Google matrix of the world trade network
Abstract
Using the United Nations Commodity Trade Statistics Database (1) we construct the Google matrix of the world trade network and analyze its properties for various trade commodities for all countries and all available years from 1962 to 2009. The trade flows on this network are classified with the help of PageRank and CheiRank algorithms developed for the World Wide Web and other large scale directed networks. For the world trade this ranking treats all countries on equal democratic grounds independent of country richness. Still this method puts at the top a group of industrially developed countries for trade in all commodities. Our study establishes the existence of two solid state like domains of rich and poor countries which remain stable in time, while the majority of countries are shown to be in a gas like phase with strong rank fluctuations. A simple random matrix model provides a good description of statistical distribution of countries in twodimensional rank plane. The comparison with usual ranking by export and import highlights new features and possibilities of our approach.
1 Introduction
The analysis and understanding of world trade is of primary importance for modern international economics (2). Usually the world trade ranking of countries is done according their export and/or import counted in USD (3). In such an approach the rich countries naturally go at the top of the listing simply due the fact that they are rich and not necessary due to the fact that their trade network is efficient, broad and competitive. In fact the trade between countries represents a directed network and hence it is natural to apply modern methods of directed networks to analyze the properties of this network. Indeed, on a scale of last decade the modern society developed enormously large directed networks which started to play a very important role. Among them we can list the World Wide Web (WWW), Facebook, Wikipedia and many others. The information retrieval and ranking of such large networks became a formidable challenge of modern society.
An efficient approach to solution of this problem was proposed in (4) on the basis of construction of the Google matrix of the network and ranking all its nodes with the help of the PageRank algorithm (see detailed description in (5)). The elements of the Google matrix of a network with nodes are defined as
(1) 
where the matrix is obtained by normalizing to unity all columns of the adjacency matrix , and replacing columns with only zero elements by . Usually for the WWW an element of the adjacency matrix is equal to unity if a node points to node and zero otherwise. The damping parameter in the WWW context describes the probability to jump to any node for a random surfer. The value gives a good classification for WWW (5). By construction the Google matrix belongs to the class of PerronFrobenius operators and Markov chains (5), its largest eigenvalue is and other eigenvalues have . According to the PerronFrobenius theorem the right eigenvector, called the PageRank vector, has maximal and nonnegative elements that have a meaning of probability attributed to node . Thus all nodes can be ordered in a decreasing order of probability with the corresponding increasing PageRank index . The presence of gap between and ensures a convergence of a random initial vector to the PageRank after about 50 multiplications by matrix . Such a ranking based on the PageRank algorithm forms the basis of the Google search engine (5). It is established that a dependence of PageRank probability on rank is well described by a power law with . This is consistent with the relation corresponding to the average proportionality of PageRank probability to its indegree distribution where is a number of ingoing links for a node (6); (5). For the WWW it is found that for the ingoing links (with ) while for outdegree distribution of outgoing links a power law has the exponent (7); (8). We note that PageRank is used for ranking in various directed networks including citation network of Physical Review (9); (10) and for rating of the total importance of scientific journals (11).
The PageRank performs ranking determined by ingoing links putting at the top most known and popular nodes. However, in certain networks outgoing links also play an important role. Recently, on an example of procedure call network of Linux Kernel software, it was shown (12) that a relevant additional ranking is obtained by taking the network with inverse link directions in the adjacency matrix corresponding to and constructing from it an additional Google matrix according to relation (1) at the same . The examples of matrices and for the world trade network are shown in Fig. 1. The eigenvector of with eigenvalue gives then a new inverse PageRank with ranking index . This ranking was named CheiRank (13) to mark that it allows to chercher l’information in a new way. While the PageRank rates the network nodes in average proportionally to a number of ingoing links, the CheiRank rates nodes in average proportionally to a number of outgoing links. The results obtained in (12); (13) confirm this proportionality with the exponent .
Since each node belongs both to CheiRank and PageRank vectors the ranking of information flow on a directed network becomes twodimensional. While PageRank highlights how popular and known is a given node, CheiRank highlights its communication and connectivity abilities. The examples of Linux and Wikipedia networks show that the rating of nodes based on PageRank and CheiRank allows to perform information retrieval and to characterize network properties in a qualitatively new way (12); (13).
In this work we apply CheiRank and PageRank approach to the World Trade Network (WTN) using the enormous and detailed United Nations Commodity Trade Statistics Database (UN COMTRADE) (1). Using these data we analyze the world trade flows both in import and export for all commodities for all years 1962  2009 available there at SITC1 and HS96 databases. We also performed analysis for specific commodities taken from SITC Rev. 1 database, mainly for year 2008: crude petroleum (S133101, ”Crude petroleum”), natural gas (S13411, ”Gas, natural”), barley (S10430, ”Barley, unmilled”), cars (S17321, ”Passenger motor cars, other than buses”), food (S10, ”Food and live animals”), cereals (S104, ”Cereals and cereal preparations”). Their codes and official UN names are given in brackets. In few cases, when certain countries were nonreporting their export, we complemented the WTN data from the import database.
For a given year we extract from the UN COMTRADE money transfer (in USD) from country to country that gives us money matrix elements (for all types of commodities noted above). These elements can be viewed as a money mass transfer from to . In contrast to the adjacency matrix of WWW, where all elements are only or , here we have the case of weighted elements. This corresponds to a case when there are in principle multiple number of links from to and this number is proportional to USD amount transfer. Such a situation appears for rating of scientific journals (11), Linux PCN (12) and for Wikipedia English articles hyperlink network (13), where generally there are few citations (links) from a given article to another one. In this case still the Google matrix is constructed according to the usual rules and relation (1) with and , if for a given all elements . Here is the total export mass for country . The matrix is constructed from transposed money matrix with . In this way we obtain the Google matrices and of WTN which allow to treat all countries on equal grounds independently of the fact if a given country is rich or poor. A similar choice was used in rating of scientific journals (11), PCN Linux (12) and Wikipedia network (13). The main difference appearing for WTN is a very large variation of mass matrix elements related to the fact that there is very strong variation of richness of various countries. Due to these reason we think that it is important to use the ranking based on the Google matrix which treats in a democratic way all world countries that corresponds to the democratic standards of the UN. For the WTN CheiRank and PageRank are naturally linked to export and import flows for a given country and hence it is very natural to use these ranks for characterization of country trade abilities. The Google matrix can be constructed in the same way not only for all commodities but also for a given specific commodity.
We note that recently the interest to the analysis of the world trade as a network becomes more and more pronounced with a few publications in this area (14); (15); (16); (17). Thus, the global network characteristics were considered in (14); (15), degree centrality measures were analyzed in (16) and time evolution of network global characteristics was studied in (17). Topological and clustering properties of multinetwork of various commodities were discussed in (18). Here we present a systematic study of directed WTN on the basis of new combination of PageRank and CheiRank methods using the Google matrix constructed for the enormous UN COMTRADE database.
The paper is composed as follows: in Section 2 we describe the global properties of the Google matrix of WTN, in Section 3 we analyze distribution of countries in PageRankCheiRank plane for all time period 1962  2009 and propose a random matrix model of WTN (RMWTN) which describes the statistical properties of this distribution in the case of all commodities; comparison with ranking based on import and export for various commodities is presented in Section 4; discussion of the results is given in Section 5. More detailed information and data are given in Appendix and at the website (19).
2 Properties of Google matrix of WTN
An example of the Google matrix of WTN in 2008 is shown in Fig. 1 for all commodities and crude petroleum. The matrices and are shown in the bases where all countries are ordered by the PageRank index of matrix constructed for corresponding commodity (left and right columns). The matrix elements of are distributed over all values being roughly homogeneously in , even if the left top corner at small , values is filled in a more dense way. In contrast the density drops at large values of . Such a structure is visible both for all commodities and crude petroleum but clearly the global density is smaller in the later case since there are less number of links there (see data in next Section). The structure of is approximately the same (we will see in next Section that rich countries are located at low , values). In contrast to and the structure of money matrix is rather different. For all commodities matrix elements drop very rapidly at large values of and that corresponds to the fact that the main amount of world money circulates only between rich countries with top ranks . In contrast to that for crude petroleum the matrix elements of are located at intermediate values. Indeed, in this case PageRank index orders countries by their crude petroleum trade where richest countries are not necessarily at the top ranks.
From the Google matrices and we find the probability distributions PageRank and CheiRank which are shown in Fig. 2 for the same commodities as Fig. 1. One of the main features of these distributions is that both and depend on their indexes in a rather similar way from, that is in contrast to the results found for the WWW (7); (8), PCN Linux (12) and Wikipedia network (13), where these distributions are different having different exponents in the power law decay. Here, up to fluctuations, we have . The size of WTN is rather small compared to usual sizes of WWW, Linux or Wikipedia networks. However, still we find that the power law gives a quite good fit of our data. The fit gives at and at (for all commodities) and and respectively (for crude petroleum) for all 227 countries in Fig. 2. For the fit of top 100 countries we have respectively () and () for all commodities and () and () for crude petroleum. There is a certain change of the exponent with a decrease of the fit interval which, however, is not very large. We attribute this to visible deviations at the tail of , with small countries (see discussion in next Section). In average the exponent value is not very far from the value corresponding to the Zipf law (20).
It is useful to compare the behavior of probabilities and with respective ranking related to import and export. To do that we rank the countries by probability import defined as a ratio of import in USD for a given country to the total world import in USD for a given year with ordering of all countries in decreasing probability order index of ImportRank . By construction we have and , where is the import mass of a given country and is the total world money mass for a given year. In the same way we construct export probability with the ExportRank . The dependence of these probabilities on their indexes is shown in Fig. 2. In the range of it can be well described by a power law with for all commodities corresponding to the Zips law (for crude petroleum we obtain for this range ). At larger values of order index we find a sharp drop with an exponential type decay on the tail. For crude petroleum this exponential decay starts at smaller values of due to a significantly smaller total number of links that gives an increase of for the range of 50 countries. The exponential decay at large results from a strong variation of richness of countries which changes more than by four orders of magnitude. From the comparison of ranks shown in Fig. 2 it is clear that PageRank and CheiRank give more equilibrated and democratic description of trade flows.
We should note that due to a small size of the WTN the fluctuations are stronger compared to large size networks like the WWW. It is especially visible for specific commodities where the total number of links is by factor 30 smaller then for all commodities (see next Section). These fluctuations are smaller for the damping factor value in agreement with the results presented in (21); (22). In fact this value was also used in (9) for PhysRev citation network. Due to that reasons in next Sections we show data for ranking at .
Finally let us discuss the spectrum of the Google matrix which follows from the equation for right eigenvectors :
(2) 
It is known that the dependence on is rather simple: all eigenvalues, except one with , are multiplied by (5). Due to that we show the spectrum of at in Fig. 3. Compared to the spectrum studied for other networks (see examples in (12); (22); (23); (24); (25); (26)) we find that the WTN spectrum is very close to real line especially for three top commodities in Fig. 3. We explain this by the fact that here an average number of links per country is very large for these commodities and that the matrix elements are not very far from the symmetric relation at which the spectrum is real. We only note that for barley the spectrum has quasidegeneracy at that signifies the existence of slow relaxation modes. We attribute this to the fact that there are certain countries which practically do not use barley that leads to appearing of isolated subspaces with corresponding quasidegenerate modes. We will return to the discussion of spectrum properties of in next Section.
3 CheiRank versus PageRank for WTN
We start from examples of distributions of countries in the PageRankCheiRank plane shown in Fig. 4 for 5 different commodities in 2008. The first case of all commodities corresponds to trade flows between countries integrated over all type of products. Even if the Google matrix approach is based on a democratic ranking of international trade, being independent of total amount of exportimport for a given country, we still find at the top ranks and the group of industrially developed countries (see more details in Table 1 in Appendix). This means that these countries have efficient trade networks with optimally distributed trade flows. Another pronounced feature of global distribution is that it is concentrated along the main diagonal . This feature is not present in other networks studied before (e.g. PCN Linux (12) and Wikipedia (13)). The origin of this density concentration is based on simple economy reason: for each country the total import is approximately equal to export since each country should keep in average an economic balance. Thus for a given country its trade is doing well if its so that the country exports more than it imports. The opposite relation corresponds to a bad trade situation. We also can say that local minima in the curve of correspond to a successful trade while maxima mark bad traders. In 2008 most successful were China, Rep. of Korea, Russia, Singapore, Brasil, South Africa, Venezuela (in order of for ) while among bad traders we note UK, Spain, Nigeria, Poland, Czech Rep., Greece, Sudan with especially strong export drop for two last cases. The comparison of our ranking with the importexport ranking will be analyzed in next Section.
Even if there is a concentration of density along the main diagonal (Fig. 4a) we still have a significant broadening of distribution especially at middle values of . This means that the gravity model of trade, often used in economy (see e.g. (2); (16)), has only approximate validity. Indeed, in this model the mass matrix is symmetric that would placed all countries on diagonal that is definitely not the case.
If we now turn to the distribution of countries for a trade in a specific commodity than it becomes absolutely clear that the symmetry approximately visible for all commodities is absolutely absent: the points are scattered practically over the whole square . The reason of such a strong scattering is clear: e.g. for crude petroleum some countries export this product while other countries import it. Even if there is some flow from exporters to exporters it remains relatively low (see more discussion in next Section). This makes the Google matrix to be very asymmetric. Indeed, the asymmetry of trade flow is well visible in Fig. 4h where arrows show the trade directions between countries within top ranks for barley.
It is also useful to use 2DRank discussed in (13), which orders all nodes according to the order of their appearance inside squares of size going from to for each of four specific commodities shown in Fig. 4. In a certain sense top countries in 2DRank are those which are active traders even not being among large exporters or importers of this product (all ranks for commodities of Fig. 4 are given in Tables 15 in Appendix). As an example, we note Singapore which is at the third position in (Table 2): it is a small country which cannot export or import a large amount of the commodity, but its trade network is very active redistributing flows between various countries that places it at a high rank.
The images of Fig. 4 allow to understand qualitatively the reasons of density concentration around diagonal for the case of all commodities: this trade is composed from hundreds of specific commodities which behave randomly and the averaging over them gives effective coarsegraining and produces a certain symmetry for matrix elements due to the central limit theorem for a sum of many positive contributions. The fact that the increase of coarsegraining cell gives more and more symmetry is well seen in Fig. 3 where the spectrum becomes more and more close to a real one, and hence there is more and more symmetry in elements , when we go from barley to cereals, food and all commodities.
We will return to the analysis of specific country ranking in the next Section while now we turn to analysis of time evolution of WTN.
The variation of global parameters of WTN during the database period 1962  2009 is shown in Fig. 5. The number of countries is increased by 38%, while the number of links per country for all commodities is increased in total by 140% with a significant increase from 50% to 140% during the period 1993  2009 corresponding to economy globalization. At the same time for a specific commodity the average number of links per country remains on a level of 35 links being by a factor 30 smaller compared to all commodities trade. During the whole period the total amount of trade in USD shows an average exponential growth by 2 orders of magnitude.
To understand the physical properties of the WTN we consider the distribution of money mass transfer matrix elements shown versus their transposed values in Fig. 6. This distribution is symmetric by the construction. In the case of symmetric matrix , corresponding to the gravity model of trade or undirected network, all elements should be located on one diagonal line that is definitely not the case. For crude petroleum the distribution is even more broad showing definite absence of symmetry of . In fact for all commodities the distribution forms a rather broad cone which form remains stable in time according to the comparison of data in 1962 and 2008 years (the density is higher in the later case since there are more countries). Keeping in mind that according to data of Fig. 2 we have the Zipf law for we propose the random matrix model of WTN (RMWTN) with the mass matrix elements given by
(3) 
where are random numbers homogeneously distributed in interval and are indexes in the ImportRank index . The distribution given by this simple model reproduces quite well the actual distribution found for all commodities (see right panels in Fig. 6). With this RMWTN distribution of we construct the Google matrices and according to the usual recipes (1) and then determine the distribution of points in plane.
To have a statistical comparison between the RMWTN and real WTN data we construct the density distribution of countries in the plane using all available years 1962  2009 at the UN COMTRADE database for all commodities. The coarsegrained distribution of about WTN data points is shown in Fig. 7. We present the data directly in plane (top left panel) and in rescaled variables plane, which takes into account that the number of countries grown by 38% during this time period. The distribution has a form of spindle with maximum density at the vertical axis . We remind that good exporters are on the left side of this axis at , while the good importers (bad exporters) are on the right side at .
The comparison of WTN data with the results produced by RMWTN model (3) are shown in bottom panels of Fig. 7: there is a good agreement between both without any fit parameters for the half of all countries with top ranks (). For countries with the RMWTN model does not succeed to describe correctly the upper part of spindle distribution found for the WTN and hence further improvements of the RMWTN are needed. However, a simple description of the distribution for a half top countries is rather successful.
A remarkable feature of the WTN spindle distribution of Fig. 7 (top right) is the appearance of high density domains at with and . They give an impression of two solid phases emerging in these two regions while the other part looks like a gas phase. This view gets additional confirmation by data of Fig. 8 where we present the velocity square , averaged over the whole period 1962  2009, as a function of . This local quantity is defined as via a one year displacement of a given country in plane with further averaging over all times and all countries. These data clearly show that for we have very small square velocity (small effective temperature) corresponding to a solid phase of rich countries, while for we have large square velocity (large effective temperature) corresponding to a gas phase with rapid rank fluctuations. There is a similar visible drop of temperature at another limit of most poor countries with which indicates a formation of solid phase of poor countries (the data are not so exact for this region due to variation of number of UN countries with time).
The presence of solid phase of rich countries and gas phase of other countries is also visible from analysis of rank variation in time for individual countries shown in Fig. 9: for the curves are almost flat while for we see strong fluctuation of curves. It is interesting to note that sharp increases in mark crises in 1991, 1998 for Russia and in 2001 for Argentina (import is reduced in period of crises). We also see that in recent years the solid phase is perturbed by entrance of new countries like China and India. However, the results presented in Fig. 10 for the variation of square velocity with time for three regions of show that the top 10, and even top 20, countries have rather small velocities , compared to those with . For we have which remains constant in time. In a certain sense it looks that the countries with protect those with (approximately corresponding to G20 major economies (27)), so that their temperature at remains unaffected even by a very larger fluctuation well visible for the range during the period of 1992  1998 with a few financial crises of Black Wednesday, Mexico crisis, Asian crisis and Russian crisis.
The presented results for distribution of countries and analysis of their time evolution in the PageRankCheiRank plane confirm a well known statement that “the poor stay poor and the rich stay rich”.
Finally let us discuss an additional parameter which characterizes the correlation between PageRank and CheiRank vectors. The correlator between PageRank and CheiRank is defined as
(4) 
and in a similar way the correlator between ImportRank and ExportRank is given by
(5) 
Recently it has been found that there are networks with small correlator, like PCN Linux (12), and large correlator, as Wikipedia (13). Here we find that for all commodities we have large values of and , which have rather similar dependence on time (see Fig. 11). In contrast, there are almost zero or even negative correlations for crude petroleum. Indeed, for crude petroleum there is no correlation between export and import while for all commodities they are strongly correlated.
4 Comparison with Import  Export ranking
It is important to compare rating based on PageRank and CheiRank with the useful way of country rating based on ImportRank and ExportRank (see e.g (3)). With this aim we present the distribution of country positions on the planes and shown for top 100 for the same commodities as in Fig. 4 for year 2008. For all commodities there is a clear correlation between PageRank and ImportRank since the distribution of points is centered along the diagonal . It starts to spread only around . At the same time for CheiRank and ExportRank such a spreading from diagonal starts significantly earlier at .
For other commodities shown in Fig. 12 the correlations between ranking based on Google matrix and corresponding Export or/and Import ranking are practically absent showing very broad scattering of points around almost the whole plane. Only for cars there is a certain level of correlation for approximately the first 10 ranks. Natural products like crude petroleum, natural gas and agriculture products like barley show no correlations.
The similar conclusions can be also drawn from the comparison of country distributions in the plane (Fig. 4) and in the plane (Fig. 13), which show data on the same scales. Clearly, the distributions are rather different and only for all commodities we can see visible correlations (we note that appearance of ordered short line segments in panels (c,g) around is due to degeneracy of and values, for those countries which e.g. do not use barley, and thus their ordering becomes somewhat arbitrary).
Let us discuss in more detail few concrete examples shown in Tables 15 in Appendix. For all commodities first 5 positions are very close in both ways of ranking. As a significant change we note Canada which moves from down to and Mexico with respective change from to : the export of these two countries is too strongly oriented on USA that becomes directly visible through CheiRank analysis. In contrast Singapore moves up from to that shows the stability and broadness of its export trade, a similar situation appears for India moving up from to .
Even more strong changes of ranking appear for specific commodities. For example for crude petroleum Russia moves up from to showing that its trade network in this product is better and broader than the one of Saudi Arabia. Iran moves in opposite direction from down to showing that its trade network is restricted to a small number of nearby countries. A significant improvement of ranking takes place for Kazakhstan moving up from to . The direct analysis shows that this happens due to an unusual fact that Kazakhstan is practically the only country which sells crude petroleum to the CheiRank leader in this product Russia. This puts Kazakhstan on the second position. It is clear that such direction of trade is more of political or geographical origin and is not based on economic reasons.
For natural gas there are also significant differences between two ways of ranking. Thus, USA moves strongly up from to due its broad trade network in this product. Canada moves down from to due to its too strong trade orientation on USA. A small country Trinidad and Tobago moves up from to since it provides about 70% of import of top leader USA.
Significant reordering appears also for barley trade. Thus, the leader Ukraine moves down from to due to too narrow trade network and USA moves up from to due to its broad trade network.
For trade of cars we have France going up from to due to its broad export network. Also Thailand goes strongly up from to due to its broad trade links. We note that on the side of import we have strong change for Nigeria which moves from up to . This is the most populated country in Africa with a strong income due to oil trade which provides a large fraction of USA import. With such oil income Nigeria buys cars from all over the world and thus becomes at the top of PageRank.
Finally we note that among top 20 countries ranked in 2DRank by all commodities in 2008 (see Table 1) there 14 among G20 major economies (27). At the same time ExportRank gives 13, and ImportRank gives 14 countries from 19 of G20list. We attribute a difference in 5 countries to political and geographical aspects of G20selection.
5 Discussion
In this work we constructed the Google matrix of the WTN using the enormous UN COMTRADE database. From this matrix we obtained PageRank and CheiRank of all world countries in various types of trade products for years 1962  2009. This new approach gives a democratic type of ranking being independent of the trade amount of a given country. In this way rich and poor countries are treated on equal democratic grounds. In a certain sense PageRank probability for a given country is proportional to its rescaled import flows while CheiRank is proportional to its rescaled export flows inside of the WTN.
The global characteristics of the world trade are analyzed on the basis of this new type of ranking. Even if all countries are treated now on equal democratic grounds still we find at the top rank the group of industrially developed countries approximately corresponding to G20 (74%). Our studies establish the existence of two solid state domains of rich and poor countries which remain stable during the years of consideration. Other countries correspond to a gas phase with ranking strongly fluctuating in time. We propose a simple random matrix model which well describes the statistical properties of rank distribution for the WTN.
The comparison between usual ImportRank–ExportRank (see e.g. (3)) and our PageRank–CheiRank approach shows that the later highlights the trade flows in a new useful manner which is complementary to the usual analysis. The important difference between these two approaches is due to the fact that ImportRank–ExportRank method takes into account only global amount of money exchange between a country and the rest of the world while PageRank–CheiRank approach takes into account all links and money flows between all countries. We hope that this new approach based on the Google matrix will find further useful applications to investigation of various flows in trade and economy.
Acknowledgments: We thank Gilles SaintPaul (TSE, Toulouse) for useful discussions on economics and Olga Chepelianskaia (UN Bangkok) for insights on UN databases. Our special thanks are addressed to Arlene Adriano and Matthias Reister (United Nations Statistics Division) for provided help and friendly access to the UN COMTRADE database (1).
6 Appendix
Ran  

1  USA  China  USA  USA  China 
2  Germany  USA  China  Germany  Germany 
3  China  Germany  Germany  China  USA 
4  France  Japan  Japan  France  Japan 
5  Japan  France  France  Japan  France 
6  UK  Italy  Italy  UK  Netherlands 
7  Italy  Russian Fed.  UK  Netherlands  Italy 
8  Netherlands  Rep. of Korea  Netherlands  Italy  Russian Fed. 
9  India  UK  India  Belgium  UK 
10  Spain  Netherlands  Rep. of Korea  Canada  Belgium 
11  Belgium  Singapore  Belgium  Spain  Canada 
12  Canada  India  Russian Fed.  Rep. of Korea  Rep. of Korea 
13  Rep. of Korea  Belgium  Canada  Russian Fed.  Mexico 
14  Russian Fed.  Australia  Spain  Mexico  Saudi Arabia 
15  Nigeria  Brazil  Singapore  Singapore  Singapore 
16  Thailand  Canada  Thailand  India  Spain 
17  Mexico  Spain  Australia  Poland  Malaysia 
18  Singapore  South Africa  Brazil  Switzerland  Brazil 
19  Switzerland  Thailand  Mexico  Turkey  India 
20  Australia  U. Arab Emir.  U. Arab Emir.  Brazil  Switzerland 
Ran  

1  USA  Russian Fed.  USA  USA  Saudi Arabia 
2  Canada  Kazakhstan  India  Japan  Russian Fed. 
3  Netherlands  U. Arab Emir.  Singapore  China  U. Arab Emir. 
4  Belgium  USA  UK  Italy  Nigeria 
5  India  Ecuador  South Africa  Rep. of Korea  Iran 
6  China  Saudi Arabia  Canada  India  Venezuela 
7  Germany  India  Australia  Germany  Norway 
8  Japan  South Africa  U. Arab Emir.  Netherlands  Canada 
9  Rep. of Korea  Nigeria  Colombia  France  Angola 
10  UK  Sudan  Azerbaijan  UK  Iraq 
11  Singapore  Azerbaijan  Malaysia  Spain  Libya 
12  Italy  Venezuela  Brazil  Singapore  Kazakhstan 
13  Australia  Norway  Belgium  Canada  Kuwait 
14  Malaysia  Iran  Trinidad and Tobago  Thailand  Azerbaijan 
15  Spain  Algeria  France  Belgium  Algeria 
16  France  Singapore  Netherlands  Brazil  Mexico 
17  Brazil  Kuwait  Kenya  Turkey  UK 
18  Sweden  UK  Angola  South Africa  Qatar 
19  South Africa  Angola  China  Poland  Oman 
20  Thailand  Canada  Thailand  Australia  Netherlands 
Ran  

1  USA  USA  USA  Japan  Norway 
2  Japan  Trinidad and Tobago  France  USA  Canada 
3  Rep. of Korea  Norway  Belgium  France  Algeria 
4  Spain  UK  South Africa  Rep. of Korea  Russian Fed. 
5  France  Russian Fed.  Italy  Spain  Qatar 
6  Italy  Oman  Canada  Belgium  Belgium 
7  Nigeria  Australia  UK  UK  Indonesia 
8  China  Canada  Malaysia  Italy  Malaysia 
9  Poland  France  Germany  Germany  Netherlands 
10  Portugal  Algeria  China  Ukraine  USA 
11  El Salvador  South Africa  Nigeria  Netherlands  Australia 
12  Kenya  Kazakhstan  Greece  Mexico  Nigeria 
13  Belgium  Qatar  Turkey  China  Saudi Arabia 
14  Guatemala  Saudi Arabia  Kenya  India  U. Arab Emir. 
15  Germany  U. Arab Emir.  Netherlands  Hungary  Trinidad and Tobago 
16  Mexico  Belgium  Rep. of Korea  Czech Rep.  Germany 
17  Ecuador  Pakistan  Spain  Canada  Oman 
18  Malaysia  Singapore  Russian Fed.  Brazil  Egypt 
19  South Africa  Netherlands  India  Turkey  UK 
20  Slovenia  Italy  Japan  Thailand  Turkmenistan 
Ran  

1  Saudi Arabia  France  USA  Saudi Arabia  Ukraine 
2  Yemen  Canada  Germany  Germany  France 
3  Germany  USA  Netherlands  Japan  Australia 
4  U. Arab Emir.  Australia  Denmark  China  Canada 
5  USA  Germany  Italy  Belgium  Germany 
6  Israel  Ukraine  Belgium  Netherlands  Russian Fed. 
7  Japan  Rep. of Moldova  China  Syria  Argentina 
8  Netherlands  UK  U. Arab Emir.  Iran  USA 
9  Oman  Argentina  UK  Jordan  Denmark 
10  Greece  Spain  Spain  USA  Kazakhstan 
11  Italy  Denmark  Singapore  Denmark  Romania 
12  Croatia  Kazakhstan  Rep. of Korea  Italy  UK 
13  Syria  Netherlands  Malaysia  Tunisia  Hungary 
14  Kuwait  Russian Fed.  Russian Fed.  Israel  Spain 
15  Cyprus  India  Austria  Colombia  Bulgaria 
16  Denmark  Hungary  Poland  Algeria  Netherlands 
17  Occ. Palestinian Terr.  Romania  Brazil  Kuwait  Lithuania 
18  Switzerland  Belgium  Ireland  Brazil  Sweden 
19  Bosnia Herzegovina  Lithuania  France  Morocco  Belgium 
20  Jordan  Sweden  South Africa  Turkey  India 
Ran  

1  Nigeria  Germany  Germany  USA  Germany 
2  Germany  Japan  USA  Germany  Japan 
3  France  USA  France  UK  USA 
4  USA  Rep. of Korea  UK  France  Canada 
5  Russian Fed.  France  Belgium  Italy  Rep. of Korea 
6  UK  UK  Spain  Russian Fed.  UK 
7  Belgium  Belgium  Italy  Belgium  France 
8  Ukraine  Spain  Japan  Canada  Spain 
9  Italy  South Africa  Australia  Spain  Belgium 
10  Greece  Thailand  Canada  China  Mexico 
11  Venezuela  Mexico  China  Netherlands  Italy 
12  Spain  Italy  Netherlands  Australia  Slovakia 
13  China  Canada  U. Arab Emir.  U. Arab Emir.  Czech Rep. 
14  Netherlands  U. Arab Emir.  Austria  Saudi Arabia  Poland 
15  Australia  Czech Rep.  South Africa  Austria  Sweden 
16  Japan  Slovakia  Poland  Mexico  Turkey 
17  Albania  Hungary  Thailand  Poland  Hungary 
18  Romania  Australia  Russian Fed.  Switzerland  Austria 
19  Sudan  Austria  Turkey  Finland  Thailand 
20  Canada  China  Portugal  Ukraine  South Africa 
References
 United Nations Commodity Trade Statistics Database http://comtrade.un.org/db/
 P.R. Krugman, M. Obstfeld, M. Melitz, International economics: theory & policy, Prentic Hall, New Jersey (2011).
 Central Intelligence Agency, The CIA Wold Factbook 2010, Skyhorse Publ. Inc. (2009)
 S. Brin and L. Page, Computer Networks and ISDN Systems 30, 107 (1998).
 A. M. Langville and C. D. Meyer, Google’s PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press (Princeton, 2006).
 N. Litvak, W.R.W. Scheinhardt, and Y. Volkovich, Lecture Notes in Computer Science, 4936, 72 (2008).
 D. Donato, L. Laura, S. Leonardi and S. Millozzi, Eur. Phys. J. B 38, 239 (2004).
 G. Pandurangan, P. Raghavan and E. Upfal, Internet Math. 3, 1 (2005).
 S. Redner, Physics Today 58, 49 (2005).
 F. Radicchi, S. Fortunato, B. Markines, and A. Vespignani, Phys. Rev. E 80, 056103 (2009).
 J. West, T. Bergstrom and C.T. Bergstrom, arXiv:0911.1807 (to appear in J. American Soc. Info. Sci. Tech. (2011)); http://eigenfactor.org/

A. D. Chepelianskii,
Towards physical laws for software architecture,
arXiv:1003.5455[cs.SE] (2010);
http://www.quantware.upstlse.fr/QWLIB/linuxnetwork/  A.O. Zhirov. O.V. Zhirov and D.L.Shepelyansky, Eur. Phys. J. B 77, 523 (2010); http://www.quantware.upstlse.fr/QWLIB/2drankwikipedia/
 D. Garlaschelli and M.I. Loffredo, Physica A: Stat. Mech. Appl. 355, 138 (2005).
 M.A. Serrano, M. Boguna and A. Vespignani, J. Econ. Interac. Coor. 2, 111 (2007).
 L. De Benedictis and L. Tajoli, The World Trade Network, working paper (2009); available at http://www.eief.it/files/2010/10/lucadebenedictis.pdf
 J. He and M.W. Deem, Phys. Rev. Lett. 105, 198701 (2010).
 M. Barigozzi, G. Fagiolo and D. Garlaschelli, Phys. Rev. E 81, 046104 (2010).

http://www.quantware.upstlse.fr/QWLIB/
tradecheirank/  G.K. Zipf, Human Behavior and the Principle of Least Effort, AddisonWesley, Boston (1949).
 K. Avrachenkov, N. Litvak and K. Pham, Internet Math. 5, 47 (2005).
 B.Georgeot, O.Giraud and D.L.Shepelyansky, Phys. Rev. E 81, 056109 (2010).
 D.L.Shepelyansky and O.V.Zhirov, Phys. Rev. E 81, 036213 (2010).
 L.Ermann and D.L.Shepelyansky, Phys. Rev. E 81, 036221 (2010).
 L.Ermann and D.L.Shepelyansky, Eur. Phys. J. B 75, 299 (2010).
 L.Ermann, A.D. Chepelianskii and D.L.Shepelyansky, Eur. Phys. J. B 79, 115 (2011).
 Wikipedia contributors. G20 major economies. Wikipedia, The Free Encyclopedia, Web. 25 Mar. 2011.