivia, Dominica, Haiti, and Paraguay,trading respectively with 29 , , , and 27 countries.The American continent trade network, shown in Figure 4, was drawn with SAGE using the matrix D AB of direct influences on offer from Definition 6.The first thing one notices from the table in Figure 5 is that the US, as expected, leads both indirect and indirect influences. One, however, may wonder why its direct influences are higher than itsindirect influences? This happens, we believe, for a couple of reasons: 1) Because of its leading role,the US is more likely to trade with countries otherwise isolated with little influence on other countries.2) Since influences are normalized, the decreasing indirect influences of the US may also be explainedby the opposite behaviour in countries such as Canada and Mexico, whose indirect influences increaseto a good extend precisely because of their special relation (NAFTA agreement) with the leading country.Note that Brazil has higher direct influences than both Canada and Mexico, this is to be expected Antigua and Barbuda, Argentina, Bahamas, Barbados, Belize, Bolivia, Brazil, Canada, Chile, Colombia, Costa Rica,Cuba, Dominica, Dominican Republic, Ecuador, El Salvador, United States, Grenada, Guatemala, Guyana, Haiti, Jamaica,Mexico, Nicaragua, Panama, Paraguay, Peru, Saint Kitts and Nevis, Saint Lucia, Saint Vincent and the Grenadines,Suriname, Trinidad and Tobago, Uruguay, and Venezuela.
The world trade network containing 177 countries distributed all around the globe, shown schematicallyin Figure 7, was drawn feeding SAGE with the matrix of direct influences on offer from Definition 6.Unsurprisingly, the world trade network contains a big highly connected component containing thevast majority of countries in the world, with the exception of a handful of commercially isolated countriessuch as Buthan, Guinea Bissau, Kiribati, Samoa, Tonga, and Tuvalu. On average a country trades with131 .
46 countries, and there are 24 countries each trading with at least 173 countries.We proceed to address a couple of questions concerning the world trade network: Are GDP anddirect or indirect influences correlated? Are there significative differences between direct and influences?These question are both analyzed by looking first to influences on offer, and then influences on trade.Instead of a statistical analysis we simply highlight interesting cases regarding the top 20 countries inindirect influences.Naturally, we expect high GDP countries to have high direct and indirect influences on offer, andindeed there is a generic tendency for this correlation to hold, however the actual data reveals a far moresubtle situation, see Figure 8. 11igure 6: Influences on Trade in the American Continent Trade NetworkJapan has the 3rd largest world GDP, but occupies the 8th and 7th places in direct and indirectinfluences, respectively. Canada is in the 11th place in the GDP ranking, but occupies the 20th and21st positions in the direct and indirect influences rankings, respectively. Similarly, Mexico is in the14th place in the GDP ranking, but 34th and 31st positions in direct and indirect influences. The dropin influences compared to GDP in the latter cases is probably consequence of Canada’s and Mexico’sstrong economic ties to the US, so their influences are somewhat diluted in the US high GDP.Comparing direct and indirect influences on offer, see Figure 8, we find countries with similar resultsin both rankings, for example Brazil, Italy, Switzerland, and Korea. There are countries with dimin-ishing indirect influences as Australia, China, India, Russia, Singapore, South Africa, Thailand, UnitedArab Emirates and the US. In the case of South Africa and even more so for Thailand the decreasesare quite sharp. There are also countries with raising indirect influences as Germany, France, Japan,United Kingdom, Netherlands, Belgium and Spain. As a rule European countries tend to have increasingindirect influences, the case of Germany being remarkable because a sharp increase occurs.Next we consider direct and indirect influences on trade in the world trade network, see Figure 9.In this case we expect less correlation with the GDP ranking, since we are looking at influences oninternational trade and disregarding domestic trade. For example, Netherlands climbs from position 17in GDP to position 7 in indirect influences on trade. Similarly, Belgium occupies the 24th position inGDP, but raises to the 12th position in indirect influences on trade. In the opposite trend, Brazil is inthe 6th place in GDP, but is in the 17th position in indirect influences on trade.From Figures 8 and 9, it is quite clear that influences on trade measure a different thing that influ-ences on offer, even if the US, China, Germany and France, in this order, remain the top 4 countriesin both rankings. For example, South Africa, Thailand, and the United Arab Emirates are not longerin the top 20 countries in influences on trade, and are replaced by Canada, Mexico and Poland. Ob-serve that Brazil, Japan, Korea, Netherlands, Spain, and the United Kingdom climb in their rankingpositions, Belgium remains stable, and Australia, India, Italy, Russia, Singapore, and Switzerland fallin their rankings positions. 12imilarly, direct and indirect influences on trade are weaklier correlated than direct and indirectinfluences on trade. Only France, among the top 20 countries, have nearly identical direct and indirectinfluences on trade. Australia, Brazil, India, Italy, Russia, Singapore, Spain, Switzerland, and the UShave diminishing indirect influences, whereas Belgium, Canada, China, Germany, Japan, Korea, Mexico,Netherlands, and the United Kingdom have increasing indirect influences. The cases of Germany andKorea are interesting because of their sharp increments.Below we show the indirect dependence-influence planes, both for trade and offer, of the world tradenetwork. Note that the United States and China have high influence and low dependence on offer, whilethey have both high influence and dependence on trade, this is due to their high GDPs. In contrast,Germany have high influence and high dependence both on offer and on trade.
In this work we developed a network viewpoint leading to a couple of ways of ranking direct influencesof countries in the international trade market, namely, a ranking based on trade and another rankingbased on offer. The first ranking measures relative international trade trough exports and imports, whilethe second ranking also takes into account domestic trade trough GDPs.As far as we can tell a network approach to international trade taking into account indirect influenceshas not been designed so far. Most studies found in the literature gauge direct influences as evidencedby trade accounts. In this regard, they compare to our networks of direct influences in trade, ratherthan to our networks of indirect influences in trade. We showed that taking in consideration indirectinfluences may lead to more realistic indexes, as relevant cases suggest.We computed indirect influences using the PWP method, and applied it to the American continenttrade network and to the World trade network. Thus we obtained new rankings and the correspondingdependence-influence planes.Part of the motivation for this work is the high public interest in ranking countries for their central-ity in international trade. Various rankings have been introduced, some quite sophisticated such as theDHL Global Connectedness Index [9]. This ranking however results from the integration of many fea-tures, whereas our rankings focus on a single feature enhanced by taking into account indirect influences.The problem of further integrating the various rankings proposed in the literature may be attacked bythinking of the set of rankings as a finite metric space. For example, we can define a suitable normalizedEuclidean distance between indices r, l : V −→ [1 , n ] (with r ( A ) and l ( A ) being the ranking positions ofcountry A according to r and l , respectively; and n is the total number of countries) given by d ( r, l ) = 1( n − √ n s X A ∈ V ( r ( A ) − l ( A )) . Furthermore, we also have the problem of consolidating the various indexes into just one index.We applied our methods for computing indirect influences to trade networks at the level of countries.It should however be clear that these methods may also be applied at the business level, and even atthe individual level. In the latter cases, finding reliable data at a global scale and managing such hugedata are daunting problems. Nevertheless, our methods can be readily applied if one focusses on specificbusiness sectors, just like we made a focus study of indirect influences on trade and offer in the Americancontinent. 13he models presented in this work were static, in the sense that our data referred to just one year,namely 2011. Nevertheless, our techniques may be extended to a multiple years dynamical model inwhich influences are time depended functions.
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