Topology of the world trade web

Topology of the World Trade Web Ma Ángeles Serrano and Marián Boguñá arXiv:cond-mat/0301015v1 [cond-mat.dis-nn] 2 Jan 2003 Departament de Fı́sica Fonamental, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain (Dated: February 1, 2008) Economy, and consequently trade, is a fundamental part of human social organization which, until now, has not been studied within the network modelling framework. Networks are mathematical tools used in the modelling of a wide variety of systems in social and natural science. Examples of these networks range from metabolic and cell networks to technological webs. Here we present the first empirical characterization of the world trade web, that is, the network built upon the trade relationships between different countries in the world. This network displays the typical properties of complex networks, namely, scale-free degree distribution, the small world property, a high clustering coefficient and, in addition, degree-degree correlation between different vertices. All these properties make the world trade web a complex network, which is far from being well-described through a classical random network description. PACS numbers: 89.75.-k, 87.23.Ge, 05.70.Ln The world is facing a challenging era. Social, political and economic arrangements initiated after the end of the Second World War are now culminating in the recognition of globalization, a process which has been accelerated by the new technological advances. When applied to the international economic order, globalization involves control of capital flow and liberalization of trade. As a consequence, economies around the world are becoming more and more interrelated, in other words, the world is becoming a global-village [1, 2]. In this scenario, trade plays a central role as one of the most important interaction channels between countries. The relevance of the international trade system goes beyond the fundamental exchange of goods and services. For instance, it can also be the channel for crises spreading [3]. A good example is found in the recent Asiatic crisis, which shows how economic perturbations originated in a country can somehow propagate elsewhere in the world [4, 5]. Thus, it seems natural to analyze the world trade system at a global level, every country being important regardless of its size or wealth. Despite the extremely complex nature of the problem, relevant structural information can be extracted from modelling the system as a network, where countries are represented as vertices and trade channels as links between these vertices. In this way, the global trade system can be examined under a topological point of view. This analysis will reveal complex properties which cannot be explained by the classical random graph theory. Complex networks have been the subject of an intense research activity over the last years [6, 7]. This great interest is fully justified by the extremely important role that this class of systems play in many different fields. Examples range from metabolic networks, where cell functionality is sustained by the network structure, to technological webs, where topology determines the system ability transmitting information [8, 9, 10, 11]. The term complex network typically refers to networks showing the following properties: (i) scale-free (SF) degree distribution, P (k) ∼ k −γ with 2 < γ ≤ 3, where the degree, k, is defined as the number of edges emanating from a vertex, (ii) the small-world property [12], which states that the average path length between any pair of vertices grows logarithmically with the system size and (iii) a high clustering coefficient, that is, the neighbors of a given vertex are interconnected with high probability. In addition, degree-degree correlation has been recently added to this list since it appears as a common feature in many real-world networked systems [10, 13, 14, 15]. This correlation accounts for the probability that a vertex of degree k is connected to a vertex of degree k ′ and is a key issue for the correct description of the hierarchical organization within the network. This correlation is found to be assortative, that is, highly connected vertices tend to attach to other highly connected vertices in social networks such as scientific collaboration networks [13]; conversely, the correlation is disassortative in technological networks, such as the Internet [10]. Classical random graph theory, first studied by Erdös and Rényi [16, 17], does not provide a good framework to fit all the above properties. This fact has posed the question of the origin of these anomalous topological features. Two possible mechanisms could explain their appearance: either the network is the result of macroscopic constraints, that is, the network is made ad hoc so that those properties are satisfied, or there is a self-organized evolution process leading, in the stationary state, to complex structures. This idea is at the core of the preferential attachment mechanisms, first introduced by Barabási and Albert [18], where a dynamical process of creation of new links, or rewiring of the existing ones, using global (or quasi global) information leads to SF networks displaying complex properties. Among all the studied networks, the social and technological networks have become the paradigmatic example of complex networks. Perhaps, the reason lies in the fact that each type exemplifies human activity working at one of two different levels: the cooperative level, where concepts such as friendship are dominant, and the competitive level, where the activity is governed by optimiza- 2 1 0 10 ck Cumulative distribution 10 −1 Pc(k) Pc(kout) Pc(kin) 1−γ Fit of the form Pc(k) α k Fit of the form ck=k ; γ=2.6 −ω with ω=0.7 Random network with <k>=43 10 0.2 −2 1 10 2 Connectivity 10 FIG. 1: Cumulative in- and out-degree distributions, Pc (kin ) and Pc (kout ), and undirected, Pc (k), corresponding to the import/export world trade web. The solid line is a power law fit of the form Pc (k) ∼ kγ−1 with γ = 2.6 ± 0.1. It is also shown the cumulative distribution of the equivalent random network with the same average degree tion criteria. These two different levels will probably lead to different attachment mechanisms which could be the origin of the assortative/disassortative degree-degree correlation observed in these networks. As an example of social network working at the competitive level, we study the network of trade relationships between different countries in the world, hereafter referred to as the world trade web (WTW). The topological characterization of the WTW is of primary interest for the modelling of crisis propagation at the global level as well as for the understanding of the effects that the new liberalist policies have on the world trade system. Moreover, we shall show that the WTW is a complex network sharing many properties with technological networks. In order to perform our analysis, we extracted data from aggregated trade statistics tables in the International Trade Center site [19], which are based on the COMTRADE database of the United Nations Statistics Division. These tables contain, for each country, an import and an export list detailing the forty more important exchanged merchandises in the year 2000. Primary and secondary markets are also reported for each product. If we consider imports as in-degrees and exports as out-degrees, it is possible to construct a directed network where vertices represent countries and directed links represent the import/export relations between them. The fact that the number of merchandises is bounded is, a priori, a limitation for the analysis. However, it is possible to overcome this problem taking advantage of the symmetry between in and out degrees. Let Ãimp and ij exp Ãij be the import/export adjacency matrices calculated from the import/export databases. Each adjacency matrix is defined so that Ãij = 1 if the country i imports from/exports to the country j and zero otherwise. These 10 1 10 2 k FIG. 2: Clustering coefficient of single countries as a function of their degree for the undirected version of the WTW. The solid line is a power law fit of the form ck ∼ k−ω with ω = 0.7 ± 0.05 matrices account only for a subset out of the total number of actual connections between countries. The import/export connections that are of little relevance for a given country are not considered in the matrices Ã, although they may be relevant to the partners as the symmetric export/import links. In fact, imports and exports definitions can apply to the same trade flow depending on whether origin or destination is considered. This implies that the complete adjacency matrices satisfy the symmetry relation Aimp = Aexp ji , which can be used in order to ij recover missing information from the original matrices. Thus, we can write h i 1 imp exp = Aimp (1) Ã + Ã ij ij ji 1 + δ(Ãimp +Ãexp ),2 ij ji where δ·,· is the Kronecker delta function. In this way we obtain an adjacency matrix where each connection is relevant, at least, to one of the two involved countries. At this point, it is worth noticing that we consider the unweighed version of the WTW. Since the weight of a link can be different depending on the import/export point of view, it is unclear how these weights should be assigned. After this symmetrization procedure, we obtain a directed network with 179 vertices representing countries and 7510 directed links representing commercial channels among them. The average degree of this network is hkin i = hkout i = 30.9. The question that first arises refers to the directed nature of the WTW. In fact, the in- and the out-degree of a given vertex are random quantities which may be correlated. A complete description should involve the knowledge of the joint probability p(kin , kout ). This function is often difficult to obtain although relevant information can be extracted from the correlation coefficient r = (hkin kout i − hki2 )/σin σout , where σin /σout are the in/out standard deviations (notice that hkin i = hkout i = hki). 3 10 2 <knn(k)> For the WTW this coefficient is r = 0.91, pointing to a strong similarity in the number of in and out connections. However, not all these connections run in both directions, that is, the fact that country A imports from country B does not necessary imply that country B imports from country A. In order to quantify this effect we compute the reciprocity of the network, defined as the fraction of links pointing simultaneously to both ends of the link. In our case the reciprocity is 0.61. These results suggest that, actually, the WTW may be thought of as an undirected network without losing relevant topological information. The average degree for the undirected version of the WTW is hki = 43. One of the most important topological properties of a network is the degree distribution, P (k). This quantity measures the probability of a randomly chosen vertex to have k connections to other vertices. In our case, due to the directed nature of the network, we have to distinguish between in- and out-degree distributions. Fig.1 shows the in, out, and undirected cumulative distributions for the P WTW, defined as Pc (k) ≡ k′ =k P (k ′ ). In all cases, the cumulative distribution shows a flat approach to the origin, indicating the presence of a maximum in P (k), at k ∼ 20, and, at this respect, similar to the Erdös-Rényi network. However, for k > 20 the cumulative distribution is followed by a power law decay Pc (k) ∼ k 1−γ , with γ ≈ 2.6, showing a strong deviation from the exponential tail predicted by the classical random graph theory. The exponent γ is found to be within the range defined by many other complex networks and, thus, we can state that the WTW belongs to the recently identified class of SF networks [6, 7]. The SF property implies an extremely high level of degree heterogeneity. Indeed, the second moment of the degree distribution, hk 2 i, diverges in the thermodynamic limit for any SF network. It may be surprising that, in fact, the SF region does not extend to the whole degree domain and that, for small values of the degree, the distribution is similar to the Erdös-Rényi network. In fact, the underlying preferential attachment mechanisms could differ depending on the particular political and economic situation of a country. Low-degree countries, most of which turn out to be the poorest, are basically constrained to subsistence trade flows and, therefore, preferential attachment mechanisms could not hold. As expected, there exists a positive correlation between the number of trade channels of a country and its wealth, measured by the per capita Gross Domestic Product (GDP) [20]. This correlation is found to be high, 0.65, which means that, indeed, most low-connected countries are poor countries –Angola, Somalia, Rwanda, Cambodia...– and most high-connected countries are rich countries –the USA, Japan, Germany and the UK, for example. However, there also exists a significant number of cases in the reversal situation, that is, low per capita GDP countries with a large number of connections and high per capita GDP countries with a relatively low number of trade channels. A germane example for the first circumstance is that of Norway or <knn(k)> <knn(kin)> <knn(kout)> −νk Fit of the form <knn(k)>=k , νk=0.5 1 10 k 10 2 FIG. 3: Average in (out, undirected) nearest neighbors degree as a function of the in (out, undirected) degree of the vertex. The solid line is a fit of the form hknn (k)i ∼ k−νk with νk = 0.5±0.05. The dashed line is the theoretical value of the hknn i corresponding to an uncorrelated network, that is, hk2 i/hki = 70.18. Iceland, which are between the top ten wealthier countries but only have 56 and 24 trade channels respectively. For the second case Brazil, China or Russia are typical examples. Thanks to a number of recent studies, it is becoming more and more evident that real networks are not completely random but they are organized according to a hierarchical structure [9, 10, 15, 21]. This hierarchy is usually analyzed by means of the local clustering coefficient and the degree-degree correlation. The clustering coefficient of the vertex i, of degree ki , is defined as ci ≡ 2ni /ki (ki − 1), where ni is the number of neighbors of i that are interconnected. If hierarchy was not present in the system, the local clustering coefficient should be a random quantity independent of any other property. Fig. 2 shows the local clustering coefficient of the undirected WTW as a function of the vertex’s degree. As it is clearly seen, this function has a strong dependence on the vertex’s degree, with a power law behavior ck ∼ k −ω , with ω = 0.7 ± 0.05. The clustering coefficient averaged over the whole network is C = 0.65, greater by a factor 2.7 than the value corresponding to a random network of the same size. Hierarchy is also reflected on the degree-degree correlation through the conditional probability P (k|k ′ ), which measures the probability of a vertex of degree k ′ to be linked to a vertex of degree k. Again, this function is difficult to measure due to statistical fluctuations. In order to characterize this correlation, it is more useful to work with the average nearest degree (ANND), P neighbors ′ ′ defined as hknn (k)i ≡ k′ k P (k |k) [14]. For uncorrelated networks, this function reads hknn i = hk 2 i/hki, independent of k. However, recent studies have revealed that almost all real networks show degree-degree correlation [10, 13] which translates into a k dependence in the 4 TABLE I: size hki hdi C WTW 179 43 1.8 0.65 Internet 5287 3.8 3.7 0.24 RG 179 43 1.73 0.24 γ ω νk 2.6 0.7 0.5 2.2 0.75 0.5 – 0 0 ANND. This correlation can be assortative or disassortative depending on whether the ANND is an increasing or decreasing function of the degree. Fig. 3 reports the ANND for the directed and undirected versions of the WTW. It can be observed a clear dependency on the vertex’s degree, with a power law decay hknn (k)i ∼ k −νk with νk = 0.5±0.05. This result means that the WTW is a disassortative network where highly connected vertices tends to connect to poorly connected vertices. This result, together with the scaling law ck ∼ k −ω reveals a hierarchical architecture of highly interconnected countries that belong to influential areas which, in turn, connect to other influential areas through hubs. Surprisingly, these results point to a high similarity between the WTW and the Internet. Indeed, the Internet is a SF network, with a critical exponent γ = 2.2, which is also organized in a hierarchical fashion. The functional behavior found for the clustering coefficient and the ANND is a power law decay as a function of the degree with exponents ωint = 0.75 and νk = 0.5 [10], exponents that turn out to be very similar to the ones reported here for the WTW. In some sense, these results are not surprising since both are competitive systems evolving in a quasi free market and, in both cases, there exists, for instance, a geographic limitation that increases the connection costs and, thus, acts as a constraint in the optimization process of each vertex. Table I presents a summary of the main characteristics of the WTW, the Internet [10] and a random graph of the same size and average degree as the WTW. As a final remark, the average path length, defined as the average of the shortest distances between all the pairs of vertices, is hdi = 1.8, which, in this case, is very [1] P. Krugman, The Accidental Theorist, and other Dispatches from the Dismal Science, (W. W. Norton&Company, New York, 1998). [2] S. Amin, Capitalism in the Age of Globalization: The Management of Contemporary Society, (Zed Books, London, 1997). [3] R. Glick and A. Rose, Journal of International Money and Finance 18, 603-617 (1999). [4] M. Noland, L-G Liu, S. Robinson and Z. Wang, Global Economics Effects of the Asian Currency Devaluations, (International Institute for Economics, Washington D. C., 1998). [5] M. Goldstein, The Asian Financial Crisis: Causes, Cures and Systemic Implications, (International Institute similar to the corresponding random network of the same size and average degree. In conclusion, this first approach to the topology of the WTW points out some previously unnoticed features which are of primary importance in the understanding of the new international order. Our research suggests that the network’s evolution is guided by collective phenomena, and that self-organization plays a crucial role in structuring the WTW scale-free inhomogeneities and its hierarchical architecture. It remains an open question if these properties could also be made apparent at other different scales, for instance, in the trade relations between regions, cities or even individuals. At the country level, the activity is driven by competition. In the same way as it occurs with the Internet, optimization criteria are applied to local decisions made by the individual vertices. Resolutions are based on information that is biased toward the more visible vertices, and may be influenced by geographical or other convenience constraints. The findings in this paper may lead to consider that there exist underlying growing mechanisms common to all competitive systems, characterized by disassortative associations, and such mechanisms may differ from the evolutionary processes in social cooperative networks, characterized by assortative associations. Further modelling efforts must be done for acquiring a more realistic representation of the WTW, where inward flows differ from outward flows and where their weights depend on the exchanged quantities. It is also essential to do further research on the underlying formation mechanisms and on the dynamic processes running on top of the WTW, such as economic crises spreading. Acknowledgments Acknowledgements are due to Romualdo PastorSatorras for helpful discussion and advices. This work has been partially supported by the European commission FET Open project COSIN IST-2001-33555. for Economics, Washington D. C., 1998). [6] R. Albert and A.-L. Barabási, Rev. Mod. 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[19] Import/Export databases are publicly avalable at the address http://www.intracen.org/menus/countries.htm [20] Gross domestic product data are publicly avalable at the address http://www.tswoam.co.uk/world [21] E. Ravasz, and A.-L. Barabási, Hierarchical Organization in Complex Networks, e-print cond-mat/0206130.