The International Trade Network: weighted network analysis and modelling
Abstract
Tools of the theory of critical phenomena, namely the scaling analysis and universality, are argued to be applicable to large complex weblike network structures. Using a detailed analysis of the real data of the International Trade Network we argue that the scaled link weight distribution has an approximate lognormal distribution which remains robust over a period of 53 years. Another universal feature is observed in the powerlaw growth of the trade strength with gross domestic product, the exponent being similar for all countries. Using the ’richclub’ coefficient measure of the weighted networks it has been shown that the size of the richclub controlling half of the world’s trade is actually shrinking. While the gravity law is known to describe well the social interactions in the static networks of population migration, international trade, etc, here for the first time we studied a nonconservative dynamical model based on the gravity law which excellently reproduced many empirical features of the ITN.
pacs:
89.65.Gh, 89.75.Hc, 89.75.Fb,Are the large real or manmade networks with complicated and heterogeneous connections scale invariant and obey the well known scaling analysis and universality concepts of Statistical Physics? In the recent years, extensive research effort have been devoted to analyzing the structure, function and dynamics of complex networks relevant to multidisciplinary fields of science Barabasi (); Dorogov (); social (). Indeed the scalefree networks reflect such scale invariance in the linknode structures of the electronic communication networks like the Internet, worldwide web, protein interaction networks and even in research collaboration networks. More recently it has been observed that strengths of links of the networks, as called weights in graph theory, also have very interesing properties and can shed much light into the understanding of the details of the network BBVPNAS (); Barrat1 (); Manna (); WL (); Havlin (). Lately it has been proposed that the International Trade Network (ITN) i.e., the system of mutual trading between different countries in the world can also be viewed as an interesting example of realworld network Serrano (); Garlaschelli (); Garlaschelli1 (); Serrano1 (); Park (); JariClustering ().
In this paper we study the ITN as an excellent example of the weighted networks obeying the scale invariance and universality properties where the extent of trade between a pair of countries can be treated as the link weight Data (); Gleditsch (). It is observed that the suitably scaled link weight distribution over many years can be approximated well by a lognormal law. The nodal strength measuring the total volume of trade of a country is seen to depend nonlinearly on the country’s Gross Domestic Product with a robust exponent. Many features observed in the analysis can be explained with a simple dynamical model, which has the wellknown gravity model of international trade as its starting point Tinbergen ().
In the ITN, a node depicts a country and an undirected link exists between any pair of nodes if the trade volume between the corresponding countries is nonzero. Both the number of nodes and the number of links in the ITN’s show annual variation and grow almost systematically over 53 years, e.g. in 1948 and whereas in 2000 and . On the other hand the link density is observed to remain roughly constant with values around 0.52 for the same period. The annual trade data are expressed in millions of dollars () of imports and exports between countries and using four different quantities , , and Data (); Gleditsch (). Due to differences in the reporting procedures, there are usually small deviations between exports from to and imports to from . Therefore, we define the link weight , as a measure of the total trade volume between the two countries (in ), as follows
(1) 
This quantity tends to average out the aforementioned discrepancies. The distribution of weights is observed to be broad with the smallest nonzero trade volume being less than 1 and the largest of the order of . The number of links with very small weights is quite large, whereas only a few links with very large weights exist (Fig. 1). The tail of the distribution consists of links connecting very few highincome countries HDR (). The average weight per link is observed to grow during the investigated period, from 15.54 in 1948 to 308.8 in 2000.
The probability density distribution of the link weights is defined as the probability that a randomly selected link has the weight between and . The probability decays systematically with increasing weight and has a long tail with considerable fluctuation. We note that inferring the form of a broad probability distribution based on a relatively small number of data points has its difficulties; in the case of ITN, loglog plots of the distributions (not shown) display small intermediate linear regions which are (too) often interpreted as power laws. However, the tails of the distributions show clear curvature and much better fits to data are obtained by using a lognormal distribution of the form
(2) 
where the constants defined as and are observed to have different values for the weight distributions for different years. However, one can get a plot independent of and by drawing as a function of , which gives a simple parabola for all years (Note that implies ). In Fig. 2 we show such a plot, where the data has been aggregated for the fiveyear periods to reduce noise, i.e., 195155, 195660, …, 19962000, each represented with its own symbol. It is clearly seen that the data points for each period fall close to the parabola , displayed as the solid line. Therefore, we conclude that the annual weight distributions are reasonably well approximated by the lognormal distribution. This lognormal distribution of trade volumes has been discussed also in Fagiolo (). Note also that the trade imbalances have been reported to follow a lognormal distribution Serrano1 ().
For weighted complex networks, the strength of node is defined as BBVPNAS (), which in the case of ITN corresponds to the total volume of annual trade associated with a node. Intuitively, one can expect that in general the strengths of highGDP. countries are higher than those of the lowGDP countries. To see this in a quantitative fashion, we have utilized an elastic constant to measure how changes in strength respond to changes in GDP (we have used total Gross Domestic Product of the country and not the GDP per capita without taking into account of inflation). Hence, we define as: , which results in a powerlaw relationship . In Fig. 3 we plot the strengths vs. GDP for 22 different countries, representing a good mix of economic strengths, i.e., 10 high, 5 middle and 7 low income countries. It is observed that the strength of each country grows nonlinearly with its GDP with approximately the same slope. In the inset of this figure we show the probability distribution of values for 168 different countries. The distribution has a long tail and the values of 12 countries are found to be larger than 2. A detailed inspection reveals that majority of these 12 countries are those originated after Soviet Union, Yugoslavia and Czechoslovakia were fragmented. An overall average of the growth exponents has been estimated to be which comes down to 1.06 if these 12 countries are not considered for the averaging. The peak value of the distribution occurs very close to . Interestingly, Irwin Irwin () has observed earlier that the total world export volume varies as a function of the total world real GDP to the power of 1.16, along with other factors. In comparison, however, our observation reveals a more detailed picture, indicating that the total trade volumes of individual countries are also approximately power laws as function of their individual GDPs, with exponents close to this value.
Nontrivial correlations among the nodes of different realworld networks have been observed. For an unweighted network this means that large degree nodes are connected among themselves forming a club. More precisely such a club consists of a subset of nodes whose degree values are at least . A richclub coefficient (RCC) is measured as where is the number of links actually exists in the club and is the total number of node pairs in the club Colizza (). A high value of implies that members are indeed tightly connected. However it has been realized that only this definition is not enough, since with this measure even uncorrelated random graphs show some richclub effect as well Colizza (). It is suggested that one needs to define a ‘null model’ or the maximally random network (MRN) keeping the nodal degree values preserved, measure the corresponding RCC and observe the variation of ratio . We have executed the same analysis for the ITN of the year 2000 and generated the MRN using the pairwise linkend exchange method Colizza (). However it is observed that the variations of and with are nearly same and consequently is nearly equal to unity for the whole range of degree values.
Next we studied the richclub effect of the same ITN but now considering it as a weighted network. The richclub is now defined as the subset of nodes whose strengths are at least controlling a major share of the world’s trade dynamics. The RCC of the weighted network is defined as:
(3) 
The corresponding maximally random weighted network (MRWN) has been generated keeping both the nodal degrees as well as the nodal strength values preserved.
To generate the MRWN from the MRN of the ITN of the year 2000 described above a selfconsistent iteration procedure is used to obtain the link weight distribution consistent with the nodal strength list . We start assigning arbitrary random numbers as the weights to all links maintaining that the weight matrix is always symmetric i.e., . For an arbitrary node the difference is calculated. Weights of all links meeting at the node are then updated as
(4) 
to balance and . By repeated iterations the link weights quickly convergence and attain consistency with nodal strengths . It is checked that relation is well satisfied for this MRWN.
In Fig. 4(a) we plot both for the ITN and for the MRWN with the scaled nodal strength . The two measures are found to be nearly same, grow like for large values and their ratio is nearly equal to one except for a few values of near (Fig. 4(b)).
To explain this we observe that only 15 elements of the adjacency matrices of the ITN and the MRN are different. Therefore a typical node of ITN retains the links to most of its neighbors even after maximal randomization. This is because of the high value of the link density ( of the ITN for the year 2000. As a result as well as are nearly equal to unity. This implies that the pairwise link connections and the associated link weights of the original ITN are very close to those of the corresponding MRWN. In fact one can say that the original ITN is a typical member of the different random configurations of the MRWN when the as well as sets of the ITN are preserved. These results on the randomized unweighted ensemble confirms the results in Garlaschelli () when compared with randomized networks with fixed degree sequence as in Park ().
Zhou and Mondragón Zhou () observed a very similar behavior for the Internet statistics. Following them we conclude that rich nodes in the original ITN and in its corresponding MRN and MRWN are tightly connected and the similarity of the richclub connectivity in the ITN structure (with and without weights) does not imply that ITN lacks a richclub structure.
In fact the presence of the richclub effect is evident even if we simply analyze the variation of the fraction of the total volume of trade taking place among the members of the club. This is depicted in Fig. 4(c) which shows that remains close to unity until high values of , then decreases gradually to 1/2 at , and finally drops sharply. Very interestingly we also observe that the size of the rich club containing 50 of the total volume of annual trade shrinks almost systematically from 19 in 1948 to 8 in 2000.
The very heterogeneous distribution of trade volumes in the ITN is also reflected in the average pair correlation function of nodal strengths and in its power law dependence on the link weights . Links with high weights obviously must connect pairs of nodes of high strength, and for them . On the other hand for links of weights around unity, . Assuming that itself is of the order of , we find an upper bound for the exponent describing the variation of . Our analysis of the ITN yields, however, a somewhat smaller value of being between 0.65 and 0.90 for different financial years between 1948 and 2000. The dependence of the weight distribution on the underlying topological structure of the ITN is studied by measuring the average strength of a node as a function of its degree, which turn out to exhibit strong degree of nonlinearity: where varies between 3.4 and 3.7 for the same period.
We now develop a dynamical model based on the wellestablished gravity model Tinbergen () used in social and economic sciences to describe the flow of social interaction between two economic centers and as a function of their economic sizes and distance : . This equation has been generalized to the parametric form Head ()
(5) 
where the exponents and usually range between 0.7 and 1.1 where as is observed to be around 0.6 Head ().
In our model, we assume a unit square to represent the world and N points distributed at random positions representing the capital cities of different countries. Initially the GDP values are randomly assigned with uniform probability such that the total GDP is unity: . Then we let the dynamics start, which is essentially a series of pairwise interactions. There are different pairwise wealth exchange models studied in the literature which calculate the distribution of wealths in a society Wealth (). A pair of countries is randomly selected for a transaction and time is measured as the number of such transactions. In a transaction the selected countries invest the amounts and calculated using Eq. (5). Then the total amount of investment is randomly shared between the two countries as a result of this transaction, as follows:
(6)  
(7) 
Here is a random fraction freshly drawn for every transaction. The random sharing of is justified by the fact that while the Gravity law describes the average interaction in terms of the strengths and distances of separation, the actual amount of trade depends on other factors, many of them are political. In this idealized model countries are not allowed to make debt, which in turn makes the dynamics nonconservative through the parameters and . It holds for these parameters that if and if . However, if for some transaction or then we add or such that after the transaction, GDP balance does not become negative. Also after the transaction the individual GDP’s are rescaled for the total GDP to remain unity. It is observed that a large number of pairwise transactions leads to a stationary state where fluctuates with time around a steady mean value. Any time after reaching the stationary state, the dynamics is used to construct a model ITN such that links are established between countries and whenever there is a transaction between them. We let the dynamics run until a preassigned link density (typically 0.30.5) has been reached. For example to generate a network corresponding to the ITN of the year 2000, we take = 187 and continue the exchange dynamics till = 10252 distinct links are dropped corresponding to the link density . The weight of a link is then defined as the total amount of investment between pairs of countries in all transactions.
For comparison the weight distribution of our model networks is analyzed in the same way as the real ITN data, and it turns out that an excellent consistency with the simple parabola is observed for parameter values , within a tolerance of 0.2 for all exponents, see Fig. 5. We also find that the GDP distribution is broad showing a power law like behavior for a short interval of (with exponent 1.9) before finite size effects set in. This is to be compared with the realworld data where the GDP distribution of different countries has been argued to be consistent with the Pareto law Pareto (). Finally, as in the real ITN, the twopoint strength correlation is seen to grow like with in the large weight limit, compared to the range of values 0.65  0.90 in the real ITN. In addition we find that the cumulative degree distribution is consistent with the results on the real ITN.
To summarize, we have shown that the weighted International Trade Network can be looked upon as an excellent example of a complex network obeying scaleinvariance and universality features. The scaled distributions of annual world trade volumes between countries collapse well to a lognormal distribution and it remains unchanged over a span of 53 years implying robustness or universality. Secondly, the nodal strength measuring the total trade volume associated with a country grows nonlinearly with its GDP with a robust exponent. Also a large fraction of the global trade is controlled by a club of few rich countries which shrinks its size as time goes on. Finally, the main features of the realworld ITN have been reproduced by using a simple nonconservative dynamical model starting from the wellknown gravity model of social and economic sciences.
We thank K. S. Gleditsch, A. Chakrabarti for useful communications. SSM thanks LCE, HUT for a visit and hospitality. JS and KK acknowledge the support from the Academy of Finland’s Centers of Excellence programs for 20002005 and 20062011.
Electronic Address: manna@boson.bose.res.in
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