Google matrix analysis of the multiproduct world trade network

4.6 MB·English

by  Leonardo Ermann

EPJ manuscript No.
(will be inserted by the editor)
Google matrix analysis of the multiproduct world trade network
L.Ermann1 and D.L.Shepelyansky2
1 Departamento de F´ısica Teo´rica, GIyA, CNEA, Av. Libertador 8250, (C1429BNP) Buenos Aires, Argentina.
2 Laboratoire de Physique Th´eorique du CNRS, IRSAMC, Universit´e de Toulouse, UPS, F-31062 Toulouse, France
Dated: January 14, 2015
5 Abstract. Using the United Nations COMTRADE database [1] we construct the Google matrix G of
1 multiproduct world trade between the UN countries and analyze the properties of trade flows on this
0 network for years 1962 - 2010. This construction, based on Markov chains, treats all countries on equal
2 democraticgroundsindependentlyoftheirrichnessandatthesametimeitconsidersthecontributionsof

n tradeproductsproportionallytotheirtradevolume.Weconsiderthetradewith61productsforupto227
a countries.Theobtainedresultsshowthatthetradecontributionofproductsisasymmetric:someofthem
J areexportorientedwhileothersareimportorientedeveniftherankingbytheirtradevolumeissymmetric

4 in respect to export and import after averaging over all world countries. The construction of the Google
1 matrix allows to investigate the sensitivity of trade balance in respect to price variations of products, e.g.
petroleum and gas, taking into account the world connectivity of trade links. The trade balance based on

] PageRank and CheiRank probabilities highlights the leading role of China and other BRICS countries in
T
theworldtradeinrecentyears.WealsoshowthattheeigenstatesofGwithlargeeigenvaluesselectspecific
S trade communities.
.
n
i PACS. 89.75.Fb Structures and organization in complex systems – 89.65.Gh Econophysics – 89.75.Hc
f
Networks and genealogical trees – 89.20.Hh World Wide Web, Internet
-
q
[

1 Introduction tradewherehiddenlinksandinteractionsbetweencertain
1
v countries and products are not taken into account since
1 only a country global import or export are considered.
AccordingtothedataofUNCOMTRADE[1]andthein-
7 Thus the statistical analysis of these multiproduct trade
3 ternationaltradestatistics2014oftheWorldTradeOrga- datarequiresautilizationofmoreadvancedmathematical
3 nization(WTO)[2]theinternationalworldtradebetween and numerical methods.
0 world countries demonstrates a spectacular growth with
. anincreasingtradevolumeandnumberoftradeproducts. In fact, in the last decade, modern societies developed
1
Itiswellclearthattheworldtradeplaysthefundamental enormous communication and social networks including
0
5 role in the development of world economy [3]. According the World Wide Web (WWW), Wikipedia, Twitter etc.
1 to the WTO Chief Statistician Hubert Escaith “In recent (see e.g. [5]). A necessity of information retrieval from
: yearswehaveseengrowingdemandfordataontheworld suchnetworksledtoadevelopmentofefficientalgorithms
v
economy and on international trade in particular. This for information analysis on such networks appeared in
i
X demand has grown in particular since the 2008-09 crisis, computer science. One of the most spectacular tools is
r whose depth and breadth surprised many experts” [2]. In the PageRank algorithm developed by Brin and Page in
a globalthedataoftheworldtradeexchangecanbeviewed 1998 [6], which became a mathematical foundation of the
as a large multi-functional directed World Trade Network Googlesearchengine(seee.g.[7]).Thisalgorithmisbased
(WTN)whichprovidesimportantinformationaboutmul- ontheconceptofMarkovchainsandaconstructionofthe
tiproduct commercial flows between countries for a given Google matrix G of Markov transitions between network
year.AtpresenttheCOMTRADEdatabasecontainsdata nodes. The right eigenvector of this matrix G, known as
for N = 227 UN countries with up to N ≈ 104 trade PageRank vector, allows to rank all nodes according to
c p
products. Thus the whole matrix of these directed trade their importance and influence on the network. The stud-
flows has a rather large size N = N N ∼ 106. A usual ies of various directed networks showed that it is useful
p c
approach is to consider the export and import volumes, to analyze also the matrix G∗ constructed for the same
expressed in US dollars (USD). An example of the world network but with an inverted direction of links [8,9]. The
mapofcountriescharacterizedbytheirimportandexport PageRank vector of G∗ is known as the CheiRank vec-
trade volume for year 2008 is shown in Fig. 1. However, tor. The spectral properties of Goggle matrix for various
suchanapproachgivesonlyanapproximatedescriptionof networks are described in [10].
2 L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network
tributions to economy are linked with their trade volume.
Thus,accordingtoTable2,inyear2008thetradevolume
of Petroleum and petroleum products (code 33 in Table 1)
is by a factor 300 larger than those of Hides, skins and
fur skins (undress.) (code 21 in Table 1). To incorporate
these features in our mathematical analysis of multiprod-
uct WTN we developed in this work the Google Person-
alized Vector Method (GPVM) which allows to keep a
democratic treatment of countries and at the same time
toconsiderproductsproportionallytotheirtradevolume.
As a result we are able to perform analysis of the global
multiproduct WTN keeping all interactions between all
countriesandallproducts.ThisisanewstepintheWTN
analysis since in our previous studies [11] it was possible
to consider a trade between countries only in one product
or only in all products summed together (all commodi-
ties). The new finding of such global WTN analysis is an
asymmetric ranking of products: some of them are more
orientedtoimportandothersareorientedtoexportwhile
therankingofproductsbythetradevolumeisalwayssym-
metricaftersummationoverallcountries.Thisresultwith
Fig. 1. World map of countries with color showing country asymmetric ranking of products confirms the indications
import (top panel) and export (bottom panel) trade volume obtained on the basis ecological ranking [12] which also
expressed in millions of USD given by numbers of the color give an asymmetry of products in respect to import and
bars.Thedataareshownforyear2008withNc =227countries export.Ourapproachalsoallowstoanalyzethesensitivity
fortradeinallN =61products(fromUNCOMTRADE[1]).
p of trade network to price variations of a certain product.
Names of countries can be find at [4].
WethinkthattheGPVMapproachallowstoperforma
most advanced analysis of multiproduct world trade. The
previousstudieshavebeenrestrictedtostudiesofstatisti-
TheapproachofGooglematrixtotheanalysisofWTN calcharacteristicsofWTNlinks,patternsandtheirtopol-
was started in [11]. The striking feature of this approach ogy (see e.g. [13,14,15,16,17,18,19]). The applications of
is that it treats all UN countries on equal democratic PageRank algorithm to the WTN was discussed in [20],
grounds, independently of richness of a given country, in the approach based on HITS algorithm was used in [21].
agreement with the principles of UN where all countries In comparison to the above studies, the approach devel-
areequal.ThispropertyofGmatrixisbasedontheprop- opedhereforthemultiproductWTNhasanadvantageof
erty of Markov chains where the total probability is con- analysis of ingoing and outgoing flows, related to PageR-
served to be unity since the sum of elements for each col- ank and CheiRank, and of taking into account of multi-
umn of G is equal to unity. Even if in this approach all product aspects of the WTN. Even if the importance to
countries are treated on equal grounds still the PageR- multiproduct WTN analysis is clearly understood by re-
ank and CheiRank analysis recover about 75% of indus- searchers (see e.g. [22]) the Google matrix methods have
trially developed countries of G20. However, now these not been efficiently used up to now. We also note that
countries appear at the top ranking positions not due to the matrix methods are extensively used for analysis of
their richness but due to the efficiency of their trade net- correlations of trade indexes (see e.g. [23,24]) but these
work.Anotherimportantaspectfoundin[11]isthatboth matrices are Hermitian being qualitatively different from
PageRankandCheiRankvectorsappearverynaturallyin those appearing in the frame of Markov chains. Here we
the WTN corresponding to import and export flows. makethestepsinmulti-functionalormultiproductGoogle
In this work we extend the Google matrix analysis for matrix analysis of the WTN extending the approach used
the multiproduct WTN obtained from COMTRADE [1] in [11].
with up to N = 61 trade products for up to N = 227
p c
countries. The global G matrix of such trade flows has a
size up to N = N N = 13847 nodes. The names and
p c 2 Methods
codes of products are given in Table 1 and their trade
volumes, expressed in percent of the whole world trade
2.1 Google matrix construction for the WTN
volume,aregiveninTable2foryears1998and2008.The
main problem of construction of such a matrix is not its
size, which is rather modest compared to those studied in Foragivenyear,webuildNp moneymatricesMcp,c(cid:48) ofthe
[10],butthenecessitytotreatallcountriesondemocratic WTN from the COMTRADE database [1] (see [11]).
grounds and at the same time to treat trade products on
the basis of their trade volume. Indeed, the products can- Mp =product p transfer (in USD) from country c(cid:48) to c
c,c(cid:48)
not be considered on democratic grounds since their con- (1)
L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network 3
Herethecountryindexesarec,c(cid:48) =1,...,N andaprod- 2.2 PageRank and CheiRank vectors from GPVM
c
uctindexisp=1,...,N .AccordingtotheCOMTRADE
p
database the number of UN registered countries is N = PageRank and CheiRank (P and P∗) are defined as the
c
227 (in recent years) and the number of products is N = right eigenvectors of G and G∗ matrices respectively at
p
10 and N = 61 for 1 and 2 digits respectively from the eigenvalue λ=1:
p
Standard International Trade Classification (SITC) Rev.
(cid:88) (cid:88)
1. For convenience of future notation we also define the G ψ =λψ , G∗ ψ∗ =λψ∗ . (6)
ij j i ij j j
volume of imports and exports for a given country and
j j
product respectively as
For the eigenstate at λ = 1 we use the notation P =
i
Vcp =(cid:88)Mcp,c(cid:48), Vc∗p =(cid:88)Mcp(cid:48),c. (2) Fψoi,rPo∗th=erψe∗igiewnsitthattehsewneourmseatlhizeantioornm(cid:80)aliPzait=ion(cid:80)(cid:80)iP|∗ψi =|2 =1.
c(cid:48) c(cid:48) (cid:80) |ψ∗|2 =1.AccordingtothePerron-Frobeniustiheoirem
i i
The import and export volumes V = (cid:80) Vp and V∗ = the components of Pi, P∗i are positive and give the prob-
(cid:80) V∗p areshownfortheworldmacpofcoupntcriesinFcig.1 abilities to find a random surfer on a given node [7]. The
p c PageRank K and CheiRank K∗ indexes are defined from
for year 2008.
thedecreasingorderingofP andP∗ asP(K)≥P(K+1)
InordertocomparelaterwithPageRankandCheiRank and P∗(K)≥P∗(K∗+1) with K,K∗ =1,...,N.
probabilities we define volume trade ranks in the whole
If we want to compute the reduced PageRank and
trade space of dimension N =N ×N . Thus the Impor-
p c CheiRank probabilities of countries for all commodities
tRank (Pˆ) and ExportRank (Pˆ∗) probabilities are given (or equivalently all products) we trace over the product
by the normalized import and export volumes space getting P = (cid:80) P = (cid:80) P (p+(c−1)N ) and
c p pc p p
P∗ = (cid:80)P∗ = (cid:80) P∗(p+(c−1)N ) with their corre-
c pc p p
Pˆi =Vcp/V , Pˆi∗ =Vc∗p/V , (3) sponding Kc and Kc∗ indexes. In a similar way we ob-
tain the reduced PageRank and CheiRank probabilities
where i=p+(c−1)N , i=1,...,N and the total trade for products tracing over all countries and getting
p
volume is V =(cid:80) Mp =(cid:80) Vp =(cid:80) V∗p. P =(cid:80) P (p+(c−1)N )(cid:80) P and
p,c,c(cid:48) c,c(cid:48) p,c c p,c c p c p p pc
The Google matrices G and G∗ are defined as N ×N Pp∗ =(cid:80)cP∗(p+(c−1)Np)(cid:80)Pp∗c withtheircorrespond-
real matrices with non-negative elements: ing product indexes Kp and Kp∗.
InsummarywehaveK ,K∗ =1,...,N andK ,K∗ =
p p p c c
Gij =αSij+(1−α)viej, G∗ij =αS∗ij+(1−α)vi∗ej, (4) 1,...,Nc. A similar definition of ranks from import and
exporttradevolumecanbedoneinastraightforwardway
via probabilities Pˆ ,Pˆ∗,Pˆ ,Pˆ∗,Pˆ ,Pˆ∗ and correspond-
where N = Np × Nc, α ∈ (0,1] is the damping factor p p c c pc pc
(0<α<1),e istherowvectorofunitelements(e =1), ing indexes Kˆ ,Kˆ∗,Kˆ ,Kˆ∗,Kˆ,Kˆ∗.
j j p p c c
andv isapositivecolumnvectorcalledapersonalization TocomputethePageRankandCheiRankprobabilities
i
vector with (cid:80) v =1 [7]. We note that the usual Google from G and G∗ keeping democracy in countries and pro-
i i
matrixisrecoveredforapersonalizationvectorv =e /N portionality of products to their trade volume we use the
i i
In this work, following [11], we fix α = 0.5. As discussed GPVMapproachwithapersonalizedvectorin(4).Atthe
in [7,10,11] a variation of α in a range (0.5,0.9) does not first iteration of Google matrix we take into account the
significantlyaffecttheprobabilitydistributionsofPageR- relative product volume per country using the following
ank and CheiRank vectors. We specify the choice of the personalization vectors for G and G∗:
personalization vector a bit below.
The matrices S and S∗ are built from money matrices v = Vcp , v∗ = Vc∗p , (7)
Mp as i N (cid:80) Vp(cid:48) i N (cid:80) V∗p(cid:48)
cc(cid:48) c p(cid:48) c c p(cid:48) c
(cid:26)Mp δ /V∗p if V∗p (cid:54)=0 usingthedefinitions(2)andtherelationi=p+(c−1)N .
S = c,c(cid:48) p,p(cid:48) c(cid:48) c(cid:48) p
i,i(cid:48) 1/N if V∗p =0 This personalized vector depends both on product and
c(cid:48)
(cid:26)Mp δ /Vp if Vp (cid:54)=0 countryindexes.Inordertohavethesamevalueofperson-
S∗ = c(cid:48),c p,p(cid:48) c(cid:48) c(cid:48) (5) alization vector in countries we can define the second it-
i,i(cid:48) 1/N if Vp =0
c(cid:48) eration vector proportional to the reduced PageRank and
CheiRank vectors in products obtained from the GPVM
wherec,c(cid:48) =1,...,Nc;p,p(cid:48) =1,...,Np;i=p+(c−1)Np; Google matrix of the first iteration:
i(cid:48) = p(cid:48) +(c(cid:48) −1)N ; and therefore i,i(cid:48) = 1,...,N. Note
p
that the sum of each column of S and S∗ are normalized P P∗
tounityandhencethematricesG,G∗,S,S∗ belongtothe v(cid:48)(i)= Np , v(cid:48)∗(i)= Np . (8)
c c
class of Google matrices and Markov chains. The eigen-
values and eigenstates of G,G∗ are obtained by a direct In this way we keep democracy in countries but weighted
numerical diagonalization using the standard numerical products. This second iteration personalized vectors are
packages. used for the main part of computations and operations
4 L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network
with G and G∗. This procedure with two iterations forms
our GPVM approach. The difference between results ob-
tained from the first and second iterations is not very
large (see Figs. 2, 3) but a detailed analysis of ranking of
countries and products shows that the personalized vec-
tor for the second iteration improves the results making
them more stable and less fluctuating. In all Figures be-
low(exceptFigs.2,3)weshowtheresultsafterthesecond
iteration.
Fig. 3. ProbabilitydistributionsofPageRankandCheiRank
for products P (K ), P∗(K∗) (left panel) and countries
p p p p
P (K ), P∗(K∗) (right panel) in logarithmic scale for WTN
c c c c
from Fig. 2. Here the results for the 1st and 2nd GPVM itera-
tionsareshownbyred(blue)curvesforPageRank(CheiRank)
with dashed and solid curves respectively. The probabilities
fromthetradevolumerankingareshownbyblackcurve(left)
and dotted red and blue curves (right) for ImportRank and
ExportRank respectively.
Wealsocharacterizethelocalizationpropertiesofeigen-
states of G,G∗ by the inverse participation ration (IPR)
defined as ξ = ((cid:80) |ψ |2)2/(cid:80) |ψ |4. This characteristic
i i i i
determines an effective number of nodes which contribute
to a formation of a given eigenstate (see details in [10]).
2.3 Correlators of PageRank and CheiRank vectors
Fig. 2. Dependence of probabilities of PageRank P(K), Followingpreviousworks[8,9,11]thecorrelatorofPageR-
CheiRank P∗(K∗), ImportRank Pˆ(Kˆ) and ExportRank ank and CheiRank vectors is defined as:
Pˆ∗(Kˆ∗) as a function of their indexes in logarithmic scale for
N
WTN in 2008 with α=0.5 at Nc =227, Np =1, N =13847. κ=N(cid:88)P(i)P∗(i)−1. (9)
Here the results for GPVM after 1st and 2nd iterations are
shown for PageRank (CheiRank) in red (blue) with dashed i=1
and solid curves respectively. ImportRank and ExportRank The typical values of κ are given in [10] for various net-
(trade volume) are shown by red and blue thin curves re-
works.
spectively. The fit exponents for PageRank and CheiRank are
ForglobalPageRankandCheiRanktheproduct-product
β =0.61,0.7forthefirstiteration,β =0.59,0.65forthesecond
correlator matrix is defined as:
iteration,andβ =0.94,1.04forImportRankandExportRank
(for the range K ∈[10,2000]). κpp(cid:48) =Nc(cid:88)Nc (cid:20)(cid:80) PP(p(p++(c((cid:48)c−−11))NNp))(cid:80)P∗(pP(cid:48)+∗((pc(cid:48)+−(1c)(cid:48)N(cid:48)−p)1)N )(cid:21)−1
c=1 c(cid:48) p c(cid:48)(cid:48) p
(10)
The obtained results show the distribution of nodes
on the PageRank-CheiRank plane (K,K∗). In addition Then the correlator for a given product is obtained
to two ranking indexes K,K∗ we use also 2DRank in- from (10) as:
κ =κ δ , (11)
dex K which combines the contribution of these indexes p pp(cid:48) p,p(cid:48)
2
as described in [9]. The ranking list K2(i) is constructed where δp,p(cid:48) is the Kronecker delta.
by increasing K → K + 1 and increasing 2DRank in- We also use the correlators obtained from the proba-
odfexfirKst2(Ki)∗b<y KoneenitfriaesnoefwCehnetiRryanisk,ptrheesnentthienotnheeulnisitt bilitiestrace(cid:80)doverproducts(Pc =(cid:80)pPpc)andovercoun-
tries (P = P ) which are defined as
step is done in K∗ and K is increased by one if the new p c pc
2
entry is present in the list of first K < K∗ entries of (cid:88)Nc (cid:88)Np
CheiRank. More formally, 2DRank K2(i) gives the order- κ(c)=Nc PcPc∗−1, κ(p)=Np PpPp∗−1. (12)
ing of the sequence of sites, that appear inside the
c=1 p=1
squares [1,1; K =k,K∗ =k; ...]whenonerunsprogres-
sively from k =1 to N. Additionally, we analyze the dis- Intheaboveequations(9)-(12)thecorrelatorsarecom-
tributionofnodesforreducedindexes(K ,K∗),(K ,K∗). puted for PageRank and CheiRank probabilities. We can
p p c c
L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network 5
alsocomputethesamecorrelatorsusingprobabilitiesfrom PageRank and CheiRank probabilities while there is no
the trade volume in ImportRank Pˆ and ExportRank Pˆ∗ difference between ImportRank and ExportRank proba-
defined by (3). bilities since they are equal after tracing over countries.
WediscussthevaluesofthesecorrelatorsinSection4. After tracing over products we obtain probability dis-
tributionsP (K ),P∗(K∗)overcountriesshowninFig.3.
c c c c
We see that the probability of volume ranking varies ap-
3 Data description proximately by a factor 1000 while for PageRank and
CheiRank such a factor is only approximate 10. Thus the
All data are obtained from the COMTRADE database democracyincountriesinducedbytheGooglematrixcon-
[1]. We used products from COMTRADE SITC Rev. 1 struction reduces significantly the variations of probabili-
classification with number of products N = 10 and 61. ties among countries and inequality between countries.
p
We choose SITC Rev.1 since it covers the longest time Both panels of Fig. 3 show relatively small variations
interval. The main results are presented for N =61 with between1stand2ndGPVMiterationsconfirmingthesta-
p
up to N = 227 countries. The names of products are bility of this approach. In next sections we present the
c
given in Table 1, their ImportRank index K and their results only for 2nd GPVM iteration. This choice is con-
fraction (in percent) of global trade volume in years 1998 firmed by consideration of ranking positions of various
and 2008 are given in Table 2. The data are collected nodes of global matrices G,G∗ which show less fluctua-
and presented for the years 1962 - 2010. Our data and tions compared to the results of the 1st GPVM iteration.
resultsareavailableat[25],thedataforthematricesMp From the global ranking of countries and products we
c,c(cid:48)
are available at COMTRADE [1] with the rules of their canselect agivenproductand thendeterminelocalrank-
distributionpolicy.Following[11]weuseforcountriesISO ing of countries in a given product to see how strong is
3166-1 alpha-3 code available at Wikipedia. their trade for this product. The results for three selected
products are discussed below for year 2008. For compar-
ison we also present comparison with the export-import
4 Results ranking from the trade volume.
Weapplytheabovemethodstothedescribeddatasetsof
COMTRADE and present the obtained results below.
4.1 PageRank and CheiRank probabilities
The dependence of probabilities of PageRank P(K) and
CheiRankP∗(K∗)vectorsontheirindexesK,K∗areshown
in Fig. 2 for a selected year 2008. The results can be ap-
proximately described by an algebraic dependence P ∝
1/Kβ with the exponent values given in the caption. It is
interesting to note that we find approximately the same
β ≈ 0.6 both for PageRank and CheiRank in contrast to
the WWW, universities and Wikipedia networks where
usually one finds β ≈ 1 for PageRank and β ≈ 0.6 for
CheiRank[7,10].Weattributethistoanintrinsicproperty
ofWTNwherethecountriestrytokeepeconomybalance Fig. 4. Country positions on PageRank-CheiRank plane
oftheirtrade.Thedatashowthattherangeofprobability (Kc,Kc∗) obtained by the GPVM analysis (left panels),
ImportRank-ExportRankoftradevolume(centerpanels),and
variation is reduced for the Google ranking compared to
forPageRank-CheiRankofallcommodities (rightpanels,data
the volume ranking. This results from a democratic rank-
from [11]). Top panels show global scale (K ,K∗ ∈ [1,200])
ing of countries used in the Google matrix analysis that c c
andbottompanelsshowzoomontopranks(K ,K∗ ∈[1,40]).
givesareductionofrichnessdispersionbetweencountries. c c
Eachcountryisshownbycirclewithitsownflag(forabetter
The results also show that the variation of probabilities
visibility the circle center is slightly displaced from its integer
for 1st and 2nd GPVM results are not very large that
position (K ,K∗) along direction angle π/4). Data are shown
demonstrates the convergence of this approach. c c
for year 2008.
After tracing probabilities over countries we obtain
probability distributions P (K ), P∗(K∗) over products
p p p p
shown in Fig. 3. The variation range of probabilities is
the same as for the case of volume ranking. This shows
that the GPVM approach correctly treats products keep- 4.2 Ranking of countries and products
ing their contributions proportional to their volume. The
difference between 1st and 2nd iterations is rather small After tracing the probabilities P(K),P∗(K∗) over prod-
and is practically not visible on this plot. The impor- ucts we obtain the distribution of world countries on the
tant result well visible here is a visible difference between PageRank-CheiRank plane (K ,K∗) presented in Fig. 4
c c
6 L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network
foratestyear2008.Inthesamefigurewepresenttherank
distributionsobtainedfromImportRank-ExportRankprob-
abilities of trade volume and the results obtained in [11]
for trade in all commodities. For the GPVM data we see
theglobal featuresalready discussedin [11]: thecountries
are distributed in a vicinity of diagonal K = K∗ since
c c
each country aims to keep its trade balanced. The top 20
list of top K countries recover 15 of 19 countries of G20
2
major world economies (EU is the number 20) thus ob-
taining 79% of the whole list. This is close to the percent
obtained in [11] for trade in all commodities.
TheglobaldistributionsoftopcountrieswithK ≤40,
c
K∗ ≤ 40 for the three ranking methods, shown in Fig. 4,
c
aresimilaronaverage.Butsomemodificationsintroduced
by the GPVM analysis are visible. Thus China (CHN)
moves on 2nd position of CheiRank while it is in the
1st position for trade volume ranking and CheiRank of
all commodities. Also e.g. Saudi Arabia (SAU) and Rus- Fig. 5. Two dimensional ranking of products on the
sia (RUS) move from the CheiRank positions K∗c = 21 PageRank-CheiRank plane (Kp,Kp∗). Each product is repre-
and K∗ = 7 in all commodities [11] to K∗ = 29 and sented by its specific combination of color and symbol: color
c c
K∗ = 6 in the GPVM ranking, respectively. Other ex- illustrates the first digit of COMTRADE SITC Rev. 1 code
c
ample is a significant displacement of Nigeria (NGA). We withthecorrespondingnameshowninthelegendontheright;
explain such differences as the result of larger connectiv- symbols correspond to product names listed in Table 1 with
their code numbers; the order of names on the right panel of
ity required for getting high ranking in the multiproduct
this Fig. is the same as in Table 1. The trade volume ranking
WTN. Indeed, China is more specialized in specific prod-
viaImportRank-ExportRankisshownbysmallsymbolsatthe
uctscomparedtoUSA(e.g.nopetroleumproductionand
diagonal Kˆ =Kˆ∗, after tracing over countries this ranking is
export) that leads to its displacement in K∗ . We note p p
c symmetric in products. Top left and right panels show years
that the ecological ranking gives also worse ranking po-
1963 and 1978, while bottom left and right panels show years
sitions for China comparing to the trade volume ranking
1993 and 2008 respectively.
[12].InasimilarwaythetradeofSaudiArabiaisstrongly
dominatedbypetroleumandmoreoveritspetroleumtrade
is strongly oriented on USA that makes its trade network
tions, 64 Paper, paperboard and manuf., 65 Textile yarn,
concentrated on a few links while Russia is improving its
fabrics, etc., 86 Scientific & control instrum, photogr gds,
positioninK∗ duetosignificanttradelinkswithEUand
c clocks).
Asia.
It is interesting to note that the machinery products
In global, the comparison of three ranks of countries
71,72.73arelocatedonleadingimportorientedpositions
shown in Fig. 4 confirms that the GPVM analysis gives a
in 1963, 1978, 1993 but they become more close to sym-
reliable ranking of multiproduct WTN. Thus we now try
metricpositionsin2008.Weattributethistodevelopment
to obtain new features of multiproduct WTN using the
ofChinathatmakesthetradeintheseproductsmoresym-
GPVM approach.
metric in import-export. It is interesting to note that in
The main new feature obtained within the GPVM ap- 1993 the product 33 Petroleum and petroleum products
proach is shown in Fig. 5 which gives the distribution of loses its first trade volume position due to low petroleum
products on the PageRank-CheiRank plane (K ,K∗) af- prices but still it keeps the first CheiRank position show-
p p
ter tracing of global probabilities P(K),P∗(K∗) over all ingitstradenetworkimportanceforexport.Eachproduct
world countries. The data clearly show that the distribu- moveson(K ,K∗)withtime.However,apartoftheabove
p p
tion of products over this plane is asymmetric while the points, we can say that the global distribution does not
ranking of products from the trade volume produces the manifest drastic changes. Indeed, e.g. the green symbols
symmetric ranking of products located directly on diago- offirstdigit2remainexportorientedforthewholeperiod
nal K = K∗. Thus the functions of products are asym- 1963 - 2008. We note that the established asymmetry of
p p
metric: some of them are more oriented to export (e.g. products orientation for the world trade is in agreement
03 Fish and fish preparations, 05 Fruit and vegetables, 26 with the similar indications obtained on the basis of eco-
Textile fibers, not manuf. etc., 28 Metalliferous ores and logicalrankingin[12].However,theGPVMapproachused
metalscrap,84Clothing);inlastyears(e.g.2008)34Gas, herehavemoresolidmathematicalandstatisticalfounda-
natural and manufactured also takes well pronounced ex- tionswithareducedsignificanceoffluctuationscomparing
portorientedfeaturecharacterizedbylocationinthelower to the ecological ranking.
righttriangle(K∗ <K )ofthesquareplane(K ,K∗).In The comparison between the GPVM and trade vol-
p p p p
contrasttothattheproductslocatedintheupperlefttri- ume ranking methods provides interesting information.
angle(K∗ >K )representimportorientedproducts(e.g. Thus in petroleum code 33 we have on top positions Rus-
p p
02Dairyproductsandeggs,04Cerealsandcerealprepara- sia, Saudi Arabia, United Arab Emirates while from the
L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network 7
in PageRank-CheiRank positions comparing to their po-
sitions in the trade volume ranking. This reflects their
strong commercial relations in the world trade. In the
trade volume ranking the top positions are taken by 33
petroleum and digit 7 of machinery products. This re-
mains mainly true for PageRank-CheiRank positions but
we see the spectacular improvement of positions of 84
Clothing for China (K∗ = 2) and 93 Special transact.
for USA (K = 4) showing thus these two products have
strongcommercialexchangeallovertheworldeveniftheir
trade volume is not so dominant.
Fig. 6. ToppanelsshowresultsoftheGPVMdataforcoun-
trypositionsonPageRank-CheiRankplaneoflocalrankvalues
K,K∗ orderedby(K ,K∗ )forspecificproductswithp=33
cp cp
(left panel), p = 72 (center panel) and p = 03 (right panel).
Bottom panels show the ImportRank-ExportRank planes re-
spectively for comparison. Data are given for year 2008. Each
country is shown by circle with its own flag as in Fig. 4.
CheiRank order of this product we find Russia, USA, In- Fig. 7. Global plane of rank indexes (K,K∗) for PageRank-
dia(seeFig.6andTable3).Thismarkstheimportanceof CheiRank (left panel) and ImportRank-ExportRank (right
the role of USA and India played in the WTN and in the panel) for N = 13847 nodes in year 2008. Each country and
product pair is represented by a gray circle. Some countries
redistribution of petroleum over nearby region countries,
are highlighted in colors: USA with black, South Korea with
e.g. around India. Also Singapore is on a local petroleum
red,Chinawithgreen,Russiawithred,Francewithyellowand
position just behind India and just before Saudi Arabia,
Brazil with orange.
see Table 3. This happens due to strong involvement of
India and Singapore in the trade redistribution flows of
petroleum while Saudi Arabia has rather restricted trade
We show the plane (K,K∗) for the global world rank-
connections strongly oriented on USA and nearby coun-
ing in logarithmic scale in 2008 in Fig. 7. The positions
tries.
of trade nodes of certain selected countries are shown by
Forelectricalmachinery72therearelessmodifications
color. We observe that the trade volume gives a higher
inthetopexportorCheiRankpositions(seeFig.6)butwe
concentration of nodes around diagonal comparing to the
observe significant broadening of positions on PageRank-
GPVM ranking. We attribute this to the symmetry of
CheiRank plane comparing to ImportRank-ExportRank.
trade volume in products.
Thus, Asian countries (China, Japan, S. Korea, Singa-
In Fig. 8 we show the distributions of top 200 ranks
pore) are located on the PageRank-CheiRank plane well
of the PageRank-CheiRank plane (zoom of left panel of
below the diagonal K = K∗ showing a significant trade Fig. 7). Among the top 30 positions of K∗ there are 8
advantages of these countries in product 72 comparing to
products of USA, 6 of China, 3 of Germany and other
Western countries (USA, Germany, France, UK).
countries with less number of products. The top position
Another product, shown in Fig. 6, is 03 Fish and fish atK∗ =1correspondstoproduct33ofRussiawhileSaudi
preparations. According to the trade volume export rank- Arabia is only at K∗ = 12 for this product. The lists of
ing the top three positions are attributed to China, Nor- all N =13847 network nodes with their K,K ,K∗ values
2
way,Thailand.However,fromCheiRankofproduct03we are available at [25].
findanotherorderwithThailand,USA,China.Thisresult
stresses again the broadness and robustness of the trade
connectionsofThailandandUSA.Asanotherexamplewe 4.3 Time evolution of ranking
note a significant improvement of Spain CheiRank posi-
tion showing its strong commercial relations for product ThetimeevolutionofindexesofproductsK ,K∗isshown
p p
03. On the other side Russia has relatively good position in Fig. 9. To obtain these data we trace PageRank and
inthetradevolumeexportof03productbutitsCheiRank CheiRank probabilities over countries and show the time
index becomes worse due to absence of broad commercial evolution of rank indexes of products K ,K∗ for top 15
p p
links for this product. rank products of year 2010. The product 33 Petroleum
Theglobaltop20positionsofindexesK,K∗,K ,Kˆ,Kˆ∗ and petroleum products remains at the top CheiRank po-
2
are given in Table 3 for year 2008. We note a signif- sition K∗ = 1 for the whole period while in PageRank
p
icant improvement of positions of Singapore and India it shows significant variations from K = 1 to 4 being
p
8 L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network
Fig. 9. Time evolution of PageRank K and CheiRank K∗
p p
indexesforyears1962to2010forcertainproductsmarkedon
the right panel side by their codes from Table 1. Top panels
Fig. 8. Top 200 global PageRank-CheiRank indexes (K,K∗)
show top 5 ranks of 2010, middle and bottom panels show
distributions for year 2008. Each country (for different prod-
ranks 6 to 10 and 11 to 15 for 2010 respectively. Colors of
ucts) is represented by its flag.
curvescorrespondtothecolorsofFig.5markingthefirstcode
digit.
at K = 4 at 1986 - 1999 when the petroleum had a low
p
price.Productswithfirstdigit7havehighranksofK but
p
especially strong variation is observed for K∗ of 72 Elec-
p
trical machinery moving from position 26 in 1962 to 4 in
2010.Amongotherindexeswithstrongvariationswenote
58 Plastic materials, 84 Clothing, 93 Special transact., 34
Gas, natural and manufactured.
Thetimeevolutionofproducts33and72ontheglobal
index plane (K,K∗) is shown in Fig. 10 for 6 countries
from Fig. 7. Thus for product 72 we see a striking im-
provementofK∗ forChinaandS.Koreathatisattheori-
ginoftheglobalimportanceimprovementofK∗ inFig.9.
p
Fortheproduct33inFig.10Russiaimprovessignificantly
its rank positions taking the top rank K∗ = 1 (see also
Table 3). Fig. 10. Time evolution of ranking of two products 72 and
ThevariationofglobalranksK,K∗withtimeisshown 33 for 6 countries of Fig. 7 shown on the global PageRank-
CheiRankplane(K,K∗).Leftandrightpanelsshowthecases
for 4 products and 10 countries in Fig. 11. For products
of 72 Electrical machinery, apparatus and appliances and 33
72,73onascaleof50yearsweseeaspectacularimprove-
petroleum and petroleum products respectively. The evolution
ment of K∗ for China, Japan, S.Korea. For the product
in time starts in 1962 (marked by cross) and ends in 2010
33 we see strong improvement of K∗ for Russia in last 15
(marked by square).
years.Itisinterestingtonotethatattheperiod1986-1992
ofcheappetroleum33USAtakesthetoppositionK∗ =1
with a significant increase of its corresponding K value.
Wethinkthatthisisaresultofpoliticaldecisiontomake
aneconomicalpressureonUSSRsincesuchanincreaseof ciency of trade network is also at the origin of significant
export of cheap price petroleum is not justified from the improvementofPageRankandCheiRankpositionsofSin-
economical view point. For the product 33 we also note gapore comparing to the trade volume ranking. Thus the
a notable improvement of K∗ of India which is visible in development of trade connections of certain countries sig-
CheiRank but not in ExportRank (see Table 3). We at- nificantly improves their Google rank positions. For the
tribute this not to a large amount of trade volume but to product 03 we note the improvement of K∗ positions of
a significant structural improvements of trade network of ChinaandArgentinawhileRussiashowsnoimprovements
India in this product. We note that the strength and effi- in this product trade for this time period.
L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network 9
Fig. 12. PageRank-CheiRank correlators κ (11) from the
p
GPVM are shown as a function of the product index p with
the corresponding symbol from Fig. 5. PageRank-CheiRank
and ImportRank-ExportRank correlators are shown by solid
and dashed lines respectively, where the global correlator κ
(9) is shown in black, the correlator for countries κ(c) (12)
is shown by red lines, the correlator for products κ(p) (12) is
shownbybluelines.Hereproductnumberpiscountedinorder
Fig. 11. Time evolution of global ranking of PageRank and ofappearanceinTable1.Thedataaregivenforyear2008with
CheiRankindexesK,K∗ forselected10countriesand4prod- N =61,N =227,N =13847.
p c
ucts. Left and right panels show K and K∗ as a function of
yearsforproducts:03Fishandfishpreparations;33Petroleum
and petroleum products; 72, Electrical machinery, apparatus
and appliances;and73Transport equipment (fromtoptobot-
tom).Inallpanelstheranksareshowninlogarithmicscalefor
10givencountries:USA,SouthKorea,China,Russia,France,
Brazil,GreatBritain,Japan,GermanyandArgentinamarked
by curve colors.
4.4 Correlation properties of PageRank and CheiRank
ThepropertiesofκcorrelatorofPageRankandCheiRank
vectors for various networks are reported in [8,10]. There
are directed networks with small or even slightly negative
values of κ, e.g. Linux Kernel or Physical Review citation
networks, or with κ∼4 for Wikipedia networks and even
larger values κ≈116 for the Twitter network.
The values of correlators defined by Eqs. (9)-(12) are
showninFigs.12,13foratypicalyear2008.Fortheglobal Fig. 13. Product PageRank-CheiRank correlation matrix
PageRank-CheiRank correlator we find κ ≈ 5.7 (9) while κp,p(cid:48) (10) for year 2008 with correlator values shown by color.
forImport-Exportprobabilitiesthecorrespondingvalueis The code indexes p and p(cid:48) of all N =61 products are shown
p
significantly larger with κ≈33.7. Thus the trade volume onxandyaxesbytheircorrespondingfirstdigit(seeTable1).
ranking with its symmetry in products gives an artificial
increase of κ by a significant factor. A similar enhance-
ment factor of Import-Export remains for correlators in uparticles,etc.and83Travelgoods,handbagsandsimilar
products κ(p) and countries κ(c) from Eq. (12) while for articles that all are related with transportation of prod-
PageRank-CheiRankweobtainamoderatecorrelatorval- ucts.
ues around unity (see Fig. 12). The PageRank-CheiRank
correlatorκ (11)forspecificproductshaverelativelylow
p
valueswithκ <1forpracticallyallproductswithp≤45 4.5 Spectrum and eigenstates of WTN Google matrix
p
(we remind that here p counts the products in the order
of their appearance in the Table 1, it is different from Above we analyzed the properties of eigenstates of G and
COMTRADE code number). G∗ atthelargesteigenvalueλ=1.However,intotalthere
The correlation matrix of products κ (10) is shown are N eigenvalues and eigenstates. The results obtained
pp(cid:48)
in Fig. 13. This matrix is asymmetric and demonstrates for the Wikipedia network [26] demonstrated that eigen-
theexistenceofrelativelyhighcorrelationsbetweenprod- states with large modulus of λ correspond to certain spe-
ucts73Transportequipment,65Textileyarn,fabrics,made cific communities of the network. Thus it is interesting to
10 L.Ermann and D.L.Shepelyansky: Google matrix analysis of the multiproduct world trade network
study the spectral properties of G for the multiproduct
WTN. The spectra of G and G∗ are shown in Fig. 14 for
year2008.ItisinterestingtonotethatforGthespectrum
showssomesimilaritieswiththoseofWikipedia(seeFig.1
in [26]). At α=1 there are 12 and 7 degenerate eigenval-
ues λ = 1 for G and G∗ respectively. Thus the spectral
gap appears only for α < 1. The dependence of IPR ξ of
eigenstates of G on Reλ is shown in Fig. 15. The results
showthatξ (cid:28)N sothattheeigenstatesarewelllocalized
on a certain group on nodes.
Fig. 15. Inverseparticipationratio(IPR)ξ ofalleigenstates
of G as a function of the real part of the corresponding eigen-
valueλfromthespectrumofFig.14.Theeigenvaluesmarked
by color circles are those from Fig. 14
Fig. 14. Spectrum of Google matrices G (left panel) and G∗
(right panel) represented in the complex plane of λ. The data
are for year 2008 with α = 1, and N = 13847, N = 227,
c
N =61. Four eigenvalues marked by colored circles are used
p
for illustration of eigenstates in Figs. 15,16.
The eigenstates ψ can be ordered by their decreas-
i
ingamplitude|ψ |givingtheeigenstateindexK withthe
i i
largest amplitude at K =1. The examples of four eigen-
i
states are shown in Fig. 16. We see that the amplitude is
mainlylocalizedonafewtopnodesinagreementofsmall
values of ξ ∼4 shown in Fig. 15. The top ten amplitudes
of these four eigenstates are shown in Table 4 with cor-
responding names of countries and products. We see that
for a given eigenstate these top ten nodes correspond to
one product clearly indicating strong links of trade be-
tween certain countries. Thus for 06 Sugar we see strong
Fig. 16. Eigenstate amplitudes |ψ | ordered by its own de-
link between geographically close Mali and Guinea with i
creasing amplitude order with index K for 4 different eigen-
furtherlinkstoUSA,Germanyetc.Inasimilarwayfor56 i (cid:80)
values of Fig. 14 (states are normalized as |ψ | = 1). Top
Fertilizers there is a groups of Latin American countries i i
panel shows two example of real eigenvalues with λ = 0.9548
Brazil, Bolivia, Paraguay linked to Argentina, Uruguay
and λ = 0.9345 while bottom panel shows two eigenval-
etc. We see a similar situation for products 57 and 52.
ues with large imaginary part with λ = 0.452+i0.775 and
These results confirm the observation established in [26]
λ=0.424+i0.467.Nodenames(country,product)fortopten
forWikipediathattheeigenstateswithlargemodulusofλ largest amplitudes of these eigenvectors are shown in Table 4.
select interesting specific network communities. We think
that it would be interesting to investigate the properties
of eigenstates in further studies. task to which we hope to return in our further investiga-
tions. However, the knowledge of the global WTN struc-
tureisanessentialbuildingblockofthistaskandwethink
4.6 Sensitivity to price variations that the presented results demonstrate that this block is
available now.
Above we established the global mathematical structure Using the knowledge of WTN structure, we illustrate
ofmultiproductWTNandpresentedresultsonitsranking herethatitallowstoobtainnontrivialresultsonsensitiv-
andspectralproperties.Suchrankingpropertiesbringnew itytopricevariationsforcertainproducts.Weconsideras
interesting and important information about the WTN. an example year 2008 and assume that the price of prod-
However, from the view point of economy it is more im- uct 33 Petroleum and petroleum products is increased by
portant to analyze the effects of crisis contamination and a relative fraction δ going from its unit value 1 to 1+δ
price variations. Such an analysis represents a complex (or δ = δ ). Then we compute the derivatives of proba-
33