COMPLEX NETWORKS AND MINIMAL SPANNING TREES IN INTERNATIONAL TRADE NETWORK

Workshop on Econophysics of the International Conference on Statistical Physics (SigmaPhi2011) International Journal of Modern Physics: Conference Series Vol. 16 (2012) 51 60 c World Scientific Publishing Company DOI: 10.1142/S2010194512007775 COMPLEX NETWORKS AND MINIMAL SPANNING TREES IN INTERNATIONAL TRADE NETWORK SEONG EUN MAENG, HYUNG WOOC CHOI and JAE WOO LEE Department of Physics, Inha University, 100 Inharo Incheon, 402-751, Korea jaewlee@inha.ac.kr The wealth of a nation is changed by the internal economic growth of a nation and by the international trade among countries. Trade between countries are one of their most important interactions and thus expects to affect crucially the wealth distribution over countries. We reviewed the network properties of the international trade networks (ITN). We analyzed data sets of world trade. The data set include a total number of 190 countries from 1950 to 2000. We observed that the world trade network showed the uneven trading relationships which are measured by the disparity. The effective disparity followed a power law, <D(k) > t δ, for the import and export network. We also construct the minimal spanning tree(mst) of international trade network, where each node is a country and directed links connecting them represent money flow from a source node to a target one. The topology of the MST shows the flow patterns of the international trades. From the MST we can identify the sub-economic zone if we delete the hub node. We observed that the cumulative degree distribution functions follow the power law, P >(k) k α, with the average exponent α =1.1(1)). We also calculated the betweenness centrality(bc) of the MST. The cumulative probability distribution of the betweenness centrality follows the power law, P >(BC) BC β, with the average exponent β = 1.09(7). Keywords: Complex network; minimal spanning tree; international trade network; power law. 1. Introduction In econophysics the complex networks were extracted from the complicated interactions among the economic agents such as the countries, stocks, firms, goods, etc. 1,2,3,4,5,6,7. For example, the inter-correlation network was generated by using the cross correlation between the time series of the stock prices 8,9,10,11,12,13,14,15,16,17. The international trade network (ITN) is a well known networks in economics. The international trade network is also called as the world trade web (WTW). In the ITN the nodes are countries that trade goods and services each other. The international trade network is defined by the trading relationship among countries such as import-export relationship. Each country exports or imports from other countries in a specified period. The links in the ITN correspond to the existence of the trade 51

52 S. E. Maeng, H. W. Choi & J. W. Lee between the pairs of the countries. The ITN is close to the fully connected network because a country trades to almost all countries. The analysis of the ITN was based on these densely connected networks 18,19,20,21,22,23,24,25,26,27,28,29. The wealth of a country represented by the gross domestic product (GDP). In the highly globalized economics, the countries interact heavily each other and their wealths depend on the trading volume among the countries. When there are some perturbation in economics, for example, some financial crisis, the impacts of the purturbation spread to all interacting countries. Understanding the network structures in economics can help overcoming the economic crisis. Many topological properties have been studied for the highly connected ITN 20,21,22,23,24,25. The ITN showed strongly dissortativity for the binary ITN and weakly disassortativity for the weighted ITN 20,21,22,23,24,25. To understand the patterns of the dominant money flows in the ITN, we have to apply some criteria to extract the sparsely connected networks from the ITN. Many authors introduced the extracting methods to get the skeleton (backbone) network from the densely connected ITNs 18,19,30,31,32,33. In this work we applied the minimal spanning tree (MST) extracted from the highly connected ITNs. We generated the international trade network each year from 1950 to 2000. If we consider the highest link strength, this minimal spanning tree is equivalent to the maximal spanning tree. We applied Kruskals algorithm to extract the minimal spanning tree from densely connected ITN 34. In section 2 we reviewed the network properties of the international trade network. We considered the binary network and weighted network for the ITN. In section 3 we considered the minimal spanning tree (MST) in the ITN. We observed the backbone network in the ITN. In section 4 we remarked the conclusions. 2. International Trade Network We consider data-sets of the international trades and the global domestic products (GDP) among countries from year 1948 to year 2000 35. The wealth of a country, i, is represented by its GDP, G i (t), where t is a year. The money flows from country j to country i by the export of the country i. The amount of money, f ij (t), denotes the money flowing into the country i from country j by the export of the country i to the country j at the year t. Similarly, f ji (t) corresponds to the money flowing out from country i to country j. We defined the link strength, w ij (t) between the country i and the country j as w ij (t) =(f ij (t) +f ji (t))/2. The large w ij (t) meansthat two countries trade heavily and they interact strongly by the trading relationship. When two countries trade each other, i.e. w ij (t) 0, we set the adjacency matrix a ij (t) =1atayeart. Let s consider the topological properties of the ITN. In the densely connected ITN, the node degree is defined as k i = j a ij, (1)

Complex Networks and Minimal Spanning Trees in International Trade Network 53 where j sums over the nearest neighbors of the node i. The node degree is equal to the number of the trading partners at the year. The node strength is defined as s i = j w ij, (2) where j also sums over the nearest neighbors of the node i. The node strength is a measure of the weighted connectivity. The probability distribution functions of the node degree are never the normal or log-normal distribution in the ITN 24.The average of the node degree in the ITN is relatively high. On average each country has about 90 trading partners. The probability distribution of the node degree exhibits some bimodality with majority of countries trading low degree countries and a bunch of countries trading with almost everyone else 24,25. The probability distribution function of the node degree is relatively left-skewed. The binary ITN is characterized by an extremely high density. The density of the network is defined by ρ(t) = 1 N(N 1) a ij (t), (3) where N is the total number of the trading country at the year t. Theobserved network density are ranging from 0.5385 to 0.6441 form year 1958 to year 2000 24. The probability distribution of the node strength is also left-skewed. Many weak trading relations coexist with a few strong trading relations. The trading relationships between the countries is not even. Some countries are important trading partners, but others contribute to small parts of the trading value. We measure the uneven distributions of the trade value by the trading activities of the countries. We defined the disparity of the trade for each country i as 33 and DS ex i DS im i = j i j ( ) 2 fij k f, (4) ik ( ) 2 fji k f. (5) ki = j The disparity of a countries approaches unity if f ij of a specific country is very large compared to those of the other countries. If all the f ij of the countries are comparable, then the disparity DS i is proportional to 1/k i. Let s define the effective disparity as D i = k i DS i. Then the average effective disparity is given as <D> k for the uneven trading relationship, and <D> const. for the pair trading relationships. In Fig. 1 we plot the average effective disparity <D>versus the node degree k for the export and import of the international trade 32.Weobserved that the effective disparity followed a power law as <D> k δ, (6) where the exponent δ characterize the uneven trading relationships. We obtained the exponents δ =0.49(2) for the export and δ =0.51(1) for the import. Therefore,

54 S. E. Maeng, H. W. Choi & J. W. Lee 40 export import 30 disparity 20 10 0 0 50 100 150 200 degree Fig. 1. The average effective disparity as a function of the node degree. The effective disparity follows the power law, <D> k δ with δ =0.49(2) for the export and δ =0.50(1) for the import. we concluded that the international trades are uneven because the effective disparity is not constant. The power law of the effective disparity indicates that the trading relations evolve by some mechanisms of the self-organized criticality. The exponents of the effective disparity for the import and the export are very close because the heavy importing country of a country is the heavy exporting country of the trading partners. The hierachical structures were characterized by the clustering coefficient C(k) and the average nearest neighbors degree <k nn (k) >. The clustering coefficient of avertex,i, with the degree k i is defined by 2n i C i = k i (k i 1), (7) where n i is the number of connecting links among the neighbors of a node i. The degree-degree correlation is characterized by the average nearest neighbor degree, k nn (k), which is defined as <k nn (k) >= k k P (k k), (8) where P (k k) is the conditional probability which measures the probability of a vertex of degree k to be linked to a vertex of degree k.

Complex Networks and Minimal Spanning Trees in International Trade Network 55 The clustering coefficient of the densely connected ITN showed a power law, C(k) k ω,withω =0.7 18,19. In the densely connected ITN, the <k nn (k) > followed a power law, <k nn (k) > k ν k, with the average exponent ν k =0.5 18,19. This result means that the ITN is a disassortative network where the high degree nodes tend to connect to the low degree nodes. The network properties for the binary and weighted structures were also reported for the densely connected network of the ITN 25. Fagiolo et al. showed that the binary network structure of the ITN followed the strong assortativity and high clustering coefficient 25. The weighted network structure showed the weak disassortativity and weak clustering coefficient 25.The degree and strength distribution function of the binary and weighted ITN do not followed the power law. The degree distribution of the binary ITN followed the binomial function 24,30. The left peak is greater than the right peak in the binomial distribution. However, the strength distribution of the weighted ITN showed skewed distribution. The money flows by the international trade depend strongly on the wealths (GDP) of the countries. The money flows show positive correlations with the source and the target countries wealths. Cha et al. showed that the international trade volumes, f ij, depend on the GDP of the importing country, G i,andthegdp of the exporting country, G j 31,32. The trading volume followed a power law, f ij G α(target) i,withα(target) = 0.67 and f ij G α(source) j, withα(source) = 0.61, for the ITN 32. The exponent α(target) is greater than α(source) by 10%, and the difference reveals a stronger dependence of the trading flow on the GDP of the target country that on the GDP of the source country. The original international trading network is a densely connected network. Therefore, it is very difficult to identify the important trading partner of a country in the densely connected ITN. There are a few studies for the sparsely connected ITN. We need to some criteria to extract the sparse subnetworks from the densely connected ITN. Bhattacharya et al. extract some sparse subnetwork by considering only links with the highest weights above 4% in the ITN 26. Tzekina et al. considered the evolution of the community structure in the world trade web. They extracted the sparsely connected scale-free network keeping the significant trading links greater than 5% of the weights 36. They identified the economic zones isolated by this constructing rule. Serrano and Boguna reported the undirected network constructed by the import and export relationship among the countries by selecting the highest 4% of weights. The cumulative in-degree and out-degree distribution, P > (k in )and P > (k out ), followed the power law, P > k 1 γ with γ =2.6 18. In the sparse network the weights of the link correspond to the average of the importing value and the exporting value in a country. They observed that the weight distribution followed a log-normal distribution, P (w) = ( ) 1 1 2πσ 2 w exp ln2 (w/w o ) 2σ 2, (9)

56 S. E. Maeng, H. W. Choi & J. W. Lee where ln w o =< ln w>is the average of the logarithmic weight and σ 2 =< (ln w) 2 > < ln w> 2 is the standard deviation of the logarithmic weight. We extracted the skeleton network in the international trade network. We just kept the most important importing link of a country. Then we obtain the skeleton network with links of the heaviest money flowing out from a country 32.This network was a directed weighted network. From this skeleton network we observed the dynamic behaviors of the wealth flows. For the skeleton trading network the probability distribution of the node and the strength showed the power law 32. 3. Minimal spanning tree We generated the minimal spanning tree from the densely connected ITN. We set the strength of the link as g ij (t) =1/w ij (t). Then we extract the minimal spanning tree by using Kruskal s algorithm 34. When we obtained the minimal spanning tree, we select a link with the least strength. We repeat the similar procedure for the remaining links. When we arrive to the N 1 nodes, we stop the selecting process. Therefore, in the MST there are N nodes and N 1 links. The obtained MST becomes loopless network. Fig. 2 represents the MST of the ITN at year 2000. In the figure the size of the node is proportional to the relative value of the GDP among the countries. The thickness of the link corresponds to the relative trading strength of each country. From MST we identify the dominant interacting trading partner among the countries. We also observed the economic zones in the ITN. In year 2000, the United State of America (USA) is a dominant hub node in the ITN as we expect. If we cut the connections between USA and its connecting countries, we observe the economic blocks such as Japanese economic zone (JPN), Chinese economic zone (CHN) German economic zone (JFR), etc. The hub (USA) and secondary hub (JPN, JFR, CHN, etc.) are well developed countries except China, India and Brazil and their GDPs are larger than their terminal countries. In the secondary hub, the terminal countries connected stronger to the secondary hub than to USA. In European economic zone we observed the ternary economic hub such as Italy(ITN), France(FRN), Russia(RUS), and Spain(SPN). The European developed countries strongly connected each other and build their own economic sub-networks. The topology of the MST changed year to year. In Fig. 3 we presented the log-log plot of the cumulative probability distribution function of the degree distribution for the MST at year 2000. We observed the obvious power law, P > (k) k α. The exponent α lies in the range 0.99 α(year) 1.28. We obtained the average of the exponent as α =1.1(1). Inset in Fig. 2 we plot the exponent α as a function of the year. The exponent α fluctuates year to year. These exponents characterized the network structure of the MST. The large exponent indicates the appearance of the hub and secondary hubs with high degree. In the network the importance of a node is measured by the betweenness centrality (BC) which indicates the importance of a node with respect to the connectivity

Complex Networks and Minimal Spanning Trees in International Trade Network 57 Fig. 2. Minimal spanning tree of the international trade network at year 2000. In this network the size of the node represent the relative value of the GDP among the countries. The thickness of the link corresponds to the relative trading strength of the countries. The symbols inside nodes indicate the name of the countries: USA(the United State of America), JPN(Japan), GFR(Germany), UKG(the United Kingdom), RUS(Russia), IND(India), ITA(Italy), FRN(France), SPN(Spain), BRA(Brazil), etc. between other nodes of the network. The betweenness centrality is defined as BC(i) = j k i σ jk(i)/σ jk,whereσ jk is the number of shortest paths between nodes j and k, andσ jk (i) is the number of shortest paths passing the node i. Ifa node is an important node in the information flow, then the BC of that node is large value. If a node is located on the boundary of the network, then its BC is small value. In Fig. 4. we presented the cumulative probability distribution function of the betweenness centrality for the MST. The pdf of the BC showed the power law, P > (BC) BC β, with the average exponent β =1.09(7). In inset figure of Fig. 4 we presented the exponent β as a function of the time. The exponents β fluctuate heavily. The trend of the exponent β is similar to that of the exponents α. InMST the hub or secondary hubs is important role to flows of the international trade.

58 S. E. Maeng, H. W. Choi & J. W. Lee 10 0 10-1 Exponent -1.9-2 -2.1-2.2 P > (k) 10-2 1950 1960 1970 1980 1990 2000 slope=-1.1 1950 1960 1970 1980 1990 2000 Year 10-3 1 10 100 k Fig. 3. Cumulative probability distribution function of the node degree for the MST of the ITN. The cumulative pdf follows the power law, P >(k) k α. Inset represents the degree exponents of the cumulative pdf. The degree exponents are not constant and the average degree exponent is about α =1.1(1). P > (BC) 10 0 10-1 Exponent -0.8-1 -1.2 1950 1960 1970 1980 1990 2000 slope=-1.09 10-2 195019601970198019902000 Year 10 100 1000 10000 BC Fig. 4. Cumulative probability distribution function of the betweenness centrality(bc) for the MST of the ITN. The cumulative pdf shows the power law, P >(BC) BC β, with the average exponent β = 1.09(7). Inset represents exponents of the cumulative pdf of betweenness centrality for the MST.

Complex Networks and Minimal Spanning Trees in International Trade Network 59 4. Conclusions We reviewed the complex network properties of the international trade network. For the original densely connected ITN, the average node degree and the average strength were very big because a country trades to many countries. We observed that the effective disparity followed the power law to the node degree. The power law of the disparity indicate that the trading relations are uneven. We identify the backbone network of the ITN by keeping the highest trading link of the import. From the sparsely connected backbone network we can observe the main stream of the wealth flow and the economic zone if we cut the connection with the main hub. We consider the minimal spanning tree of the international trade networks. In the MST the US is a dominant hub node and we can identify the secondary economic zone by deleting the hub node. The cumulative distribution function of the degree showed the power law. The exponents of the power law fluctuate year to year. We also measured the betweenness centrality of the MST. The betweenness centrality followed the power law. Acknowledgments This work has been supported by the Research Fund of Inha University. References 1. R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, New York, 2000), p. 20. 2. J. Voit, The Statistical Mechanics of Financial Markets (Springer, Berlin, 2001), p. 59. 3. S. Sinha, A. Chatterjee, A. Chakraborti, and B. K. Chakrabarti, Econophysics (Wiley- VCH, Morlenbach, 2011). 4. S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079 (2002). 5. R. Albert and A-L Barabasi, Rev. Mod. Phys. 74, 47 (2002). 6. M. E. J. Newman, SIAM Rev. 45, 167 (2003). 7. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Phys. Rep. 424, 175 (2006). 8. R. N. Mantegna and H. E. Stanley, Nature 376, 46 (1995). 9. X. Gabaix, P. Gospikrishnan, V. Plerou, and H. E. Stanley, Nature 423, 26 (2003). 10. R. L. Axtell, Science 293, 1818 (2001). 11. R. Cont, Quant. Fin. 1, 223 (2001). 12. Z. Eisler and J. Kertesz, Physica A 343, 603 (2004). 13. V. Plerou and H. E. Stanley, Phys. Rev. E 76, 046109 (2007). 14. B. H. Hong, K. E. Lee, and J. W. Lee, Phys. Lett. A 361, 6 (2007). 15. J. W. Lee, K. E. Lee, and P. A. Rikvold, Physica A 364, 355 (2006). 16. B. H. Hong, K. E. Lee, J. K. Hwang, and J. W. Lee, Physica A 388, 863 (2009). 17. K. E. Lee and J. W. Lee, J. Korean Phys. Soc. 44, 668 (2004). 18. M. A. Serrano and B. Boguna, Phys. Rev. E 68, 015101 (2003). 19. X. Li, Y. Y. Jin, and G. Chen, Physica A 328, 287 (2003). 20. D. Garlaschelli, T. Di Matteo, T. Aste, G. Caldarelli, and M. I. Loffredo, Eur. Phys. J. B 57, 159 (2007).

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