A network analysis of countries' export flows: firm grounds for the building blocks of the economy
Guido Caldarelli, Matthieu Cristelli, Andrea Gabrielli, Luciano Pietronero, Antonio Scala, Andrea Tacchella
aa r X i v : . [ phy s i c s . s o c - ph ] A p r A network analysis of countries’ export flows: firm grounds forthe building blocks of the economy
Guido Caldarelli , , , Matthieu Cristelli , ∗ , Andrea Gabrielli , , Luciano Pietronero , , Antonio Scala , ,Andrea Tacchella ∗ E-mail: [email protected]
Abstract
In this paper we analyze the bipartite network of countries and products from UN data on country pro-duction [1, 2]. We define the country-country and product-product projected networks and introduce anovel method of filtering information based on elements’ similarity. As a result we find that countryclustering reveals unexpected socio-geographic links among the most competing countries. On the samefootings the products clustering can be efficiently used for a bottom-up classification of produced goods.Furthermore we mathematically reformulate the “reflections method” introduced by Hidalgo and Haus-mann [2] as a fixpoint problem; such formulation highlights some conceptual weaknesses of the approach.To overcome such an issue, we introduce an alternative methodology (based on biased Markov chains)that allows to rank countries in a conceptually consistent way. Our analysis uncovers a strong non-linearinteraction between the diversification of a country and the ubiquity of its products, thus suggesting thepossible need of moving towards more efficient and direct non-linear fixpoint algorithms to rank countriesand products in the global market.
Introduction
Complex Networks
Networks emerged in the recent years as the main mathematical tool for the description of complexsystems. In particular, the mathematical framework of graph theory made possible to extract relevantinformation from different biological and social systems [3, 4]. In this paper we use some concepts ofnetwork theory to address the problem of economic complexity [5–7].Such activity is in the track of a long-standing interaction between economics and physical sciences[8–12] and it explains, extends and complements a recent analysis done on the network of trades betweennations [1, 2]. Hidalgo and Hausmann (HH) address the problem of competitiveness and robustness ofdifferent countries in the global economy by studying the differences in the Gross Domestic Product andassuming that the development of a country is related to different“capabilities”. While countries cannotdirectly trade capabilities, it is the specific combination of those capabilities that results in differentproducts traded. More capabilities are supposed to bring higher returns and the accumulation of newcapabilities provides an exponentially growing advantage. Therefore the origin of the differences in thewealth of countries can be inferred by the record of trading activities analyzed as the expressions of thecapabilities of the countries.
Revealed Competitive Advantage and the country-product matrix
We consider here the Standard Trade Classification data for the years in the interval 1992 − N c = 129 different countries and N p = 772 different products.To make a fair comparison between the trades, it is useful to employ Balassa’s Revealed ComparativeAdvantage (RCA) [13] i.e. the ratio between the export share of product p in country c and the share ofproduct p in the world market RCA cp = X cp X p ′ X cp ′ / X c ′ X c ′ p X c ′ ,p ′ X c ′ p ′ (1)where X cp represents the dollar exports of country c in product p .We consider country c to be a competitive exporter of product p if its Revealed Comparative Advantage(RCA) is greater than some threshold value, which we take as 1 as in standard economics literature;previous studies have verified that small variations around such threshold do not qualitatively change theresults.The network structure of the country-product competition is given by the semipositive matrix M defined as M cp = (cid:26) if RCA cp > R ∗ if RCA cp < R ∗ (2)where R ∗ is the threshold ( R ∗ = 1).To such matrix ˆ M we can associate a graph whose nodes are divided into two sets { c } of N c nodes(the countries) and { p } of N p nodes (the products) where a link between a node c and a node p exists ifand only if M cp = 1, i.e. a bipartite graph. The matrix ˆ M is strictly related to the adjacency matrix ofthe country-product bipartite network.The fundamental structure of the matrix ˆ M is revealed by ordering the rows of the matrix by thenumber of exported products and the columns by the number of exporting countries: doing so, ˆ M assumesa substantially triangular structure. Such structure reflects the fact that some countries export a largefraction of all products (highly diversified countries), and some products appear to be exported by mostcountries (ubiquitous products). Moreover, the countries that export few products tend to export onlyubiquitous products, while highly diversified countries are the only ones to export the products that onlyfew other countries export.This triangular structure is therefore revealing us that there is a systematic relationship between thediversification of countries and the ubiquity of the products they make. Poorly diversified countries havea revealed comparative advantage (RCA) almost exclusively in ubiquitous products, whereas the mostdiversified countries appear to be the only ones with RCAs in the less ubiquitous products which ingeneral are of higher value on the market. It is therefore plausible that such structure reflects a rankingamong the nations.The fact that the matrix is triangular rather than block-diagonal suggests that, as countries becomemore complex, they become more diversified. Countries add more new products to the export mix whilekeeping, at the same time, their traditional productions. The structure of ˆ M therefore contradicts mostof classical macro-economical models predicting always a specialization of countries in particular sectorsof production (i.e. countries should aggregate in communities producing similar goods) that would resultin a more or less block-diagonal matrix ˆ M .In the following, we are going to analyze the economical consequences of the structure of the bipartitecountry-product graph described by ˆ M . In particular, we analyze the community structure inducedby ˆ M on the countries and products projected networks. As a second step, we reformulate as a linearfixpoint algorithm the HH’s reflection method to determine the countries and products respective rankingsinduced by ˆ M . In this way we are able to clarify the critical aspects of this method and its mathematicalweakness. Finally, to assign proper weights to the countries, we formulate a mathematically well definedbiased Markov chain process on the country-product network; to account for the bipartite structure ofthe network, we introduce a two parameter bias in this method. To select the optimal bias, we comparethe results of our algorithm with a standard economic indicator, the gross domestic product GDP . Theoptimal values of the parameters suggests a highly non-linear interaction between the number of differentproducts produced by each country ( diversification ) and the number of different countries producing eachproduct ( ubiquity ) in determining the competitiveness of countries and products. This fact suggests that,to better capture the essential features of economical competition of countries, we need of a more directand efficient non-linear approach.
Results
The network of countries
In order to obtain an immediate understanding of the economic relations between countries inducedby their products a possible approach is to define a projection graph obtained from the original set ofbipartite relations represented by the matrix ˆ M [14]. The idea is to connect the various countries witha link whose strength is given by the number of products they mutually produce. In such a way theinformation stored in the matrix ˆ M is projected into the network of countries as shown in Fig. 1.The country network can be characterized by the ( N C × N C ) country-country matrix ˆ C = ˆ M ˆ M T .The non-diagonal elements C cc ′ correspond to the number of products that countries c and c ′ havein common (i.e. are produced by both countries). They are a measure of their mutual competition,allowing a quantitative comparison between economic and financial systems [15]; the diagonal elements C cc corresponds to the number of products produced by country c and are a measure of the diversificationof country c .To quantify the competition among two countries, we can define the similarity matrix among countriesas S Ccc ′ = 2 C cc ′ C cc + C c ′ c ′ . (3)Note that 0 ≤ S Ccc ′ ≤ c and c ′ . Similar approaches to define a correlation between vertices or adistance [16] have often been employed in the field of complex networks, for example to detect proteincorrelations [17] or to characterize the interdependencies among clinical traits of the orofacial system [18].The first problem for large correlation networks is how to visualize the relevant structure. Thesimplest approach to visualize the most similar vertices is realized by building a Minimal SpanningTree (MST) [19, 20]. In this method, starting from an empty graph, edges ( c, c ′ ) are added in order ofdecreasing similarity until all the nodes are connected; to obtain a tree, edges that would introduce aloop are discarded. A further problem is to split the graph in smaller sub-graphs (communities) thatshare important common feature, i.e. have strong correlations. Similarity , like analogous correlationindicators, can be used to detect the inner structure of a network; while different methods for communitydetection vary in their detailed implementation [21, 22], they give reasonably similar qualitative resultswhen the indicators contain the same information.The MST method can be thus generalized in order to detect the presence of communities by addingthe extra condition that no edge between two nodes that have been already connected to some other nodeis allowed. In this way we obtain a set of disconnected sub-trees (i.e. a forest) embedded in the MST. This
Minimal Spanning Forest (MSF) method naturally splits the network of countries into separate subsets.This method allows for the visualization of correlations in a large network and at the same time performsa sort of community detection if not precise, certainly very fast.By visual inspection in Fig.2 we can spot a large subtree composed by developed countries and someother subtrees in which clear geographical correlations are present. Notice that each subtree containscountries with very similar products, i.e. countries that are competing on the same markets. In particular,developing countries seem to be mostly direct competitors of their geographical neighbors. This is ageneral feature of economics systems, even if it is not the most rationale choice [23, 24]: as an example,both banks [25] and countries [26] trade preferentially with similar partners, thereby affecting the wholerobustness of the system [27, 28]. This behavior can be reproduced by simple statistical models based onagents’ fitnesses [29].
The network of products
Similarly to countries, we can project the bipartite graph into a product network by connecting twoproducts if they are produced by the same one or more countries giving a weight to this link proportionalto the number of countries producing both products. Such network can be represented by the ( N P × N P ) product-product matrix ˆ P = ˆ M T ˆ M . The non-diagonal elements P pp ′ correspond to the number ofcountries producing both p and p ′ have in common, while the diagonal elements P pp corresponds to thenumber of countries producing p .In analogy with Eq. (3), the similarity matrix among products is defined as S Ppp ′ = 2 P pp ′ P pp + P p ′ p ′ . (4)It indicates how much products are correlated on a market: a value S Ppp ′ = 1 indicates that wheneverproduct p is present on the market of a country, also product p ′ would be present. This could be forexample the case of two products p , p ′ that are both necessary for the same and only industrial process.As in the case of countries, the MSF algorithm can be applied to visualize correlations and detectcommunities. In the case of the product network this analysis brings to an apparently contradictoryresults: let’s see why. Products are officially characterized by a hierarchical topology assigned by UN.Within this classification similar issue as “metalliferous ores and metal scraps” (groups 27.xx) are ina totally different section with respect to “non ferrous metals” (groups 68.xx). By applying our newalgorithm, based on the economical competition network ˆ M , one would naively expect that productsbelonging to the same UN hierarchy should belong to the same community and vice-versa ; therefore, ifwe would assign different colors to different UN hierarchies, one would expect all the nodes belonging toa single community to be of the same color. In Fig. 3 we show that this is not the case. Such a paradoxcan be understood by analyzing in closer detail the detected communities with the MSF method. As anexample, we show in Fig.4 a large community where most of the vertices belong to the area of “vehiclepart and constituents”. In this cluster we can spot the noticeable presence of a vertex belonging to “food”hierarchy. This apparent contradiction is solved up by noticing that such vertex refers to colza seeds, atypical plant recently used mostly for bio-fuels and not for alimentation: our MSF method has correctlypositioned this ”food” product in the ”vehicle” cluster. Therefore, methods based on community detectioncould be considered as a possible rational substitute for current top-down ”human-made” taxonomies [29]. Ranking Countries and Products by Reflection Method
Hidalgo and Haussman (HH) have introduced in [1,2] the fundamental idea that the complex set of capa-bilities of countries (in general hardly comparable between different countries) can be inferred from thestructure of matrix ˆ M (that we can observe). In this spirit, ubiquitous products require few capabilitiesand can be produced by most countries, while diversified countries possess many capabilities allowingto produce most products. Therefore, the most diversified countries are expected to be amongst thetop ones in the global competition; on the same footing ubiquitous products are likely to correspond tolow-quality products.In order to refine such intuitions in a quantitative ranking among countries and products, the authorsof [1, 2] have introduced two quantities: the n th level diversification d ( n ) c (called k c,n in [1, 2]) of thecountry c and the n th level ubiquity u ( n ) p (called k p,n in [1, 2]) of the product p . At the zero th order thediversification of a country is simply defined as the number of its products or d (0) c = N p X p =1 M cp ≡ k c (5)where k c is the degree of the node c in the bipartite country-product network); analogously the zero th order ubiquity of a product is defined as the number of different countries producing it u (0) p = N c X c =1 M cp ≡ k p (6)where k p is the degree of the node p in the bipartite country-product network. The diversification k c isintended to represent the zero th order measure of the “quality” of the country c with the idea that themore products a country exports the strongest its position on the marker. The ubiquity k p is intendedto represent the zero th order measure of the “dis-value of the product p in the global competition withthe idea that the more countries produce a product, the least is its value on the market.In the original approach these two initial quantities are refined in an iterative way via a so-called“reflections method”, consisting in defining the diversification of a country at the ( n + 1) th iteration asthe average ubiquity of its product at the n th iteration and the ubiquity of a country at the ( n + 1) th iteration as the average diversification of its producing countries at the n th iteration: d ( n +1) c = k c P N p p =1 M cp u ( n ) p u ( n +1) p = k p P N c c =1 M cp d ( n ) c (7)In vectorial form, this can be cast in the following form d ( n ) = ˆ J A u ( n − u ( n ) = ˆ J B d ( n − (8)where d ( n ) is the N c − dimensional vector of components d ( n ) c , u ( n ) is the N p − dimensional vector ofcomponents u ( n ) p , and where we have called ˆ J A = ˆ C ˆ M and ˆ J B = ˆ P ˆ M t (the upper suffix t stands for“transpose”), with ˆ C and ˆ P respectively the N c × N c and N p × N p square diagonal matrices defined by C cc ′ = k − c δ cc ′ and P pp ′ = k − p δ pp ′ .Such an approach suffers from some flaws. The first one is related to the fact that the process isdefined in a bipartite networks and therefore even and odd iterations have different meanings. In fact,let us consider the diversification d (1) c of the c th country: as prescribed by the algorithm, d (1) c is theaverage ubiquity of the products of the c th country at the 0-th iteration. Therefore countries with mostubiquitous (less valuable) products would get an highest 1 st order diversification. On the other hand,the approximately triangular structure of ˆ M tells us that these countries are the same ones with a smalldegree and therefore with a low value of the 0 − th order diversification d (0) . As shown to by [1,2], this isthe case also to higher orders; therefore the diversifications at even and odd iterations are substantiallyan anti-correlated. Conversely, successive even iterations are positively correlated so that d (2) c looks arefinement of d (0) c , d (4) c a refinement of d (2) c and so on. Same considerations apply to the iterations forthe ubiquity of products.The major flaw in the HH algorithm is that it is a case of a consensus dynamics, i.e. the state of anode at iteration t is just the average of the state of its neighbors at iteration t −
1. It is well knownthat such iterations have the uniform state (all the nodes equal) as the natural fixpoint. It is thereforepuzzling how such ”equalizing” procedure could lead to any form of ranking. To solve such a puzzle, let’swrite the HH algorithm as a simple iterative linear system and analyze its behavior.Focusing only on even iterations and on diversifications, we can write HH procedure as: d (2 n ) = ˆ J A ˆ J B d (2 n − = ( ˆ J A ˆ J B ) n d (0) = ˆ H n d (0) , (9)where ˆ H = ˆ J A ˆ J B = ˆ C ˆ M ˆ P ˆ M t is a N c × N c squared matrix.The matrix ˆ H in Eq.9 is a Markovian stochastic matrix when it acts from the right on positive vectors,in the sense that every element H cc ′ ≥ N c X c =1 H cc ′ = 1 . In particular for the given ˆ M adjacency matrix it is also ergodic. Therefore, its spectrum of eigenvaluesis bounded in absolute value by its unique upper eigenvalue λ = 1. Since ˆ H acts on d (2 n − from theleft, the right eigenvector e corresponding to the largest eigenvalue λ = 1 is simply a uniform vectorwith identical components, i.e. in the n → ∞ limit d (2 n ) converges to the fixpoint e where all countrieshave the same asymptotic diversification.It is therefore not a case that HH prescribe to stop their algorithm at a finite number of iterationsand that they introduce as a recipe to consider as the ranking of a country the rescaled version of the2 n th level diversifications [2] ˜ d (2 n ) c = d (2 n ) c − d (2 n ) σ (2 n ) d , (10)where d (2 n ) is the arithmetic mean of all d (2 n ) c and σ (2 n ) d the standard deviation of the same set. Withthese prescription, HH algorithm seems to converge to an approximately constant value after ∼
16 steps.This observed behavior can be easily be explained by noticing that, in contrast with the erroneousstatement in [2], finding the fitness by the reflection method can be reformulated as a fix-point problem(our Eq. 9) and solved using the spectral properties of a linear system. In fact,since the ergodic Markoviannature of ˆ H we can order eigenvalues/eigenvectors such that | λ N c | ≤ | λ N c | ≤ ... ≤ | λ | < λ = 1.Therefore, expanding d (0) in terms of the right eigenvectors { e , e , ..., e N c } of ˆ H the initial condition d (0) = a e + a e + ... + a N c e N c , we can write the 2 n -th iterate as d (2 n ) = a e + a λ n e + ... + a N c λ nN c e N c = a e + a λ n e + O (( λ /λ ) n ) . (11)Therefore, at sufficiently large n the ordering of the countries is completely determined by the componentsof e ; notice that such an asymptotic ordering is independent from the initial condition d (0) and thereforeshould be considered as the appropriate fixpoint renormalized fitness d ∗ for all countries.What happens to the HH scheme? At sufficiently large n , (cid:10) d (2 n ) (cid:11) ≈ a e and σ d (2 n ) ∝ a λ n e +0 (( λ /λ ) n ); therefore d (2 n ) becomes proportional to e (Eq. 10). The number of iterations it neededto converge is given by the ratio between λ and λ (( λ /λ ) it ≪
1; therefore the it ∼
16 iterationsprescribed by HH are not a general prescription but depend on the structure of the network analyzed.Notice also that when the numerical reflection method is used, the renormalized fitness representsa deviation O ( λ n ) from a constant and can be detected only if it is bigger than the numerical error;therefore only ”not too big” it can be employed. On the other hand, the spectral characterization wepropose does not suffer from such a pitfall even when. Similar considerations can be developed for theeven iterations of the reflection method for the products. Biased Markov chain approach and non-linear interactions
Having assessed the flaws of HH’s method, we investigate the possibility of defining alternative linearalgorithms able to implement similar economical intuitions about the ranking of the countries whilekeeping a more robust mathematical foundation. In formulating such a new scheme we will keep theapproximation of linearity for the iterations even though we shall find in the results hints of the non-linear nature of the problem.Our approach is inspired to the well-known PageRank algorithm [30]. PageRank (named after theWWW, where vertices are the pages) is one of the most famous of Bonacich centrality measures [31].In the original PageRank method the ranking of a vertex is proportional to the time spent on it by anunbiased random walker (in different contexts [11] analogous measures assess the stability of a firm in abusiness firm network).We define the weights of vertices to be proportional to the time that an appropriately biased randomwalker on the network spends on them in the large time limit [32]. As shown below, such weights,being the generalization of k c and k p , give a measure respectively of competitiveness of countries and“dis-quality” (or lack of competitiveness) of products. As the nodes of our bipartite network are entitiesthat are logically and conceptually separated (countries and products), we assign to the random walkera different bias when jumping from countries to products respect to jumping from products to countries.Let us call w ( n ) c weight of country c at the n th iteration and w ( n ) p fitness of product p at the n th iteration. We define the following Markov process on the country-product bipartite network w ( n +1) c ( α, β ) = P N p p =1 G cp ( β ) w ( n ) p ( α, β ) w ( n +1) p ( α, β ) = P N c c =1 G pc ( α ) w ( n ) c ( α, β ) (12)where the Markov transition matrix ˆ G is given by G cp ( β ) = M cp k − βc P Ncc ′ =1 M c ′ p k − βc ′ G pc ( α ) = M cp k − αp P Npp ′ =1 M cp ′ k − αp ′ (13)Here G cp gives the probability to jump from product p to country c in a single step, and G pc theprobability to jump from country c to product p also in a single step. Note that Eqs.(13) define a( N c + N p ) − dimensional connected Markov chain of period two. Therefore, random walkers initiallystarting from countries, will be found on products at odd steps and on countries at even ones; the reversehappens for random walkers starting from products. By considering separately the random walkersstarting from countries and from products, we can reduce this Markov chain to two ergodic Markovchains of respective dimension N c and N p . In particular, if the walker starts from a country, using avectorial formalism, we can write for the weights of countries w ( n +1) c ( α, β ) = ˆ T ( α, β ) w ( n ) c ( α, β ) (14)where the N c × N c ergodic stochastic matrix ˆ T is defined by T cc ′ ( α, β ) = N p X p =1 G cp ( β ) G pc ′ ( α ) . (15)At the same time for products we can write w ( n +1) p ( α, β ) = ˆ S ( α, β ) w ( n ) p ( α, β ) , (16)where the N p × N p ergodic stochastic matrix ˆ S is given by S pp ′ ( α, β ) = N c X c =1 G pc ( α ) G cp ′ ( β ) . (17)Given the structure of ˆ T and ˆ S , it is simple to show that the two matrices share the same eigenvaluespectrum which is upper bounded in modulus by the unique eigenvalue µ = 1. For both matrices, theeigenvectors corresponding to µ are the stationary and asymptotic weights { w ∗ c ( α, β ) } and { w ∗ p ( α, β ) } of the Markov chains. In order to find analytically such asymptotic values, we apply the detailed balancecondition: G pc w ∗ c = G cp w ∗ p ∀ ( c, p ) (18)which gives w ∗ c = A (cid:16)P N p p =1 M cp k − αp (cid:17) k − βc w ∗ p = B (cid:16)P N c c =1 M cp k − βc (cid:17) k − αp (19)where A and B are normalization constants. Note that for α = β = 0 Eq. (13) gives the completelyunbiased random walk for which ˆ T = ˆ H t where ˆ H is given in Eq. (9). Therefore, in this case Eqs. (19)become w ∗ c (0 , ∼ k c w ∗ p (0 , ∼ k p , (20)as for the case of unbiased random walks on a simple connected network the asymptotic weight of a node,is proportional to its connectivity. Thus, in the case of α = β = 0 we recover the zero th order iteration ofthe HH’s reflection method. Note that, in the same spirit of HH, w ∗ c (0 ,
0) gives a rough measure of thecompetitiveness of country c while w ∗ p gives an approximate measure of the dis-quality in the market ofproduct p . By continuity, we associate the same meaning of competitiveness/disquality to the stationarystates w ∗ c / w ∗ p at different values of α and β .To understand the behavior of our ranking respect to the bias, we have analyzed the mean correlation(square of the Pearson coefficient) for the year 1998 (other years give analogous results) between thelogarithm of the GDP of each country and its weight (Eqs. (19) for different values of α and β (seeFig. 5).It is interesting to note that the region of large correlations (region inside the contour plot in theFig. 5) is found in the positive quadrant for about 0 . < α < . . < β <
1; in particular themaximal value is approximately at α ≃ . β ≃ .
8. These results can be connected with theapproximately “triangular” shape of the matrix ˆ M . In fact, let us rewrite Eqs. (19) (apart from thenormalization constant) as: w ∗ c ∼ k − βc (cid:10) k − αp (cid:11) c w ∗ p ∼ k − αp (cid:10) k − βc (cid:11) p , where (cid:10) k − αp (cid:11) c is the arithmetic average of k − αp of the products exported by country c and (cid:10) k − βc (cid:11) p isthe arithmetic average of k − βc for countries exporting product p . Since β is substantially positive andslightly smaller of 1 and α is definitely positive with optimal values around 1, the competitive countrieswill be characterized by a good balance between a high value of k c and a small typical value of k p of itsproducts. Nevertheless, since the optimal values of α are distributed up to the region of values muchlarger than 1 (i.e. 1 − β is significantly smaller than 1), we see that the major role for the asymptoticweight of a country is played by the presence in its portfolio of un-ubiquitous products which alonegive the dominant contribution to w ∗ c . A similar reasoning leads to the conclusion that the dis-value(or ugliness) of a product is basically determined by the presence in the set of its producers of poorlydiversified countries that are basically exporting only products characterized by a low level of complexity.Our new approach based on biased Markov chain theory permits thus to implement the interestingideas developed by HH in [2], on a more solid mathematical basis using the framework of linear iteratedtransformations and avoiding the indicated flaws of HH’s “reflection method”. Interestingly, our resultsreveal a strongly non-linear entanglement between the two basic information one can extract from thematrix ˆ M : diversification of countries and ubiquity of products. In particular, this non-linear relationmakes explicit an almost extremal influence of ubiquity of products on the competitiveness of a countryin the global market: having “good” or complex products in the portfolio is more important than to havemany products of poor value. Furthermore, the information that a product has among its producers somepoorly diversified countries is nearly sufficient to say that it is a non-complex (dis-valuable) product inthe market. This strongly non-linear entanglement between diversifications of countries and ubiquitiesof products is an indication of the necessity to go beyond the linear approach in order to introduce moresound and direct description of the competition of countries and products possibly based on a suitable ab initio non-linear approach characterized by a smaller number of ad hoc assumptions [36]. We are aware that GDP is not an absolute measure of wealth [33] as it does not account directly for relevant quantitieslike the wealth due to natural resources [34]). Nevertheless, we expected that GDP monotonically increases with the wealth.What network analysis shows is that the number of products is correlated with both quantities. We envisage such kind ofanalysis in order to define suitable policies for underdeveloped countries [35]. Discussion
In this paper we applied methods of graph theory to the analysis of the economic productions of coun-tries. The information is available in the form of an N c × N p rectangular matrix ˆ M giving the differentproduction of the possible N p goods for each of the N c countries. The matrix ˆ M corresponds to a bi-partite graph, the country-product network, that can be projected into the country-country network C and the product-product network P . By using complex-networks analysis, we can attain an effectivefiltering of the information contained in C and P . We introduce a new filtering algorithm that identifiescommunities of countries with similar production. As an unexpected result, this analysis shows thatneighboring countries tend to compete over the same markets instead of diversifying. We also show thata classification of goods based on such filtering provides an alternative product taxonomy determined bythe countries’ activity. We then study the ranking of the countries induced by the country-product bipar-tite network. We first show that HH’s reflection method’s ranking is the fix-point of a linear process; inthis way we can avoid some logical and numerical pitfalls and clarify some of its weak theoretical points.Finally, in analogy with the Google PageRank algorithm, we define a biased, two parameters Markovchain algorithm to assign ranking weights to countries and products by taking into account the structureof the adjacency matrix of the country-product bipartite network. By correlating the fix-point ranking(i.e. competitiveness of countries and products) with the GDP of each country, we find that the optimalbias parameters of the algorithm indicate a strongly non-linear interaction between the diversification ofthe countries and the ubiquity of the products. Materials and Methods
Graphs
A graph is a couple G = ( V, E ) where V = { v i | i = 1 . . . n A } is the set of vertices, and E ⊆ V × V is theset of edges. A graph G can be represented via its adjacency matrix AA ij = (cid:26) v i and v j . (21)The degree k i of the node v i is the number P j A ij of its neighbors.An unbiased random walk on a graph G is characterized by a probability p ij = 1 /k i of jumping froma vertex v i to one of its k i neighbors and is described by the jump matrix J G = K − A , (22)where K is the diagonal matrix K ij = k i δ ij corresponding to the nodes degrees. Bipartite Graphs
A bipartite graph is a triple G = ( A, B, E ) where A = { a i | i = 1 . . . n A } and B = { b j | j = 1 . . . n B } aretwo disjoint sets of vertices, and E ⊆ A × B is the set of edges, i.e. edges exist only between vertices ofthe two different sets A and B .The bipartite graph G can be described by the matrix ˆ M defined as M ij = (cid:26) a i and b j . (23)In terms of ˆ M , it is possible to define the adjacency matrix A of G as A = (cid:20) MM T . (cid:21) (24)1. It is also useful to define the co-occurrence matrices P A = M M T and P B = M T M that respectivelycount the number of common neighbors between two vertices of A or of B . P A is the weighted adjacencymatrix of the co-occurrence graph C A with vertices on A and where each non-zero element of P A cor-responds to an edge among vertices a i and a j with weight P Aij . The same is valid for the co-occurrencematrix P B and the co-occurrence graph C B .Many projection schemes for a bipartite graph G start from constructing the graphs C A or C B andeliminating the edges whose weights are less than a given threshold or whose statistical significance islow. Matrix from RCA
To make a fair comparison between the exports, it is useful to employ Balassa’s Revealed ComparativeAdvantage (RCA) [13] i.e. the ratio between the export share of product p in country c and the share ofproduct p in the world market RCA cp = X cp X p ′ X cp ′ / X c ′ X c ′ p X c ′ ,p ′ X c ′ p ′ (25)where X cp represents the dollar exports of country c in product p .The network structure is given by the country-product adjacency matrix ˆ M defined as M cp = (cid:26) if RCA cp > R ∗ if RCA cp < R ∗ (26)where R ∗ is the threshold. A positive entry, M cp = 1 tells us that country c is a competitive exporter ofthe product p . Acknowledgments
We thank EU FET Open project FOC nr.255987 and CNR-PNR National Project ”Crisis-Lab” forsupport.
Author Contributions
All the Authors contributed equally to the work
References
1. Hidalgo CA, Klinger B, Barab´asi AL, Hausmann R (2007) The Product Space Conditions theDevelopment of Nations. Science 317: 482–487.2. Hidalgo CA, Hausmann R (2009) The building blocks of economic complexity. Proceedings of theNational Academy of Sciences 106: 10570–10575.3. Caldarelli G (2007) Scale-Free Networks: Complex Webs in Nature and Technology. Oxford Uni-versity Press.24. Battiston S, Delli Gatti D, Gallegati M, Greenwald B, Stiglitz JE (2007) Credit chains andbankruptcy propagation in production networks. Journal of Economic Dynamics and Control31: 2061–2084.5. Jackson MO (2008) Social and Economic Networks.6. Borgatti SP, Mehra A, Brass DJ, Labianca G (2009) Network Analysis in the Social Sciences.Science 323: 892–895.7. Haldane AG, May RM (2011) Systemic risk in banking ecosystems. Nature 469: 351–355.8. Stanley HE, Amaral LAN, Buldyrev SV, Gopikrishnan P, Plerou V, et al. (2002) Self-organizedcomplexity in economics and finance. Proceedings of the National Academy of Sciences of theUnited States of America 99: 2561–2565.9. Serrano, Bogu˜n´a M (2003) Topology of the world trade web. Phys Rev E 68: 15101.10. Schweitzer F, Fagiolo G, Sornette D, Vega-Redondo F, Vespignani A, et al. (2009) EconomicNetworks: The New Challenges. Science 325: 422–425.11. Fu D, Pammolli F, Buldyrev SV, Riccaboni M, Matia K, et al. (2005) The growth of business firms:Theoretical framework and empirical evidence. Proceedings of the National Academy of Sciencesof the United States of America 102: 18801–18806.12. Majumder SR, Diermeier D, Rietz TA, Amaral LA (2009) Price dynamics in political predictionmarkets. Proceedings of the National Academy of Sciences 106: 679–684.13. Balassa B (1965) Trade liberalization and ’revealed’ comparative advantage. Manchester School33: 99–123.14. Bellman R (1997) Introduction to matrix analysis (2nd ed.). Philadelphia, PA, USA: Society forIndustrial and Applied Mathematics. URL http://portal.acm.org/citation.cfm?id=264987 .15. Johnson N, Lux T (2011) Financial systems: Ecology and economics. Nature 469: 302–303.16. Bonanno G, Caldarelli G, Lillo F, Mantegna RN (2003) Topology of correlation-based minimalspanning trees in real and model markets. Phys Rev E 68: 46130.17. Brun C, Chevenet F, Martin D, Wojcik J, Gu´enoche A, et al. (2003) Functional classification ofproteins for the prediction of cellular function from a protein-protein interaction network. Genomebiology 5.18. Auconi P, Caldarelli G, Scala A, Ierardo G, Polimeni A (2011) A network approach to orthodonticdiagnosis. Orthodontics & Craniofacial Research 14: 189–197.19. Mantegna RN (1999) Hierarchical structure in financial markets. European Physical Journal B 11:193–197.20. Mantegna RN, Stanley HE (2000) An Introduction to Econophysics: Correlations and Complexityin Finance. Cambridge Univ. Press, Cambridge UK.21. Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proceed-ings of the National Academy of Sciences 99: 7821–7826.22. Fortunato S (2010) Community detection in graphs. Physics Reports 486: 75–174.323. Farmer JD, Lo AW (1999) Frontiers of finance: Evolution and efficient markets. Proceedings ofthe National Academy of Sciences 96: 9991–9992.24. Chi Ho Yeung YCZ (2009) Minority Games. pp. 5588–5604.25. De Masi G, Iori G, Caldarelli G (2006) Fitness model for the Italian interbank money market.Phys Rev E 74: 66112.26. Garlaschelli D, Loffredo MI (2004) Fitness-Dependent Topological Properties of the World TradeWeb. Phys Rev Lett 93: 188701.27. Podobnik B, Horvatic D, Petersen AM, Uroˇsevi´c B, Stanley HE (2010) Bankruptcy risk model andempirical tests. Proceedings of the National Academy of Sciences .28. Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of failuresin interdependent networks. Nature 464: 1025–1028.29. Caldarelli G, Capocci A, De Los rios P, Mu˜noz MA (2002) Scale Free Networks from VaryingVertex Intrinsic Fitness. Physical Review Letters 89: 258702+.30. Page L, Brin S, Motwami R, Winograd T (1999) The PageRank citation ranking: bringing orderto the web. URL http://dbpubs.stanford.edu:8090/pub/1999-66 .31. Bonacich P (1987) Power and Centrality: A Family of Measures. American Journal of Sociology92: 1170–1182.32. Zlati´c V, Gabrielli A, Caldarelli G (2010) Topologically biased random walk and community findingin networks. Physical Review E 82: 066109+.33. Arrow KJ, Dasgupta P, Goulder LH, Mumford KJ, Oleson K (2010) Sustainability and the Mea-surement of Wealth. National Bureau of Economic Research Working Paper Series : 16599+.34. Dasgupta P (2009) The Place of Nature in Economic Development. Technical report. URL http://ideas.repec.org/help.html .35. Dasgupta P (2010) Poverty traps: Exploring the complexity of causation. International Food PolicyResearch Institute (IFPRI) 2010 Vision briefs BB07 Special Edition .36. Tacchella A, Cristelli M, Caldarelli G, Gabrielli A, Pietronero L (2012) Economic complexity:a new metric for countries’ competitiveness and products’ complexity. submitted to Journal ofEconomic Dynamics and Control .4
Figure 1. The network of countries and products and the two possible projections.Figure 2. The Minimal Spanning Forest for the Countries.
The various subgraphs have adistinct geographical similarity. We show in green northern European countries and in red the “Baltic”republics. In general neighboring (also in a social and cultural sense) countries compete for theproduction of similar goods.5
Figure 3. The Minimal Spanning Forest (MSF) for the Products.
We put a different coloraccording to the first digit used in COMTRADE classification. This analysis should reveal correlationbetween different but similar products.6
Figure 4. The largest tree in the Products MSF.
When passing from classification colors to thereal products name, we see they are all strongly related. It is interesting the presence of colza seeds inthe lower left corner of the figure.7 β
0 0.5 1 1.5 2 α Figure 5. The plot of the mean Correlation (square of Pearson coefficient, R ) betweenlogarithm of GDP and fixpoint weights of countries in the biased (Markovian) randomwalk method as a function of parameters α and β . The contour plot for a level of R ..