Multi-cultural Wikipedia mining of geopolitics interactions leveraging reduced Google matrix analysis

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Multi-cultural Wikipedia mining of geopolitics

interactions leveraging reduced Google matrix analysis

Klaus Frahm, Samer El Zant, Katia Jaffrès-Runser, Dima Shepelyansky

To cite this version:

Klaus Frahm, Samer El Zant, Katia Jaffrès-Runser, Dima Shepelyansky. Multi-cultural Wikipedia

mining of geopolitics interactions leveraging reduced Google matrix analysis. Physics Letters A,

Elsevier, 2017, 381 (33), pp.2677-2685. �10.1016/j.physleta.2017.06.021�. �hal-01422586�

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This is an author’s version published in: http://oatao.univ-toulouse.fr/2 2156

To cite this version:

Frahm, Klaus and El Zant, Samer and Jaffrès-Runser, Katia and Shepelyansky, Dima Multi-cultural Wikipedia mining of geopolitics interactions leveraging reduced Google matrix analysis. (2017) Physics Letters A, 381 (33).

2677-2685. ISSN 0375-9601.

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https://doi.org/10.1016/j.physleta.2017.06.021

Open Archive Toulouse Archive Ouverte

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Multi-cultural Wikipedia mining of geopolitics interactions leveraging reduced Google matrix analysis

KlausM. Frahm^a, Samer El Zant^b,Katia Jaffrès-Runser^b,Dima L. Shepelyansky^a^,^∗

aLaboratoiredePhysiqueThéoriqueduCNRS,IRSAMC,UniversitédeToulouse,UPS,F-31062Toulouse,France bInstitutdeRechercheenInformatiquedeToulouse,UniversitédeToulouse,INPT,31061Toulouse,France

a b s t ra c t

Keywords:

Markovchains Googlematrix Wikipedianetwork

Geopoliticalinteractionsofcountries

Geopoliticsfocusesonpoliticalpowerinrelationtogeographicspace.Interactionsamongworldcountries havebeenwidelystudiedatvariousscales,observingeconomicexchanges,worldhistoryorinternational politics amongothers.Thiswork exhibits the potential ofWikipediamining for suchstudies.Indeed, Wikipediastoresvaluableﬁne-graineddependenciesamongcountriesbylinkingwebpagestogetherfor diversetypesofinteractions(notonlyrelatedtoeconomical,politicalorhistoricalfacts).Wemineherein the Wikipedianetworks of severallanguage editions usingthe recently proposed methodofreduced Googlematrixanalysis.Thisapproachallowstoestablishdirectand hiddenlinksbetweenasubsetof nodes that belongtoamuch largerdirected network.Ourstudy concentrateson40 major countries chosenworldwide. Ouraimisto offeramulticultural perspectiveontheir interactionsby comparing networks extracted fromﬁvedifferent Wikipedialanguage editions,emphasizing English,Russianand Arabicones.We demonstratethatthisapproachallowstorecovermeaningfuldirectand hiddenlinks amongthe40countriesofinterest.

1. Introduction

Political and economic interactions between regions of the worldhavealwaysbeenofutmostinteresttomeasureandpredict theirrelativeinﬂuence.Suchstudiesbelongtotheﬁeldofgeopoli- ticsthatfocusesonpoliticalpowerinrelationtogeographicspace.

Interactionsamong world countries havebeen widely studied at variousscales(worldwide,continentalorregional)usingdifferent types of information. Studies are driven by observing economic exchanges,socialchanges,history,internationalpoliticsanddiplo- macy among others [1,2]. The major ﬁnding of this paper is to showthatmeaningfulworldwideinteractionscanbeautomatically extractedfromtheglobalandfreeonlineEncyclopaediaWikipedia [3]for a given set of countries. All information gatheredin this collaborativeknowledgebasecanbeleveragedtoprovideapicture ofcountriesrelationships,fosteringanewframeworkforthorough geopoliticsstudies.

Wikipedia has become the largest open source of knowledge beingclosetoEncyclopaediaBritannica [4]by theaccuracy ofits

* Correspondingauthor.

E-mailaddresses:frahm@irsamc.ups-tlse.fr(K.M. Frahm),

scientific entries [5] andovercoming the latter by the enormous quantityofavailableinformation.Adetailedanalysisofstrongand weak features of Wikipedia is given at [6,7]. Wikipedia articles make citations to each other,providing a direct relationship be- tweenwebpagesandtopics.Assuch,Wikipediageneratesalarger directednetwork ofarticletitleswitharatherclearmeaning.For thesereasons, it is interesting to apply algorithms developed for searchenginesofWorldWideWeb(WWW)suchasthePageRank algorithm[8](seealso[9]), toanalyzetherankingpropertiesand relationsbetweenWikipediaarticles.Forvariouslanguageeditions of Wikipediait was shownthat the PageRank vector produces a reliable ranking of historical figures over 35 centuries of human history[10–14]andasolidWikipediarankingofworlduniversities (WRWU)[10,15].IthasbeenshownthattheWikipediarankingof historicalfiguresisinagoodagreementwiththewell-knownHart ranking [16], while the WRWUis in a goodagreement with the ShanghaiAcademicrankingofworlduniversities[17].

Atpresentdirected networksofrealsystemscanbeverylarge (about 4.2 million articles for the English Wikipedia edition in 2013[13] or3.5 billionweb pages(calledalso nodes)forapub- liclyaccessiblewebcrawlthatwasgatheredbytheCommonCrawl Foundation in2012 [18]). Forsome studies, one might be inter- estedonlyintheparticularinteractionsbetweenaverysmallsub- set of nodes compared to the full network size. For instance, in samer.elzant@enseeiht.fr (S. ElZant),kjr@enseeiht.fr (K. Jaffrès-Runser),

dima@irsamc.ups-tlse.fr (D.L. Shepelyansky).

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Fig. 1. Geographicaldistributionofthe 40selected countries.Color codegroups countriesinto7sets:orange(OC)forEnglishspeakingcountries,blue(BC)forfor- merSovietunionones,red(RC)forEuropeanones,green(GC)forLatinAmerican ones,yellow(YC)forMiddleEasternones,purple(PUC)forNorth-EastAsianones andﬁnallypink(PIC)forSouth-Easterncountries(seecolorsandcountrynamesin Table 1;othercountriesareshowninblack).(Forinterpretationofthereferencesto colorinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

thispaper, we are interested in capturingthe interactions ofthe 40 countries represented in Fig. 1 using the networks extracted fromﬁveWikipedia language editions covering afew millions of articleseach.However,letusassumethatthereisaratherimpor- tantperson(having hisownWikipediaarticlecorresponding toa nodeC)whowasbornincountry A andworkedthemainpartof hislife incountry B;therefore A and B mayhave linkstoC (in eitherdirection)andthus theremaybe an indirectlink between thetwonodesA andB viathenodeC (orothernodes).Inprevi- ousworks, a solutionto thisgeneralproblemhas beenproposed in[19,20]by deﬁningthereducedGooglematrixtheory.Mainel- ementsofReducedGooglematrix GR willbepresentednext, but inafewwords,itcapturesina40-by-40Perron–Frobeniusmatrix thefullcontribution ofdirect andindirectinteractions happening in the full Google matrix between the 40 nodes of interest (we tooktop 40countriesofPageRankvectorofEnWiki). Elementsof reducedmatrixG_R(i,j)canbeinterpretedastheprobabilityfora randomsurferstarting atwebpage j toarriveinwebpage i using directandindirectinteractions.Indirectinteractionsrefertopaths composed inpart ofwebpages differentfromthe 40 onesof interest.EvenmoreinterestinganduniquetoreducedGooglematrix theory,we showherethat intermediate computation stepsof GR offeradecompositionof GRintomatricesthat clearlydistinguish directfromindirectinteractions. As such,it ispossible toextract the probability for an indirect interaction betweentwo nodes to happen.

ReducedGooglematrixtheoryisaperfectcandidateforanalyz- ingthedirectandindirectinteractionsbetweencountriesselected worldwide. Inthispaper,we extractfrom GR andits decomposi- tion intodirect and indirectmatrices ofselected subset network of N_r=40 countries. The Google matrix of this subset network of Nr nodes is computed taking into account direct and hidden (i.e. indirect) directed links. More specifically, we deduce a fine- grainedclassification ofcountriesthat captures what we callthe hiddenfriendsandhiddenfollowersofagivencountry.Thestructure of these graphs provides relevant social information: communi- ties of countries withstrong tiescan be clearly exhibited while countriesactingasbridges arepresentaswell.Thisismainlythe case for the hidden interactions networks of friends (or followers)that offer newinformationcompared to thedirect networks offriends(orfollowers)whosetopologyismainlyenforcedbytop PageRank countries. The mathematical procedure of the reduced

Table 1

Listofnamesof40selectedcountrieswithPageRankK,CheiRank K^∗forEnWiki, ArWikiandRuWiki,orderedbyincreasingPageRankofEnWikiedition.Fig. 1gives colorcorrespondencedetails.AWikipediaarticlewithcountrynamerepresentsone nodeofthewholenetworkwithNnodes,e.g.https://en.wikipedia.org/wiki/France for“France”inEnWiki.(Forinterpretationofthereferencestocolorinthistable, thereaderisreferredtothewebversionofthisarticle.)

Country EnWiki ArWiki RuWiki

K K^∗ K K^∗ K K^∗

United States (US) OC 1 9 1 5 2 27

France (FR) RC 2 19 3 31 3 14

United Kingdom (UK) OC 3 25 6 20 7 3

Germany (DE) RC 4 33 8 14 4 24

Canada (CA) OC 5 26 13 19 12 26

India (IN) PIC 6 23 9 25 13 8

Australia (AU) OC 7 35 16 22 18 12

Italy (IT) RC 8 15 5 1 6 32

Japan (JP) PUC 9 4 11 9 11 7

China (CN) PUC 10 8 12 17 9 21

Russia (RU) BC 11 6 7 2 1 2

Spain (ES) RC 12 30 4 8 8 15

Poland (PL) RC 13 12 26 32 10 17

The Netherlands (NL) RC 14 37 18 33 15 31

Iran (IR) YC 15 2 14 15 30 22

Brazil (BR) GC 16 3 21 26 20 1

Sweden (SE) RC 17 22 22 7 19 5

New Zealand (NZ) OC 18 28 34 24 36 4

Mexico (MX) GC 19 40 23 38 22 37

Switzerland (CH) RC 20 38 20 34 16 18

Norway (NO) RC 21 32 35 16 27 11

Romania (RO) RC 22 10 19 6 32 36

Turkey (TR) YC 23 7 15 13 21 38

South Africa (ZA) OC 24 24 29 39 35 20

Belgium (BE) RC 25 18 27 37 29 30

Austria (AT) RC 26 39 28 28 14 28

Greece (GR) RC 27 21 10 36 25 25

Argentina (AR) GC 28 1 32 29 33 23

Philippines (PH) PIC 29 17 36 21 39 33

Portugal (PT) RC 30 36 24 12 17 9

Pakistan (PK) PUC 31 5 25 35 37 29

Denmark (DK) RC 32 16 33 10 31 19

Israel (IL) YC 33 20 17 18 28 6

Finland (FI) RC 34 14 37 4 26 16

Egypt (EG) YC 35 31 2 3 24 39

Indonesia (ID) PIC 36 13 31 11 34 10

Hungary (HU) RC 37 11 40 40 23 40

Taiwan (TW) PUC 38 27 39 27 40 34

South Korea (KR) PUC 39 34 38 30 38 35

Ukraine (UA) BC 40 29 30 23 5 13

GR matrixconstructionfor Nr nodesisdescribed indetailinSec- tion2.

The networksof GR directandhiddeninteractions canbecal- culated for different Wikipedia language editions. In this paper, reduced Google matrix analysisis applied to the same setof 40 countries on networks representing ﬁve different Wikipedia editions:English(EnWiki),Arabic (ArWiki),Russian(RuWiki), French (FrWiki) andGerman(DeWiki) editions. Wetake for analysisthe top 40 countries according to the EnWiki PageRank. Wikipedia languageeditionsareusuallymodiﬁedbyauthorswhomainlybe- long to the regionassociated withthis language.Thus ourstudy showstheimpactofthisculturalbiaswhencomparingdirectand hidden networks of friends (or followers) among different language editions. We show that part ofthe interactions are cross- cultural whileothersare clearly biasedby the culture ofthe au- thors.

InSection2weintroducethemainelementsofreducedGoogle matrix theory,Section 3describes GR calculated for40countries andforﬁvedifferentWikipediaeditions.Speciﬁcemphasisisgiven totheverydifferentEnglish,ArabicandRussianeditions.Networks offriendsandfollowersfordirectandhiddeninteractionmatrices arecreatedanddiscussedinSection4,andconclusionisdrawnin Section5.

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Fig. 2.Position of countries in the local(K,K^∗)plane of the reduced network of 40 countries in the EnWiki (left), ArWiki (middle) and RuWiki (right) networks.

2. ReducedGooglematrixtheory 2.1.Googlematrix

Itis convenient to describe thenetwork of N Wikipediaarti- cles: a network node is givenby the article name,all nodes are numbered,somenodescorrespondtoarticleswithcountrynames, seeTable 1;thenumberofcountriesismuchsmallerthantheto- talnumberofarticles N.ThentheGoogle matrixGisconstructed fromtheadjacencymatrix Ai j withelements 1 ifarticle(node) j pointstoarticle(node)iandzerootherwise.Inthiscase,elements oftheGooglematrixtakethestandardform[8,9]

G_{i j}=αS_{i j}+(1−α)/N , (1)

where S isthe matrixofMarkovtransitionswithelements Si j= A_{i j}/k_out(j), k_out(j)=_N

i=¹A_{i j}=^{0 being} ^the ^node ^j ^out-degree (number ofoutgoing links) and with Si j=¹/N if j has no out- goinglinks(danglingnode).Here0<α<1 isthedampingfactor whichfora random surferdetermines the probability (1−α) to jumptoanynode;belowwe use α=⁰.85.Therighteigenvectors ψi(j)ofG aredeﬁnedby:

j

G_{j j}ψ_i(j)=λ_iψ_i(j) . (2)

The PageRank eigenvector P(j)= ψi=⁰(j) corresponds to the largest eigenvalue λi=⁰=¹ ^[8,9]. Ît ^has ^positive êlements ^which give the probability to find a random surfer on a given node in thestationarylongtimelimitoftheMarkovprocess.Allnodescan beordered by a monotonicallydecreasing probability P(Ki) with thehighestprobabilityat K=^1.^Theîndex ^K îs^the^PageRankîn- dex.ThePageRankvectoriscomputedbythePageRankalgorithm withiterativemultiplication ofinitialrandomvector by G matrix [8,9,14]. Left eigenvectors are biorthogonal to right eigenvectors ofdifferent eigenvalues. The left eigenvector for λ=^{1 has} îden- tical(unit) entriesduetothecolumnsumnormalizationofG.In thefollowingwe usethe notationsψ_L^T andψR forleft andright eigenvectors,respectively. Notation T stands forvector ormatrix transposition.

Inaddition tothe matrix G it is usefultointroduce a Google matrix G^∗ constructed from the adjacency matrix of the same network but with inverted direction of all links [21]. The vector P^∗(K^∗) is called the CheiRank vector [10,21] and the index numbering nodes in order of monotonic decrease of probability P^∗ is notedasCheiRankindex K^∗.Thus, nodeswithmanyingo- ing (or outgoing) links have small values of K =¹,2,3... (or of

K^∗=¹,2,3,...)[9,14].Weshowthedistributionofselectedcoun- triesonthePageRank–ChiRankplane(K,K^∗)inFig. 2.

2.2. ReducedGooglematrix

We construct the reduced Google matrix fora certain subset of Nr selectednodes fromthe globalWikipedianetwork with N nodes(Nr^N).Âsâ^subset^we ^choose^top⁴⁰^countries^with^the largestPageRankprobabilitiesforEnWikinetwork(thenamesare given in Table 1). The reduced Google matrix GR is constructed onthe mathematicalbasisdescribed below.The mainelement of thisconstructionistokeepthesamePageRankprobabilitiesofN_r nodesasintheglobalnetwork(uptoafixedmultipliercoefficient) andtotakeintoaccountallindirectlinksbetweenNr nodescou- pled by transitions via N−^Nr nodesof the globalnetwork (see also[20]).

Let G be a typical Google matrix (1) for a network with N nodes such that G_{i j}≥^{0 and} ^the ^column ^sum normalization

N

i=¹Gi j=^{1 is}^veriﬁed.^We ^consider^asub-network withNr<N nodes,called“reducednetwork”.Inthiscasewecanwrite G ina blockform:

G=

Grr Grs

G_sr G_ss

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where the index“r” refers to the nodes ofthe reduced network and“s”totheotherNs=^N−^Nrnodeswhichformacomplemen- tarynetworkwhichwewillcallthe“scatteringnetwork”.ThusG_rr is given by the direct links between selected Nr nodes, Gss describes links between all other N−^Nr nodes, Grs and Gsr give linksbetweenthesetwoparts.PageRankvectorofthefullnetwork isgivenby:

P= Pr

Ps

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which satisfiesthe equation GP=^P ôrⁱⁿ ôther ^words ^P îs ^the right eigenvector of G for the unit eigenvalue. This eigenvalue equationreadsinblocknotations:

(1−^Grr)Pr−^GrsPs=⁰, (5)

−^GsrPr+(1−^Gss)Ps=⁰. (6) Here1istheunitmatrixofcorrespondingsizeNrorNs.Assuming thatthe matrix1−^Gss isnotsingular,i.e.all eigenvalues Gssare strictlysmallerthanunity(inmodulus),weobtainfrom(6)that P_s=(1−^Gss)⁻¹G_srP_r (7)