Table Of ContentTopology of the World Trade Web
Ma A´ngeles Serrano and Mari´an Bogun˜a´
Departament de F´ısica Fonamental, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain
(Dated: February 1, 2008)
Economy,andconsequentlytrade,isafundamentalpartofhumansocialorganizationwhich,until
now, has not been studied within the network modelling framework. Networks are mathematical
3 tools used in the modelling of a wide variety of systems in social and natural science. Examples
0
of these networks range from metabolic and cell networks to technological webs. Here we present
0
the first empirical characterization of the world trade web, that is, the network built upon the
2
trade relationships between different countries in the world. This network displays the typical
n properties of complex networks, namely, scale-free degree distribution, the small world property, a
a highclusteringcoefficient and,inaddition,degree-degree correlation betweendifferentvertices. All
J thesepropertiesmaketheworldtradewebacomplexnetwork,whichisfarfrombeingwell-described
2 through a classical random network description.
] PACSnumbers: 89.75.-k,87.23.Ge,05.70.Ln
n
n
- The world is facing a challenging era. Social, political degree, k, is defined as the number of edges emanating
s
i andeconomicarrangementsinitiatedaftertheendofthe from a vertex, (ii) the small-world property [12], which
d
Second World War are now culminating in the recogni- states that the average path length between any pair of
.
t tion of globalization, a process which has been acceler- vertices grows logarithmically with the system size and
a
m atedbythenewtechnologicaladvances. Whenappliedto (iii)ahighclusteringcoefficient,thatis,theneighborsof
the international economic order, globalization involves a given vertex are interconnected with high probability.
-
d control of capital flow and liberalization of trade. As a In addition, degree-degree correlation has been recently
n consequence, economies around the world are becoming added to this list since it appears as a common feature
o more and more interrelated, in other words,the world is in many real-world networked systems [10, 13, 14, 15].
c
becoming a global-village [1, 2]. In this scenario, trade This correlation accounts for the probability that a ver-
[
plays a central role as one of the most important inter- tex of degree k is connected to a vertex of degree k′ and
1 action channels between countries. The relevance of the is a key issue for the correct description of the hierar-
v
international trade system goes beyond the fundamen- chical organization within the network. This correlation
5
tal exchange of goods and services. For instance, it can is found to be assortative, that is, highly connected ver-
1
also be the channel for crises spreading [3]. A good ex- tices tend to attachto other highly connectedverticesin
0
1 ample is found in the recent Asiatic crisis, which shows social networks such as scientific collaboration networks
0 how economic perturbations originated in a country can [13]; conversely, the correlation is disassortative in tech-
3 somehow propagate elsewhere in the world [4, 5]. Thus, nological networks, such as the Internet [10].
0 it seems natural to analyze the world trade system at a
Classical random graph theory, first studied by Erdo¨s
/
t global level, every country being important regardlessof and R´enyi [16, 17], does not provide a good framework
a
its size orwealth. Despite the extremely complex nature
m to fit all the above properties. This fact has posed the
oftheproblem,relevantstructuralinformationcanbeex-
questionoftheoriginoftheseanomaloustopologicalfea-
- tracted from modelling the system as a network, where
d tures. Two possible mechanisms could explain their ap-
countries are represented as vertices and trade channels
n pearance: eitherthe networkisthe resultofmacroscopic
o as links between these vertices. In this way, the global constraints, that is, the network is made ad hoc so that
c trade system can be examined under a topological point
those properties are satisfied, or there is a self-organized
: of view. This analysis will reveal complex properties
v evolutionprocessleading,inthestationarystate,tocom-
i whichcannotbeexplainedbytheclassicalrandomgraph plex structures. This idea is at the core of the preferen-
X
theory.
tialattachmentmechanisms,firstintroducedbyBaraba´si
r
a Complex networks have been the subject of an in- andAlbert[18],whereadynamicalprocessofcreationof
new links, or rewiring of the existing ones, using global
tense research activity over the last years [6, 7]. This
(or quasi global) information leads to SF networks dis-
great interest is fully justified by the extremely impor-
playing complex properties.
tantrolethatthis classofsystemsplayinmanydifferent
fields. Examples range from metabolic networks, where Amongallthestudiednetworks,thesocialandtechno-
cellfunctionalityissustainedbythenetworkstructure,to logical networks have become the paradigmatic example
technological webs, where topology determines the sys- ofcomplexnetworks. Perhaps,the reasonlies in the fact
tem ability transmitting information [8, 9, 10, 11]. The thateachtypeexemplifieshumanactivityworkingatone
termcomplex networktypically refersto networksshow- of two different levels: the cooperative level, where con-
ing the following properties: (i) scale-free (SF) degree cepts such as friendship are dominant, and the compet-
distribution, P(k) ∼ k−γ with 2 < γ ≤ 3, where the itive level, where the activity is governed by optimiza-
2
100 1
n
o
uti
b
ri
st
e di10−1 ck
v
ulati Pc(k)
m P(k )
u c out
C PFict( koifn )the form Pc(k) α k1−γ ; γ=2.6 Fit of the form ck=k−ω with ω=0.7
Random network with <k>=43
10−2 101 102 0.2 101 102
Connectivity
k
FIG.1: Cumulativein-and out-degreedistributions, Pc(kin)
FIG.2: Clusteringcoefficientofsinglecountriesasafunction
and Pc(kout), and undirected, Pc(k), corresponding to the
of their degree for the undirected version of the WTW. The
ifimtpoofrtt/heexfpoorrmtwPocr(lkd)t∼radkeγ−w1ebw.iTthheγs=olid2.6lin±e0is.1a.pIotwiesralalswo s0o.7lid±l0in.0e5is a power law fit of the form ck ∼ k−ω with ω =
shown the cumulative distribution of the equivalent random
network with the same average degree
matrices account only for a subset out of the total num-
ber of actual connections between countries. The im-
tion criteria. These two different levels will probably port/export connections that are of little relevance for a
lead to different attachment mechanisms which could be given country are not considered in the matrices A˜, al-
the originofthe assortative/disassortativedegree-degree thoughthey may be relevantto the partnersasthe sym-
correlation observed in these networks. As an example metricexport/importlinks. Infact,importsandexports
of social network working at the competitive level, we definitionscanapplytothesametradeflowdependingon
study the network of trade relationships between differ- whether originor destination is considered. This implies
ent countries in the world, hereafter referred to as the thatthecompleteadjacencymatricessatisfythesymme-
world trade web (WTW). The topological characteriza- try relation Aimp =Aexp, which can be used in order to
ij ji
tionoftheWTWisofprimaryinterestforthemodelling recover missing information from the original matrices.
of crisis propagationat the globallevel as well as for the Thus, we can write
understanding of the effects that the new liberalist poli-
1
cies have on the world trade system. Moreover,we shall Aimp = A˜imp+A˜exp (1)
showthatthe WTWis acomplexnetworksharingmany ij 1+δ(A˜imp+A˜exp),2 h ij ji i
ij ji
properties with technological networks.
In order to perform our analysis, we extracted data where δ·,· is the Kronecker delta function. In this way
we obtain an adjacency matrix where each connection
from aggregated trade statistics tables in the Interna-
is relevant, at least, to one of the two involved coun-
tional Trade Center site [19], which are based on the
tries. At this point, it is worthnoticing that we consider
COMTRADE database of the United Nations Statistics
the unweighed version of the WTW. Since the weight of
Division. These tables contain, for each country, an im-
a link can be different depending on the import/export
port and an export list detailing the forty more impor-
point of view, it is unclear how these weights should be
tant exchanged merchandises in the year 2000. Primary
assigned. After this symmetrization procedure, we ob-
and secondary markets are also reported for each prod-
tain a directed network with 179 vertices representing
uct. If we consider imports as in-degrees and exports as
countries and 7510 directed links representing commer-
out-degrees,itispossibletoconstructadirectednetwork
cial channels among them. The average degree of this
whereverticesrepresentcountriesanddirectedlinksrep-
network is hk i=hk i=30.9.
resent the import/export relations between them. The in out
The questionthatfirstarisesrefersto the directedna-
fact that the number of merchandises is bounded is, a
ture of the WTW. In fact, the in- and the out-degree
priori, a limitation for the analysis. However, it is pos-
of a given vertex are random quantities which may be
sible to overcome this problem taking advantage of the
correlated. A complete description should involve the
symmetry between in and out degrees. Let A˜imp and
ij knowledgeofthejointprobabilityp(k ,k ). Thisfunc-
A˜eijxpbetheimport/exportadjacencymatricescalculated tionisoftendifficulttoobtainalthouginhreoluetvantinforma-
from the import/export databases. Each adjacency ma- tioncanbeextractedfromthecorrelationcoefficientr =
trix is defined so that A˜ = 1 if the country i imports (hk k i−hki2)/σ σ ,whereσ /σ arethein/out
ij in out in out in out
from/exportstothecountryj andzerootherwise. These standard deviations (notice that hk i = hk i = hki).
in out
3
For the WTW this coefficient is r = 0.91, pointing to
a strong similarity in the number of in and out connec-
tions. However, not all these connections run in both
102
directions, that is, the fact that country A imports from
country B does not necessary imply that country B im-
ports from country A. In order to quantify this effect we >
)
compute the reciprocity of the network, defined as the k
(
n
fraction of links pointing simultaneously to both ends of kn
<
thelink. Inourcasethereciprocityis0.61. Theseresults
suggest that, actually, the WTW may be thought of as
<k (k)>
nn
an undirected network without losing relevant topologi- <k (k )>
nn in
cal information. The average degree for the undirected <knn(kout)>
version of the WTW is hki=43. Fit of the form <knn(k)>=k−νk , νk=0.5
One of the most important topological properties of a 101 102
network is the degree distribution, P(k). This quantity k
measures the probability of a randomly chosen vertex to
have k connections to other vertices. In our case, due to FIG.3: Averagein(out,undirected)nearestneighborsdegree
asafunction ofthein(out,undirected)degreeofthevertex.
thedirectednatureofthenetwork,wehavetodistinguish
betweenin-andout-degreedistributions. Fig.1showsthe The solid line is a fit of the form hknn(k)i∼k−νk with νk =
in, out, and undirected cumulative distributions for the 0.5±0.05. Thedashedlineisthetheoreticalvalueoft2hehknni
correspondingtoanuncorrelatednetwork,thatis,hk i/hki=
′
WTW, defined as P (k)≡ P(k ). In allcases,the
c Pk′=k 70.18.
cumulative distribution shows a flat approachto the ori-
gin, indicating the presence of a maximum in P(k), at
k ∼ 20, and, at this respect, similar to the Erdo¨s-R´enyi Iceland, which are between the top ten wealthier coun-
network. However, for k > 20 the cumulative distribu- triesbutonlyhave56and24tradechannelsrespectively.
tionisfollowedbyapowerlawdecayPc(k)∼k1−γ,with For the second case Brazil, China or Russia are typical
γ ≈ 2.6, showing a strong deviation from the exponen- examples.
tial tail predicted by the classical random graph theory. Thanks to a number of recent studies, it is becom-
The exponent γ is found to be within the range defined ing more and more evident that real networks are not
bymanyothercomplexnetworksand,thus,wecanstate completely random but they are organized according to
that the WTW belongs to the recently identified class a hierarchical structure [9, 10, 15, 21]. This hierarchy
of SF networks [6, 7]. The SF property implies an ex- is usually analyzed by means of the local clustering co-
tremely high level of degree heterogeneity. Indeed, the efficient and the degree-degree correlation. The cluster-
second moment of the degree distribution, hk2i, diverges ing coefficient of the vertex i, of degree k , is defined as
i
in the thermodynamic limit for any SF network. c ≡2n /k (k −1), where n is the number of neighbors
i i i i i
It may be surprising that, in fact, the SF region does ofithatareinterconnected. Ifhierarchywasnotpresent
not extend to the whole degree domain and that, for in the system, the local clustering coefficient should be
small values of the degree, the distribution is similar to a random quantity independent of any other property.
the Erdo¨s-R´enyi network. In fact, the underlying pref- Fig. 2 shows the local clustering coefficient of the undi-
erential attachment mechanisms could differ depending rected WTW as a function of the vertex’s degree. As it
on the particular political and economic situation of a is clearly seen, this function has a strong dependence on
country. Low-degree countries, most of which turn out thevertex’sdegree,withapowerlawbehaviorck ∼k−ω,
tobethepoorest,arebasicallyconstrainedtosubsistence with ω = 0.7±0.05. The clustering coefficient averaged
tradeflowsand,therefore,preferentialattachmentmech- over the whole network is C = 0.65, greater by a factor
anisms could not hold. As expected, there exists a pos- 2.7thanthevaluecorrespondingtoarandomnetworkof
itive correlation between the number of trade channels the same size.
of a country and its wealth, measured by the per capita Hierarchy is also reflected on the degree-degree corre-
′
Gross Domestic Product (GDP) [20]. This correlation is lationthroughtheconditionalprobabilityP(k|k ),which
′
found to be high, 0.65, which means that, indeed, most measures the probability of a vertex of degree k to be
low-connected countries are poor countries –Angola, So- linked to a vertex of degree k. Again, this function is
malia, Rwanda, Cambodia...– and most high-connected difficult to measuredue to statistical fluctuations. Inor-
countries are rich countries –the USA, Japan, Germany der to characterize this correlation, it is more useful to
and the UK, for example. However, there also exists a workwiththeaveragenearestneighborsdegree(ANND),
′ ′
significantnumberofcasesinthe reversalsituation,that defined as hk (k)i ≡ k P(k |k) [14]. For uncor-
is, low per capita GDP countries with a large number related netwonrkns, this fuPnckt′ion reads hk i = hk2i/hki,
nn
of connections and high per capita GDP countries with independent of k. However, recent studies have revealed
a relatively low number of trade channels. A germane thatalmostallrealnetworksshowdegree-degreecorrela-
example for the first circumstance is that of Norway or tion [10, 13] which translates into a k dependence in the
4
similartothecorrespondingrandomnetworkofthesame
TABLE I:
size and average degree.
size hki hdi C γ ω ν
k In conclusion, this first approach to the topology of
WTW 179 43 1.8 0.65 2.6 0.7 0.5 theWTWpointsoutsomepreviouslyunnoticedfeatures
Internet 5287 3.8 3.7 0.24 2.2 0.75 0.5 which are of primary importance in the understanding
RG 179 43 1.73 0.24 – 0 0 of the new international order. Our research suggests
that the network’s evolution is guided by collective phe-
nomena, and that self-organization plays a crucial role
ANND. This correlation can be assortative or disassor- in structuring the WTW scale-free inhomogeneities and
tative depending on whether the ANND is an increasing its hierarchical architecture. It remains an open ques-
or decreasing function of the degree. Fig. 3 reports the tion if these properties could also be made apparent at
ANND for the directed and undirected versions of the other different scales, for instance, in the trade relations
WTW. It can be observed a clear dependency on the between regions, cities or even individuals. At the coun-
vertex’sdegree,with a powerlaw decay hknn(k)i∼k−νk try level, the activity is driven by competition. In the
withν =0.5±0.05. ThisresultmeansthattheWTWis same way as it occurs with the Internet, optimization
k
a disassortativenetworkwhere highly connected vertices criteria are applied to local decisions made by the in-
tends to connect to poorly connected vertices. This re- dividual vertices. Resolutions are based on information
sult, togetherwiththe scalinglawc ∼k−ω revealsahi- that is biased toward the more visible vertices, and may
k
erarchicalarchitectureofhighlyinterconnectedcountries be influenced by geographical or other convenience con-
that belong to influential areas which, in turn, connect straints. The findings in this paper may leadto consider
to other influential areas through hubs. thatthereexistunderlyinggrowingmechanismscommon
Surprisingly, these results point to a high similarity to all competitive systems, characterized by disassorta-
between the WTW and the Internet. Indeed, the Inter- tive associations, and such mechanisms may differ from
net is a SF network, with a critical exponent γ = 2.2, theevolutionaryprocessesinsocialcooperativenetworks,
which is also organized in a hierarchical fashion. The characterizedby assortative associations.
functional behavior found for the clustering coefficient Furthermodelling effortsmustbe done foracquiringa
and the ANND is a power law decay as a function of morerealisticrepresentationofthe WTW,whereinward
the degree with exponents ωint = 0.75 and νk = 0.5 flows differ from outward flows and where their weights
[10], exponents that turn out to be very similar to the depend on the exchanged quantities. It is also essential
ones reported here for the WTW. In some sense, these todofurtherresearchontheunderlyingformationmech-
results arenot surprisingsince both are competitive sys- anisms and on the dynamic processes running on top of
tems evolving in a quasi free market and, in both cases, the WTW, such as economic crises spreading.
there exists, for instance, a geographic limitation that
increases the connection costs and, thus, acts as a con-
straint in the optimization process of each vertex. Table
Acknowledgments
I presents a summary of the main characteristics of the
WTW,theInternet[10]andarandomgraphofthesame
size and averagedegree as the WTW. Acknowledgements are due to Romualdo Pastor-
As a final remark, the average path length, defined Satorras for helpful discussion and advices. This work
as the average of the shortest distances between all the has been partially supported by the European commis-
pairs of vertices, is hdi=1.8, which, in this case, is very sion FET Open project COSIN IST-2001-33555.
[1] P. Krugman, The Accidental Theorist, and other for Economics, Washington D. C., 1998).
Dispatches from the Dismal Science, (W. W. Nor- [6] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. 74, 47
ton&Company,New York,1998). (2002).
[2] S. Amin, Capitalism in the Age of Globalization: The [7] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51,
ManagementofContemporary Society,(ZedBooks,Lon- 1079 (2002).
don,1997). [8] H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai and A.-L.
[3] R. Glick and A. Rose, Journal of International Money Barab´asi, Nature 407, 651 (2000).
and Finance 18, 603-617 (1999). [9] E.Ravasz,A.L.Somera,D.A.Mongru,Z.N.Oltvaiand
[4] M. Noland, L-G Liu, S. Robinson and Z. Wang, Global A.-L. Barab´asi, Science 297, 1551 (2002).
Economics Effects of the Asian Currency Devaluations, [10] A. V´azquez, R. Pastor-Satorras, and A. Vespignani,
(International Institute for Economics, Washington D. Phys. Rev.E 65, 066130 (2002).
C., 1998). [11] R.Albert,H.JeongandA.-L.Barab´asi,Nature401,130
[5] M. Goldstein, The Asian Financial Crisis: Causes, (1999).
CuresandSystemicImplications,(InternationalInstitute [12] D.J.WattsandS.H.Strogatz, Nature393,440(1998).
5
[13] M. E. J. Newman, Phys.Rev.Lett. 89, 208701 (2002). [18] A.-L. Barab´asi and R.Albert, Science 286, 509 (1999).
[14] A. Pastor-Satorras, R. V´azquez, and A. Vespignani, [19] Import/Exportdatabasesarepubliclyavalableatthead-
Phys.Rev.Lett. 87, 258701 (2001). dress http://www.intracen.org/menus/countries.htm
[15] K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett 88, [20] Grossdomesticproductdataarepubliclyavalableatthe
108701 (2002). address http://www.tswoam.co.uk/world
[16] P. Erd¨os and P. R´enyi, Publicationes Mathematicae 6, [21] E.Ravasz,andA.-L.Barab´asi,HierarchicalOrganization
290-297 (1960). in Complex Networks, e-print cond-mat/0206130.
[17] P. Erd¨os and P. R´enyi, Publ. Math. Inst. Hung. Acad.
Sci. 5, 17 (1960).
