Hric, Darko; Kaski, Kimmo; Kivelä, Mikko Stochastic block model reveals maps of citation patterns and their evolution in time

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Journal of Informetrics

DOI:

10.1016/j.joi.2018.05.004 Published: 01/08/2018

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Please cite the original version:

Hric, D., Kaski, K., & Kivelä, M. (2018). Stochastic block model reveals maps of citation patterns and their

evolution in time. Journal of Informetrics, 12(3), 757-783. https://doi.org/10.1016/j.joi.2018.05.004

ContentslistsavailableatScienceDirect

Journal of Informetrics

jo u r n al hom e p ag e :w w w . e l s e v i e r . c o m / l o c a t e / j o i

Regular article

Stochastic block model reveals maps of citation patterns and their evolution in time

Darko Hric, Kimmo Kaski, Mikko Kivelä

DepartmentofComputerScience,AaltoUniversitySchoolofScience,P.O.Box12200,FI-00076,Finland

a rt i c l e i n f o

Articlehistory:

Received2May2017

Receivedinrevisedform30May2018 Accepted30May2018

Keywords:

Webofscience Citationnetworks Evolutionofscience Stochasticblockmodel

a b s t ra c t

Inthisstudywemapoutthelarge-scalestructureofcitationnetworksofsciencejournals andfollowtheirevolutionintimebyusingstochasticblockmodels(SBMs).TheSBMfit- tingproceduresareprincipledmethodsthatcanbeusedtofindhierarchicalgroupingof journalsthatshowsimilarincomingandoutgoingcitationspatterns.Thesemethodswork directlyonthecitationnetworkwithouttheneedtoconstructauxiliarynetworksbasedon similarityofnodes.WefittheSBMstothenetworksofjournalswehaveconstructedfrom thedatasetofaround630millioncitationsandfindavarietyofdifferenttypesofgroups, suchascommunities,bridges,sources,andsinks.Inadditionweusearecentgeneralization ofSBMstodeterminehowmuchamanuallycuratedclassificationofjournalsintosubfields ofscienceisrelatedtothegroupstructureofthejournalnetworkandhowthisrelationship changesintime.TheSBMmethodtriestofindanetworkofblocksthatisthebesthigh-level representationofthenetworkofjournals,andweillustratehowtheseblocknetworks(at variouslevelsofresolution)canbeusedasmapsofscience.

©2018TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Theprocessofcreatingscientificknowledgereliesonpublicationsthatareoftenstoredandarchived,withtheprimary purposeofpreservinganddistributingtheknowledgeobtainedthroughresearch.Thesearchivescanalsobeusedtostudy thesciencemakingitself,forexample,byextractinginformationofcollaborations,citations,orkeywordsofthepublished articles.Researchinthisfieldhasafairlylongandrichhistorywithwiderangeofresearchtopics,liketheassessment andpredictionofperformanceandqualityofindividualpapers,researchers,institutions,journals,fields,andevencountries (Althouse,West,Bergstrom,&Bergstrom,2009;Lehmann,Jackson,&Lautrup,2008;Nerur,Sikora,Mangalaraj,&Balijepally, 2005),aswellasidentificationofvariouslargescalestructuresofscience(Boyack&Klavans,2014;Carpenter&Narin, 1973;deSollaPrice,1965;Leydesdorff,Carley,&Rafols,2013;Small,1999;Waltman,vanEck,&Noyons,2010),journal classification(Janssens,Zhang,Moor,&Glänzel,2009;Leydesdorff,2006;Wang&Waltman,2016;Zhang,Liu,Janssens, Liang,&Glänzel,2010),followingresearchtrends(Chen,2013;Persson,2010;Porter&Rafols,2009),andrecognizingthe emergingfieldsorresearchers(Cozzensetal.,2010;Lambiotte&Panzarasa,2009;Shibata,Kajikawa,Takeda,Sakata,&

Matsushima,2011;Small,Boyack,&Klavans,2014;Small&Greenlee,1989).

Bibliographicdatabases,likeWebofScience,Scopus,andGoogleScholar,storemetadataofscientificpublications,which canbeusedtoanalysesciencemakingatalllevels,fromlargescalestructuretoperformanceofindividualpapers.Thenumber

∗ Correspondingauthor.

E-mailaddresses:darko.hric@aalto.fi(D.Hric),kimmo.kaski@aalto.fi(K.Kaski),mikko.kivela@aalto.fi(M.Kivelä).

https://doi.org/10.1016/j.joi.2018.05.004

1751-1577/©2018TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/

4.0/).

andreliablyisbecomingevenmorechallengingasthenetworksunderstudycontinuetogrow.

Conventionaldataanalysistools,suchasclusteringordimensionreductionmethods,canbeusedtosimplifythedata aboutthecomplexrelationshipsbetweenthedataentities.Representingtheentitiesasvectorsoftheirfeaturesisacommon andpracticalabstractionthatallowstheuseofclusteringmethodsinthespaceoffeatures,inwhichthemostsimilarentities aregroupedbasedonthesimilarityoftheusedfeatures.Thesevectorscancontaincitationinformationbetweentheentities, andonecandefinesimilaritymeasures,likebibliographiccoupling,co-citation,distancebetweencitationvectors(Euclidean, cosine,Jaccard,etc.),andcorrelationcoefficientsbetweenthecitationvectorsorpublicationtexts(abstracts,keywords,etc.) (Boyacketal.,2005;Carpenter&Narin,1973;Janssensetal.,2009;Kessler,1963;Leydesdorff&Rafols,2012;Marshakova, 1973;Small,1973;Wang&Koopman,2017).

Thedataofscientificprogresscanbeanalysedwithavarietyofmethodsoncethedatahasbeenpreprocessed.The dimensionalityreductiontechniquesprojectthevectorsintothemostsignificantsubspacesrevealinggroupsofcorrelated entities(multidimensionalscaling,factoranalysis,etc.)(Leydesdorffetal.,2013;Small,1999).Classicalclusteringtechniques, e.g.hierarchicalclusteringandk-means,operateonthefullspaceoffeatures,andprovideclustersofsimilarentities,based onimplicitlyorexplicitlydefinedsimilaritymeasureordistance(Boyacketal.,2005;Modha&Spangler,2000;Punj&

Stewart,1983;Silva,Rodrigues,Oliveira,da,&Costa,2013;Wang&Koopman,2017).Thefactoranalysisappliedseparately tothecitingandciteddirectionofthecompletecitationmatrix,enablesfurtherspecializationintothetypesofgroupsit finds,sincebyusingonlyonedirectionatatime,itdetectsgroupsbasedonpastandfuturecitations,separately(Leydesdorff

&Rafols,2009).Theco-citationandbibliographiccouplingusesimilaritiesincitationsinthefutureandpastrespectively, andthusprovideaseparationnaturally(Weinberg,1974).Theresultsofthistypeofanalysisdependsonthepreprocessing stepofconstructingthedatavectorsandsimilarities,andgreatcareisneededininterpretingtheresults(Boyacketal.,2005;

Gläser,Glänzel,&Scharnhorst,2017;vanEck&Waltman,2009).

Thebibliometricdatacanalsobeanalysedbyconstructingnetworks—suchasthecitationnetworkbetweenjournals—and directlyfindingstructureinthemusingthegeneralpurposetoolsforanalysingthenetworks.Thedevelopmentofsuch methodswithinnetworksciencehasexplodedsincemassiveamountsofdataonlargevarietyofnetworks—suchason socialandtransportationnetworks—havebecomeavailable(Boccaletti,Latora,Moreno,Chavez,&Hwang,2006;Newman, 2003).Aprominentwayoffindingstructureincitationnetworksusingthesemethodsistoinvestigatenetworkclustersor communities(Fortunato,2010;Fortunato&Hric,2016;Porter,Onnela,&Mucha,2009),whicharesubnetworksthathave alargenumberoflinksinsidethem(Chen&Redner,2010;Lambiotte&Panzarasa,2009;Lancichinetti&Fortunato,2012;

Radicchi,Fortunato,&Vespignani,2012;Rosvall&Bergstrom,2008).Theassumptionwithmostofthesemethodsisthatthe networkisconstructedfromdenselyconnectedcoresofnodesorjournalsthathavearelativelysmallnumberofcitationsto therestofthenetwork.Thisisincontrasttothemethodsbasedonsimilarityofjournalsthatcanfindgroupswithastrong preferenceforreceivingorgivingcitationsfromacertainsubsetofjournals,forinstanceworkofappliedresearchcancite theoreticalworks,withoutbeingcitedback.

Evenifonewouldacceptthepremisethatthecommunity-likestructuresarerelevantincitationnetworks,manycom- munitydetectionmethodsarebesiegedwithintrinsicproblems.Veryoftentheydetectstructuresevenincaseofrandom networksbymistakingnoisefordata,theymightbeverysensitivetosmallperturbations(noise),andpossesa“resolution limit”,i.e.sufferingfromtheinabilitytoidentifycommunitiesbelowacertainsizethatdependsonthetotalsizeofthe network(Fortunato&Barthélemy,2007;Guimerà,Sales-Pardo,&Amaral,2004).Theperformance,reliability,andeventhe resultstosomeextentdependonthechoiceofamethodfromthelargesetofcurrentlyavailablemethods.

Theproblemswithcommunitydetectionmethodsarewell-knowninthenetworkscienceliterature,andtheneedto findthericherstructureinnetworksthanthoseobtainedbypartitioningnodestocommunitieshasbeenacknowledged formanytypesofnetworks(Leskovec,Lang,Dasgupta,&Mahoney,2009;Palla,Derényi,Farkas,&Vicsek,2005;Rombach, Porter,Fowler,&Mucha,2014;Wang&Hopcroft,2010;Xie,Kelley,&Szymanski,2013).Veryrecently,asasolutiontothis problem,theoldideaofusingstochasticblockmodels(SBMs)asmodelsofnetworkstructure(Holland,Laskey,&Leinhardt, 1983;Lorrain&White,1971;Wasserman&Anderson,1987)hasreceivedrenewedattention,becauseofthetheoreticaland algorithmicadvancesthatenabledtheiruseinareliableandscalableway(Bianconi,2009;Karrer&Newman,2011;Peixoto, 2012a).SBMisamodelinwhichnodesbelongtoblocks(thenameforgroupsintheSBMparadigm)andedgesarecreated between(andwithin)theblockswithsomefixedprobabilitiesforeachpairsofblocks.ThemethodsbasedonSBMswork byfindingthemodelwhichbestexplainsthenetworkdata.Thebestexplanationisnotnecessarilythemodelthatwould havemostlikelyproducedthedata,butthesimplicityofthemodelmustalsobetakenintoaccount,andtheprincipledand powerfulideasfromstatisticalinferenceliteratureareusedtoavoidsuchoverfitting.Onecanconsidertheblocksas“super nodes”thatareconnectedwithweightededges,andSBMmethodsthen—bydefinition—trytofindthe“supernetwork”that isthebestsimplificationoftheoriginalnetwork.

HerewetaketheadvantageoftherecentadvancesinSBMmethodsfoundinthenetworkscienceliteratureandapplythem tolargescalecitationnetworksbetweenjournals.WeusejournalcitationnetworksfromThomsonReutersCitationIndex^® fortheyearsrangingfrom1900to2013whichcontainshundredsofmillionsofcitations.Manystudiesconcentratedon smallsubsetsofthecitationnetwork(An,Janssen,&Milios,2004;Grossman,2002;Neruretal.,2005;Pieters,Baumgartner, Vermunt,&Bijmolt,1999;Porter&Rafols,2009;Shibataetal.,2011;Zhangetal.,2010),whileotherswereinterestedin large-scalepatterns(Boyacketal.,2005;deMoya-Anegónetal.,2007;Leydesdorff&Rafols,2009).Wefocusonthelarge scalecitationnetworksthatareconstructedusingallarticlesinthisbibliographicdataset.Firstwedividethefulltime periodintothetimewindowsof5or10yearsandusethearticlesinthosewindowstoconstructnetworksofthejournals activeineachwindow.Thatis,wetakesnapshotsofthecontemporaryscienceatdifferentpointsoftimeandtrackthe importantdevelopmentsbyfittingthemwithhierarchicalSBMs.We visualizetheresultingblockstructureatmultiple levelsofhierarchy,andillustratethepresenceofblocksthatarenotcommunity-likebycategorizingthemassources,sinks, bridges,andcommunities.Moreover,wefollowtheevolutionoftheseblockcategoriesin16largestfieldsofscienceintime andreportthelarge-scalechangesinthemovermorethanahundred-yearobservationperiod.

Thecitationnetworkscanbestudiedinisolationbuttheycanalsobeaugmentedandcomparedwithmanyotherdata sourcessuchasjournalcategorizations,articlekeywords,andauthorinformation.Previousstudieshave,forexample,com- paredpredeterminedjournalcategoriestonetworkclusters(Boyacketal.,2005;Janssensetal.,2009)ortofactorsfrom factoranalysis(Leydesdorff&Rafols,2009).Theyhavealsoconstructednetworksusingcategoriesasnodes(Zhangetal., 2010)andevaluatedthequalityofcategorisationsusingcriteriathatfavourcommunity-likecategories(Wang&Waltman, 2016).HerewewillutilizearecentlydevelopedgeneralizationoftheSBMmethodthatallowstheinclusionofany“tag”

informationaboutthenodes(Hric,Peixoto,&Fortunato,2016)anduseittoanalysehowmuchinformationthepredeter- minedjournalcategorizationscarryaboutthegroupstructureofthecitationnetworks.Thisapproachdoesnotassumethat thejournalclassificationsarethegroundtruth,butdeterminesthesuitabilityofsubjectcategoriesfordescribingcitation structurebyaskinghowmuchbetterwecandoinestimatingthecitationflowswiththeclassificationsthanwecando withouttheknowledgeoftheclassifications.Theconstructionofcontemporarycitationnetworksallowsustotrackthe congruityofthesubjectcategorieswithcitationpatternsthroughoutthelastcentury.

Thepaperisorganizedasfollows.Theprocessofbuildingannotatedjournalnetworksfromrawcitationdataisdescribed inSection2.ThestochasticblockmodelsareintroducedanddescribedinSection3.Thenthevisualizationofthecitation networksisdescribedandaselectionofresultsispresentedinSection4.Moredetailedanalysisofjournalgroupsproperties isdoneinSection5,whileSection6dealswiththeirevolutionintime.Nextacomparisonbetweenthesubjectcategories andcitationstructureisdevelopedandpresentedinSection7.Conclusionsaremadein8.Somebasicpropertiesofthedata andadditionalresultsarepresentedinAppendicesAandD.

2. Data

AllthenetworksconstructedinthispaperarebasedondataonarticlesandcitationsextractedfromthreeThomsonReuters CitationIndex^®datasets(ScienceCitationIndexExpanded^TM,SocialSciencesCitationIndex^®,andArts&HumanitiesCitation Index^®).Thisdatabasecontainsinformationaboutthepublishingyearandthevenue(journal,proceeding,conference,etc.) ofarticles,andeachvenue(fromnowoncalledjournal)isassignedtonone,one,orseveralsubfields.Wejointhesubfieldsinto largerfieldssimilartoParoloetal.(2015).Thedatasetspansfromyear1900to2013andcontainsabout76,000journals, approximately5.5Marticles,andabout630Mcitationsintotal.Amoredetaileddescriptionofthedatacanbefoundin AppendixA.

Asthefulldatasetspansformorethanahundredyears,itincludesinformationneededtotrackdevelopmentofmodern science.Weaimtoinvestigatehowthecitationpatternshaveevolvedduringthistimeperiodandtothatendwesplitthe dataintomultipletimewindows,eachofwhichisthenusedtoconstructacontemporarynetworkofjournals.Thetotal volumeofpublicationsandcitationsisgrowingexponentiallyintime(Panetal.,2016),andbecauseofthiswesetthetime windowlengthtotenyearsbefore1970sandtofiveyearsafterwards.

2.1. Networkconstruction

Ineachtimewindow,anodecorrespondstoanactivejournalthathaspublicationsinthegiventimeperiod.Thecon- nectionsbetweenthejournalsareconstructedusingoutgoingcitationsfromthesejournalssuchthatthereisadirectedlink fromjournalatojournalbifanarticleinjournalacitesanarticleinjournalb,andtheweightofthislinkistakentobe thenumberofsuchcitations.Foreachtimewindowweonlycountthecontemporarycitationssatisfyingthefollowingtwo criteria:(1)thecitedarticleispublishedinajournalthatisactiveinthetimewindow,and(2)thetimedifferencebetween thecitingandthecitedarticleisshorterthanthelengthofthewindow.Thisprocedureensuresthatallarticlesinthetime windowcontributeequally(withtheircitations)tonetworklinks.

Wehavealsotestedadifferentapproachforselectingthecontemporarycitationswhereboththecitingandthecited articlewererequiredtobewithinthefixedtimewindow.Themorestrictfilteringofcontemporarycitationsbringsimbalance toincomingandoutgoingcitationsofarticlesdependingwhethertheyarepublishedatthebeginningortowardstheendof thewindow:thoseatthebeginninghavelargerpoolofarticlestheycanreceivecitationsfromthanthepooltheycancite,

versionofgraph-toolweused(2.19dev),inSection7wehadtousesimplifiednetworks(undirected,unweighted,and withoutself-loops).Anaivemethodofdiscardinglinkdirectionsandweights,andremovingself-loops,leavesthenetworks verydense,andisapoorapproximationbecauseitregardsalllinksequallyimportant,irrespectiveoftheirdirectionor weight.Ausualapproachistosetaglobalthresholdonthelinkweightsthatkeepsonlythestrongestlinks,ortouseonly thelinksthatformamaximumspanningtree(Kruskal,1956;Macdonald,Almaas,&Barabási,2005).Bothoftheseare globalmethods,meaningthatthedecisiononwhetheralinkwillbekeptorremoveddependsontheweightdistribution andthestructureofthefullnetwork.Weusealocalthresholdingmethod,inwhichstatisticalsignificanceofweightsof linksofeverynodearecalculatedbasedonanullmodeldefinedforeachnodeseparately(Serrano,Bogu ˜ná,&Vespignani, 2009).Thissignificancemeasureistheprobabilitythatlinkweightiscompatiblewiththenullhypothesis,instatistical inferenceknownasthepvalue,butheredenotedwith˛.Bykeepingonlythelinksthathave˛valuelowerthanacertain threshold,wearedismissingalllinksthatdonotsignificantlydifferfromonescreatedrandomly,whilethosethatarekept canbeconsideredsignificant(notrandom),andthis“significance”iscontrolledwiththevalueofthethreshold.Wetested therangeofthresholdsandfoundthattheresultsarerobustagainstthechangeofthethresholdvalue˛(seeSection7).

Theresultsareshownfor˛intherange0.05,...,0.25thatpreserveabout6%toabout21%ofthemostimportantlinks (representingabout23%toabout45%ofcitations)andabout51%toabout99%ofnodes,respectively.

3. Stochasticblockmodel

Networksandgraphscanbemeasuredandsummarizedatmanylevelsofgranularity,startingfromglobalormacroscopic measures—suchasthetotalnumberoflinksordiameter—tolocalormicroscopicmeasuressuchasnodedegreeorthe clusteringcoefficient(Newman,2010).Hereweconcentrateondescribingnetworksinamesoscopicscalethatisbetween thesetwoextremes.Networkanalysismethodsthatworkatthislevelofgranularityalmostexclusivelydealwithsetsof nodesandlinkscalledcommunitiesordependingonthefieldofresearch,clusters,groups,modules,etc.(Boccalettietal.,2006;

Fortunato,2010;Schaeffer,2007;Wasserman&Faust,1994).Thereisnotasingle,precisedefinitionofcommunity,butmost oftenitisdescribedasasetofnodeswithmoreconnectionsbetweenthemthantotherestofthenetwork(Fortunato,2010;

Porteretal.,2009).Thecommunityparadigmassumesthatanetworkcanbedescribedasacollectionoftightlyknitsetsof nodes,whicharelooselyconnectedtoeachother.

Stochasticblockmodelrelaxestheassumptionaboutthenatureofconstituentsetsofnodessuchthattheyonlyneed tobeequivalentinthewaytheyconnecttoothergroups(calledblocksintheSBMparadigm),whichineffectallowsfora descriptionbeyondthecommunitystructure,likebipartite,core-periphery,etc.(Barucca&Lillo,2016;Karrer&Newman, 2011).SBMisagenerativemodel,meaningthatitassumesamodeloftheunderlyingstructureandprescribesaprocedure forbuildingnetworksthathavethisstructureincommon.Themodelisdefinedbyassigningallnodestodisjointsets¹and settingthenumberoflinksbetweenandwithinblocks.Obeyingtheabovedescribedconstraints,anetworkisgeneratedby randomlyplacinglinksbetweennodes.Analternativedescriptionistosettheprobabilitiesforplacingalinkbetweenany twoblocks,butweusedthelinkcountsfollowingtheapproachlaidoutinPeixoto(2014b).

Nodeswithinblockssharetheprobabilitiesforlinkstowardsthenodesinotherblocksbutalsoincludingtheirownblock.

Injournalcitationnetworksthismeansthatallthejournalsinablockhavethesamecitationpatternstootherblocks.They can,forinstance,receivemostoftheircitationsfromonesetofjournals,andgivethemouttoanotherset,orhavehigher thanaverageprobabilitytoexchangecitationswithsomeblocksandlower-than-averageprobabilitywithotherblocks.Two blockscouldalsohaveidenticalcitationpatternstootherblocks,butdifferentnumberofinternalcitations.Allthistellsus thatthismodelgroupsnodes(journals)intoclassesbytheirroleinthenetwork,whicheverthoseare.

Oncethemodelisknown,buildingnetworkswiththeprescribedblockstructureisstraightforward.However,themore commonsituationisoppositetothis:onlyasinglerealizationofthemodeloftheempiricalnetworkathandisknown,and parametersofthemodelthatmostlikelyproducethisnetwork,needtobeinferred.Findingthemostlikelyparametersisa highlynon-trivialtask,andmanyapproachestosolveithavebeenused(Wasserman&Anderson,1987).Allapproachesuse anobjectivefunction,inoneformortheother,thatmeasurestheprobabilityofthegivenparameterstobetheonesthat producedtheobservednetwork.Theproblemwiththisnaiveapproachisthatthebestfittingmodelwillbetoodetailedand willreproducetheobservednetworkwithveryhighaccuracy,whichgoesagainstthepurposeofthemodelsinproviding agoodsimplificationofthereality.Thecauseforthisisthefactthatthesimpleapproachusesallavailabledata,including noise,forfittingtheparameters.Intheextremecaseahighlydetailedmodelendsupputtingallthenodesintheirown

1 Theassumptionaboutblocksbeingdisjointsetsofnodescanberelaxed(Peixoto,2015).

blocks,sincethisreproducesthenetworkperfectly.Asimplesolutiontopreventthisfromhappeningistointroducethe numberofblocksasaconstraintinthefittingprocedure(Karrer&Newman,2011).Thisworksfineincaseswherethe numberofblocksisknown,otherwiseitneedstobeinferredfromdata,forinstancebyfindingabalancebetweenthemodel descriptionlengthanditsgoodness-of-fittodata.SomeoftheapproachesinthisdirectionarelistedinLatouche,Birmele, andAmbroise(2012).Inthisworkwearetakingadvantageoftherecentprogresswiththistopicthatestimatesthenumber ofblocksbyincludingtheinformationnecessaryfordescribingthemodelparametersintothetotalinformationamount beingminimizedsuchthatitpenalizesatoolengthydescription(Peixoto,2013,2015).

OneofthemainissuewiththeclassicalSBMs,inadditiontooverfitting,isthattheyassumePoisson-distributednode degrees,soanydeviationfromtheexpectedstructureisconsideredafeatureofthedataandthefittingalgorithmtriesto findamodelthatwouldadequatelydescribeit.Fittingthismodelwithanetworkwithadifferentdegreedistributionwould yieldblocksthatrepresentclassesofnodesbasedontheirdegrees,notjustonwhichothernodestheyconnectto.So,for instance,highlycitedjournalswouldbeseparatedfromjournalspublishingreviewarticles,thatpresumablyciteinlarge volumesbutdonotgetcitedasmuch.Also,journalsthatplayacentralroleintheirrespectivecommunitieswouldbeput together,evenifcommunitiesmightnotbeotherwiserelated.Thedegree-correctedstochasticblockmodelis“blind”tothis kindofstructure:itseparatesnodesintoclassesbasedontheirdegree,whichremovesunfaircompetitionforlinksbetween nodesoflargelydifferentdegrees(Karrer&Newman,2011).Thiseffectivelyincludesthedegreesequenceintothemodel thatdoesincreasetheinformationneededtodescribethemodel,butthebenefitofasimplermodelfordescribingthedata lowersthetotaldescriptionlengthinmostcases.

TheSBManditsdegree-correctedvariantcanbeexpandedtodescribehierarchicalstructureofblocks(Peixoto,2014b).

Inthiscase,eachlevelconstitutesanetworkofblocksandthebestfitofSBMofthelevelbelowit,startingfromthenetwork itselfatthebottomlevel,tothetrivialone-blocklevelontop.Fittingallthelevelsisdonesimultaneouslytoobtainthe minimumdescriptionlengthofthewholestructure.Thisapproachshowsseveraladvantages,likeallowingforevensmaller blocksthatcanreliablybeinferredandprovidingamulti-resolutionviewofthenetwork.HierarchicalSBMcanbeviewed asastackofprogressivelysimplerweightednetworks—eachlevelinthehierarchyisazoomed-outversionofthelevel belowit.Thismeansthatwecaninspectconnectivitypatternsofblocksofnodesatthedesiredlevelofdetail/blocksizes.

Theexistenceofthesmallestidentifiableblocksisincommunitydetectionliteratureknownastheresolutionlimitanditis definedasthesmallestgroupthatamethodisabletoidentify(Fortunato&Barthélemy,2007).Oneofthemostcommonly usedobjectivefunctionsismodularity(Newman&Girvan,2004),forwhichithasbeenshownthatitisnotabletoseparate smallgroups,eveninobviouscases,ifthenumberoflinksinsidethemistoosmallcomparedtothenumberoflinksintherest ofthenetwork(Fortunato&Barthélemy,2007).ItshouldbenotedthatSBMsarenotcompletelyimmunetoresolutionlimit problems(Choi,Wolfe,&Airoldi,2012;Peixoto,2013),whichpresentsaprobleminthecomparativeanalysisofnetworks spanningtwoordersofmagnitude,asisthecasewithourtimeslices.Thesmallestdetectableblocksscalewiththenetwork size,whichmeansthathigh-resolutionlevelsforlargenetworkswouldremainhiddenfromus.Fittingtheblocksatall resolutionsforthehierarchicalSBMatthesametimereducesthelimit,becauseeachlowerlevelusesblocksfromthelevel aboveasaconstraint,soblockinferenceatthelowerlevelsisineffectdonelocallyineachhigher-levelblock(Peixoto, 2014b).

Thecriteriaandalgorithmsdescribedaboveareimplementedingraph-toolPythonmodule,whichweusetodoall SBMfittinginthiswork(Peixoto,2014a).

4. Visualizingcitationflowsbetweenblocks

HierarchicalSBMlevelscanbevisualizedasnetworksthatprovideuswithamulti-resolutionmapofthecitationnetwork.

Networkvisualizationisapowerfultoolforvisualinspectionofcomplexnetworkdata,butitsreadabilityandthususefulness dependsonthelevelofdetailspresentedandthetotalsizeofthenetwork.Inaddition,ifthenetworkisdense,i.e.average nodedegreeislarge,itisevenhardertomakeaclearpictureofit.Lowerlevelscontainalotofinformationinfinedetail,but theyareoftenvastandtoodenseforvisualizationtobereadable.Butnotallweightedlinksareequallyimportantifweare interestedinlarge-scalepatterns.Thismeanswecandepictonlythemostimportantones,makingthenetworksignificantly lessdense.ThisisagaindonebytheprocedureoutlinedinSection2,whichkeepsonlythelinksthatformthebackboneof thenetwork(Serranoetal.,2009).Forthepurposeofthisanalysisthedirectionsoflinksarepreserved.

ThreeexamplesofnetworksofSBMblocksforthetimeperiodsof1920s,1960s,and1995–2000,atthelevelsmost similar,i.e.theclosestmatchingnumberofblockstosubfieldsandfieldsaredepictedinFig.1.

Clusteringofblocksofsimilarfieldsisquitevisibleinallsixnetworks.Medicineformsthebiggestcluster,followed byBiology,Chemistry,and Physics.Large-scalestructureremainssimilarforallthreetimeperiods:Medicineistightly connectedtoBiology,whichisthenconnectedtoChemistryandPhysics.Thisstructureisinaccordancewiththepreviously publishedmaps(Leydesdorff&Rafols,2009;Rosvall&Bergstrom,2008).Thefiguresalsoincludeinterestingsmall-scale detailsthatvarybetweentheresolutionsandtimewindows.Forinstance,theinterdisciplinaryblocksareoftenlocatedat theboundariesbetweenotherfields,butthiseffectismorevisibleinthesubfieldresolutionlevel.

Thefactthattherearemultipleblockswiththesamedominantfieldinlevelshavingthemostsimilarnumberofblocks tothenumberofassignedfields,isaconsequenceoflargeheterogeneityofthefieldsizesandimportance.Largefields alsohaverichinternalstructurethatovershadowssmall,moresimplefieldsinprocessofinferringthebestblocksateach hierarchicallevel.Asafirstapproximation,wehavechosenthenumberoffieldsasaguideforchoosingthemostappropriate

Fig.1.NetworksofSBMblocksforthetimeperiodsof1920s,1960s,and1995–2000.Thetoprowshowsresolutionlevelsmostsimilartosubfields,bottom tofields.Thenodeshapesandcoloursrepresentthedominantfieldinthatblockandthenodesizesareproportionaltothenumberofarticles.Directed linksarecolouredandsizedlogarithmically,accordingtothenumberofcitationstheycarry.Nodeandlinksizesarenormalizedperpairsofnetworks fromthesametimewindow,sothatthesesizescanbecomparedbetweenthetworesolutionlevelsbutnotacrossthetimewindows.Thelinkcoloursare normalizedforeachnetworkseparately.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthe article.)

levelasitsimpleandintuitive,whileanother,morecomplexapproachcomparingthefullsizedistributionsdidnotprovide significantlydifferentresults(seeAppendixB).

5. Connectivitypatternsofblocks

Incontrasttotraditionalcommunitydetection,blocksinSBMarenotlimitedto“densesubgroups”wherethecitations stayinsidetheblocks,butablockcanalsorepresentastructurewherethecitationsflowoutoforintotheblock,aslongas alljournalsintheblockbehaveinsimilarway.Wesummarizethetypeofblockintermsofcitationflowsbycountingthe numberofcitationsenteringtheblocks_in(articlesintheblockarebeingcited),leavingtheblocksout(theycitearticlesin otherblocks)andinternalcitationss_int(citingarticlesinthesameblock)².

2 Onecanviewthesystemofflowsandblocksasaweightednetwork.Intheliteratureofweightedcomplexnetworks(Barrat,Barthélemy,Pastor- Satorras,&Vespignani,2004),weightedsumofanode’slinksiscalledstrength,andforthedirectednetworksitcanbein-,out-andinternalstrength:sin, soutandsint.

Fig.2.Propertiesofblocksdependingontheirlocationontheconnectivitypatternplot. ¯sinand ¯soutarerelativein-andout-flowsofcitationsto/fromthe block.Duetorelation ¯sin+s¯out≤1pointsareconfinedtoregionsoutlinedbygreylines.Regions,aswellasredpointsatspeciallocations,aremarkedwith arrowsandannotatedwithdescriptionsofthepropertiesofblocksatthoselocations.(Forinterpretationofthereferencestocolorinthisfigurelegend, thereaderisreferredtothewebversionofthearticle.)

Notethatthesumofthethreeflowcountsrepresentsthetotalactivityofthejournalsintheblock.Herewearenot interestedinthetotalactivitiesbutinthetypeofflows.Weinvestigatethesetypesbyseparatingthetotalflowfromour flowmeasuresandconcentrateonrelativein-,out-,andinternalflows.Thesenormalizedflowsaredefinedas:

¯

s_in=s_in/stot, s¯out=sout/stot,

¯

s_int =s_int/stot,

where stot=s_in+sout+s_int. (1)

Bynormalisingtheflowsbythetotalactivitywereducethenumberoffreeparametersneededtodescribeblocks’flow patternsfromthreetotwo.Thatis,thesumofthethreerelativeflowsequalstoone,andknowingtwoofthemisenoughas thethirdonecanalwaysbecalculatedbasedonthem.Thismeansthatwecanreportthetwooutofthethreeflowmeasures thatarethemostconvenientforus.

Forvisualizingtheconnectivitypatternsofblocks,thechoiceof ¯s_inand ¯soutasxandycoordinatesmakesiteasytovisually assesstheblock’spropertiesfromitspositionontheplotasillustratedinFig.2.Sincethesum ¯s_in+¯soutmustbe≤1,points canlieonlyinthetriangleboundedbythediagonal(0,1)−(1,0).Proximitytotheorigintellsushowmuch“self-centred”

or“community-like”theblockis,whilethedistancesfromtheaxessignifythebalancebetweenreceivingandgivingout citations.Ithelpstoconsiderfourextremalcasesforablock,markedwithredpointsinFig.2:(0,0)Purecommunity.Journals inthisblockareisolatedfromtherestofthenetwork.Articlespublishedinthesejournalsonlyciteandgetcitedbyarticles injournalsfromthisblock.(1,0)Puresink.Journalsinthisblockonlyreceivecitationsanddonotciteatall.(0,1)Puresource.

Journalsinthisblockdonotgetcited,butciteothers.(0.5,0.5)Purebridge.Journalsinthisblockciteandgetcitedequally,but therearenocitationswithintheblock.Inmostcases,thevaluesliesomewherebetweentheseextremes.Thetriangularspace canbedividedintothreeregionsoutlinedbygreylinesinFig.2thatcontainblockswiththefollowingproperties:Inner triangleiscommunity-like(“acommunityinaweaksense”[Radicchi,Castellano,Cecconi,Loreto,&Parisi,2004]).More thanhalfofthecitationspertainingtothisblockstaywithinit.Upperwingmostlycitesothers.Lowerwingmostlyreceives citations.

InFig.3wevisualizethetypesofconnectivitypatternsofblocksfoundinthecitationnetworks.Wedisplaythedatafor threetimeperiods(1920s,1960s,and1995–2000)andforthreelevelsofresolution.Thisgivesusanoverviewofthetypes ofmesoscalestructuresonecanfindinthecitationnetworks.AlargenumberofblocksdetectedbytheSBMmethodfall outsidetheinnertriangle,andarethusnotcommunity-likestructuresevenintheweaksense(Radicchietal.,2004).Thisis especiallyevidentforhigh-resolutionlevelswherethevastmajorityofinferredblocksarenotcommunities.Notethatthe inclinationtowardsnon-community-likestructuresisafeatureofthedataastheSBMfittingmethodweusedoesnothave apreferenceforanyparticularblockstructurethatisplausiblewithintheSBMframework.

Notethat,inthehierarchicalstructureofblocks,thelevelthatcontainslargerblockscannothavelesscommunity-like blocksthanalevelthatcontainssmallerblocks,andfortheoneblockatthetopofthehierarchyallcitationsareinternal.

Thiscanbeillustratedbyconsideringthemergeroftwoblocks:theirinternalcitationsremaininternal,butpartoftheir externalcitationsthatgobetweenthembecomeinternaltothemergedblock(c.f.AppendixE).Inconsequence,theaverage

Fig.3. Connectivitypatternsofstructuralandclassificationblocks,forthetimeperiodsof1900–1910,1950–1960,and1995–2000.Eachrowcorrespondsto adifferentblocktype:journalsthemselves;thefirstSBMlevel;subfieldsclassification;SBMlevelwiththeclosestmatchingnumberofblockstothenumber ofsubfields(SBM:∼subfields);fieldsclassification;andSBMlevelwiththeclosestmatchingnumberofblockstothenumberoffields(SBM:∼fields).Blocks arerepresentedaspoints(colouredandshapedaccordingtothedominantfieldinablock)withcoordinatesbeingtheincomingandoutgoingcitationsas fractionsoftotalcitationsforeachblock(¯sinand ¯sout,Eq.(1)).Pointareasareproportionaltothetotalcitations(strengthstot)pertinenttotheblock.Blocks withfewerthat10totalcitationsarenotshown.Blockscontainingaselectedsetofjournalsareannotated.(Forinterpretationofthereferencestocolorin thisfigurelegend,thereaderisreferredtothewebversionofthearticle.)

fractionofinternalcitationsintheblockatthehigherlevelofhierarchycanonlybeequaltoorhigherthantheweighted averageofthetwoblocksatthelowerlevelofhierarchy.

Theprocedureoutlinedinthissectioncanbeusedwithanykindofpartitionofthejournalsinto“blocks”,notjustones inferredforSBM.Thejournalsareexplicitlypartitionedinthedatabysubfieldandfieldclassification,andwewantto comparethesepartitioningstoonesgivenbytheblocksfoundbytheSBMmethod.Forourpurposesitisusefultoview bothofthesepartitionsasblockstructures,buttoavoidconfusionwenametheblocksasdeterminedbytheclassification dataclassificationblocks,andtheonesinferredbySBMfromthecitationpatterns’structuralblocks.Journalsthemselvesform elementaryblocks,whichcanbeviewedasthe“zeroth”levelofhierarchicalSBMoranyotherhierarchicalblockstructure.

Thesezerothlevelblocksgiveuspropertiesofindividualjournals.Journalsareassignedtosubfields(inthedataset),which areinturngroupedintofields.Onewouldexpectthisclassificationtobereflectedincitationpatterns,sinceitshould groupsimilarjournalstogether.Weareabletotestthisassumptionbycomparingthepropertiesofartificialblocks,defined bysubfieldsandfields,withblocksinferredfromcitations.HierarchicalSBMprovidesuswithmanylevelsatdifferent resolutions,andinFig.3wecomparethelevelwiththemostsimilarnumberofblockstothenumberofsubfieldsandfields inthatnetwork.

Individualjournalsspanthewholespaceofin-andoutflowbalance(therearemanystrongsinksaswellasstrong sources),whiletheoverwhelmingmajoritydonotpredominantlycitethemselves.Thereareannotatedjournalsofhigher prominencethatarefoundinthelowerwing,whichistobeexpectedsincetheyreceivemorecitationsthangiveout.

ThefirstSBMleveldeterminesthesmallestnon-trivialstructuralblocks,anditisthesmallestgroupingofjournalsthat cite,andarecited,inasimilarwayastheyhavesimilarcitationpatternstowardstherestofthenetwork.Comparedwiththe levelofjournalsthespreadofpointsisreducedinalltimeperiods,althoughtoavaryingdegree.Thespreadisreducedwith time:thereisalmostnochangein1920s,somechangein1960sandasignificantconstrictionofvaluesfor1995–2000.This couldbeaconsequenceoftheresolutionlimit,wheresmallerdetailsareincreasinglyhardertocaptureasthesizeofthe networkincreases,ortheoutlyingjournalscarrylessinformationinlateryears,sotheyarecombinedwithmoremoderate ones.

Atthelevelofsubfields(the3rdand4throwsinFig.3),bothstructuralblocksandclassificationblocksdisplayapattern wheremultidisciplinaryblocksarelocatedmostlyonthecitedside.Thisisexpectedgiventhattheseblocksaredominatedby highimpactjournalspublishinghighqualityarticlesfromawidespectrumoffields.Comparingthedistributionofpointsfor classificationblocksversusstructuralblocksweseethatthelatteronesaremoreevenlyspreadoutin1960sand1995–2000, whilethisisnotthecasein1920s.Therearealsomorecommunity-likeblocksinthestructuralcase,inparticularinthefields ofEconomics,Physics,andMathematicsandGeosciencesin1920s.ThismeansthatSBMcapturesabroaderspectrumof blocktypes,whileclassificationblockstendtobemoresimilartoeachotherortheyhaveapreferenceforcertainproperties.

Thehighestlevelofhierarchywefocuson—forbothclassificationblocksandstructuralblocks—istheleveloffields(5th and6throwsinFig.3).Constrictioninclassificationblocksisagainpresent,albeitnotsostrongasforthesubfields.Withtime, Multidisciplinaryfieldstronglyseparatesfromthebulkthatremainsmoreelongatedinthecommunity-bridgedirection andslightlyleanedtowardsource-likebehaviour.Similartothelevelofsubfields,Multidisciplinaryblocktendstobecome morecitedwithtimeforbothclassificationandstructuralblocks,withthisbehaviourbeingmorepronouncedforstructural blocksandthelasttimeperiod.MedicineandPhysicsgetseparatedintoacommunity-likeblock,meaningthatthemost citationsremainwithintheirfield,whichwasnotthecaseforthesubfields.TheSBMfindsmultipleblocksofmixedfields thataresource-andcommunity-likefor1960s,andonlycommunity-likefor1995–2000.Mostofthemixedblocksin1960s arecomprisedofunclassifiedjournals,numberofwhichrisesinthesecondhalfofthecentury(seeAppendixAfordetails).

Forclassificationblocksthesejournalsareallcollectedunderthe“mixed/unknown”field(largewhitesquare).

NotethatoneneedstobecarefulwhencomparingblocksacrossdifferentpanelsinFig.3,asthereisnoguaranteethat blockswithsimilarqualitiesindifferentpanelscompriseofthesamejournals.Deeperanalysisinthisdirectionwouldrequire listingalljournalsinablock,orannotatingjournalsofinterest(whichisdoneforafewjournalsinFig.3).

6. Evolutionofblockconnectivitypatternsintime

Intheprevioussectionwesummarizedthetypesofcitationflowsofindividualblocks.Wewillnextbuildonthissum- marizationmethod,andquantifytheevolutionofcitationflowsofspecificfields.Tobemoreprecise,weaskthequestion:

whatistheexpectedtypeofcitationflowoftheblockwherearandomlychosenarticleofagivenfieldbelongsto?

Theexpectedflowsforajournalinafieldcanbecalculatedbycollectingalljournalsbelongingtoafieldandtakinga weightedaverageoftheflowsoftheblockstheybelongto.Tocalculateaverageflowsforarticles,averagingneedstobe weightedbyafractionofjournals’publicationsoutofthetotalnumberinthefield:

¯ s^f_x=

ja_j¯sx

ja_j , (2)

where ¯sxisanyofthethreeaverageflows(in-,out-,andinternal)fromEq.(1),a_jisthenumberofarticlespublishedby thejournalj,andthesumsgooveralljournalsofthefieldf.Ifwedothisfornetworksinalltimewindows,inadditionto differentiatingfieldsamongthemselves,wecanfollowtheevolutionofcitationflowpatternsofindividualfieldsovertime.

Fig.4.Plotoftheevolutionoftheaveragecitationflows,fortheexamplefieldEngineering.Time(inyears)isonthehorizontalaxis.Relativecitationflows ofblockscontainingjournalsfromtherespectivefield[seeEq.(2)]areshownasshadedregions:bottompart(thedarkest)isforinternalflow ¯s^f_int,middle (lighter)forincomingflow ¯s^f_in,andtoppart(thelightest)foroutgoingflow ¯s^f_out.Theaveragevalueofeachflowismarkedwithablackline,andtheerrorof themeanisshownasatransitionalshadebetweenregions.Thecentralhorizontallinemarks50%oftheflow.

Similartotheprevioussection,werepeattheanalysisforstructuralblocksinferredbytheSBMandclassificationblocks givenasclassificationsofthedata(fieldsandsubfields).Thatis,theaverageflows ¯s^f_x(in-,out-andinternal)ofafieldfare calculatedoverallthefield’sjournals,andthevalues ¯sxaretakenfromblocks,eitherthestructuralblocksortheclassification blocks,thejournalsbelongto.

WeillustratetheapproachusedtoplottheevolutionoftheaveragecitationflowsofblockconnectivitiesinFig.4.Time (inyears)isonthehorizontalaxis,whileverticalaxisissplitbyrelativecitationflows(Eq.(2))ofblockscontainingjournals fromtherespectivefield:bottompart(thedarkest)representsinternalflow ¯s^f_int,middle(lighter)incomingflow ¯s^f_in,andtop part(thelightest)outgoingflow ¯s^f_out.Theaveragevalueofeachflowismarkedwithablackline,andtheerrorofthemean isshownasatransitionalshadebetweenregions.Thecentralhorizontallinemarks50%oftheflow.

Theevolutionoftheaveragecitationflowsorblockconnectivitiesfor16mostprominentfieldsisshowninFig.5.Fields areorderedbythesurfaceareaunderthecurvegainedbytakinglogarithmsofnumbersofjournalsineachfield.Logarithmic scaleisusedtomitigatetheimpactoftheexponentialincreaseofthenumberofjournals(c.f.AppendixA).

Forhalfofthe16fields,theinternalflowsofstructuralblocksarelargerthantheinternalflowsofclassificationblocks.

Mathematics,Economics,Geosciences,Law,Education,PoliticalScience,andAnthropologyjournalsarefoundtoresidein structuralblocksthataremorecommunity-likethanthecorrespondingclassificationblocks.ThiscouldmeanthatSBMis abletofindtheir“naturalcommunities”—onesthatcapturethemostofthecitationflows—whiletheclassificationsalone arenotabletoachieve.Thestrengthofthiseffectvaries,withthemoststrikingexamplesbeingMathematics,Law,and PoliticalScience.ClassificationblocksofEconomics,andpartiallyGeosciencesandLawarethemselvesquitecommunity- like.TheoppositeeffectispresentforMedicineandBiology,meaningthatthestructuralblocksarelesscommunity-likethan classificationones.Thiscanbeexplainedbythefactthatthesefieldshavealargenumberofbothsubfieldsandjournals,so theycontainarichinternalstructureofblockswithlargeflowofcitationsbetweenthem.Theirinternalcitationstructure mightalsobedifferentfromotherfields,astheycontainhighlycitedpapersdescribingmethodsandprocedures(Small&

Griffith,1974).

Thein-flows ¯s^f_inandout-flows ¯s^f_outarequitebalancedformostfields,withafewnotableexceptions.Multidisciplinary fieldhasnoticeablymoreincomingcitationsthanoutgoing,forallhierarchylevelsandforbothtypesofblocks.Thisdoes notnecessarilymeanthatalljournalsinMultidisciplinaryfieldattractcitations,butitcanbeduetothefactthatitcontains severalhighprofilejournals,³ likeNature,Science,andPNAS.IndividualjournalsinPoliticalSciencefieldreceivemore citationsthantheygiveout,whilethisdifferencevanishesfortheblockstheybelongto.TheoppositeistrueforMedicine, Health,andEnvironmental(theygiveoutmorecitationsthantheyreceive)andthisbehaviourremainspresentforblocks inhigherlevels.

Intimedomain,fieldsexhibitawealthofbehaviours.Somehaverelativelystablepatterns(Medicine,Chemistry,Eco- nomics,Multidisciplinary,andtosomedegreeHealthandPoliticalScience),mostoftheothershavelargechangesaround theWorldWarII,whilesomehaveuniformandsteadyshifts(EducationandAnthropologyarethemostnotable).

ThemoststrikingfeatureisthesuddenriseinshareofoutgoingflowsatthetimeoftheWWII,mostlyforclassification blocks,andtosomedegreefor journalsofsomefields.Thelargestrises areindecreasingorderfor:Law,Geosciences, Mathematics,Anthropology,Physics,Biology,andEnvironmental.InBiologyandChemistryafainteffectisalsovisibleatthe levelofjournals,whileincaseofEnvironmentalitismostlyadropininternalflow.Giventhatthiseffectisalmostinvisible forstructuralblocks,theobservedchangesmostlikelydonotarisefromthechangeinjournals’citationpatterns,butinthe waytheyareclassified.Thissuddenchangecorrelateswiththelargeincreaseofthenumberofsubfields(c.f.AppendixA).

Somefieldsexhibitslowbutsteadychangeovertime,predominantlyinthestructuralblocks.Anthropologyjournalsare citingmoreexternalliteratureandlessthemselveswithtime,whichisalsovisibleinthesmalleststructuralblocks,tolesser extentinthestructuralblocksofsizeofsubfields,andnotatallinthestructuralblocksofsizeoffields.Thismeansthat considerableamountofthecitationflowthatisincreasinglygoingoutofthejournalsisneverthelessretainedinsidethe

3 Forthisreasonsomeauthorshaveinsimilaranalysesexcludedthesejournalsaltogether(Zhangetal.,2010).

Fig.5.Evolutionofstructuralandclassificationblockconnectivitiesintime,for16largestfields.Fieldsaregroupedincolumns(coloursarethesameas inFig.3)andeachrowisfordifferenttypeofblocks,fromtop:journalsthemselves;thefirstSBMlevel;subfieldsclassification;SBMlevelwiththeclosest matchingnumberofblockstothenumberofsubfields(SBM:∼subfields);fieldsclassification;andSBMlevelwiththeclosestmatchingnumberofblocks tothenumberoffields(SBM:∼fields).Time(inyears)isonthehorizontalaxis,whileverticalaxisissplitbyrelativecitationflowsofblockscontaining journalsfromtherespectivefield:outgoing,incomingandinternalflows(fromtoptobottom,respectively).(Forinterpretationofthereferencestocolor inthisfigurelegend,thereaderisreferredtothewebversionofthearticle.)

structuralblocks.Biology,Chemistry,Multidisciplinary,andEducationjournalsshowsimilarbehaviour,buttheiroutgoing flowsremainstablealsointhestructuralblocksofsizeofsubfields.Aplausibleexplanationisthatasanewjournalappears inthefield,it“steals”someofthecitationflowsfromtheoldjournalswhilemimickingtheircitationpatterntowardstherest ofthenetwork,whichmeansthattheywillallbeneverthelessputintothesameblockbySBM.Provingsuchexplanations wouldrequireamoredetailedanalysis.

Fig.6.Schematicrepresentationofjointjournal-annotationsmodel.Journalswithcitationsconnectingthem(bluecirclesandblacklines)areaugmented withannotations(redsquaresandgreylines).SBMisfittedontothewholenetworksuchthatblocksofjournalsareseparatefromblocksofannotations (bluecirclesandredsquaresrespectively).Notethatajournalcanhavemultipleannotations,oritcanbeunannotated.Hereweusesubfieldandfield classificationsastheannotationsofjournals,butanyotherdataonjournalscanbeused.(Forinterpretationofthereferencestocolorinthisfigurelegend, thereaderisreferredtothewebversionofthearticle.)

7. Predictivepowerofsubjectcategorisations

Citationnetworkscanbeaugmentedwithawealthofinformationaboutarticles,journals,andauthors.Theseinclude subfields,tags,keywords,authoraffiliations,etc.Inthisworkweuseclassificationofjournalsintosubjectcategories(sub- fields)providedinthedataset,andweareinterestedinhowmuchdoesthisclassificationcorrespondtothestructuralblocks foundinthecitationpatterns.

Thecomparisonoftwopartitions—suchasclassifications,clusterings,orblockstructures—isingeneraloftendoneusing somecomparisonmeasure,suchasJaccardindex,Omegaindex,andVariationofInformation(Meil˘a,2007).Itistypical tocomparepartitionsarisingfromaclassificationgiveninthedataandthegroupsarisingfromthenetworkstructureas returnedbysomecommunitydetectionmethod(Bommarito,Katza,&Zelnerd,2010;Chen&Redner,2010;Hric,Darst,&

Fortunato,2014;Lancichinetti&Fortunato,2009;Yang&Leskovec,2015).Thisisaviableoptioninourcaseaswell,butwe wouldhavetomakeachoiceofacomparisonmeasure.Becausethequestionofhowsimilartwopartitionsareisill-defined, eachcomparisonmeasurerealisesitdifferentlyandcanevenreturndifferentresults(Fortunato&Hric,2016;Meil˘a,2007;

Traud,Kelsic,Mucha,&Porter,2011).

Insteadofaskinghowsimilarthetwopartitionsare,weaskthequestion:whatcanweknowaboutthecitationsofajournal fromitsclassification?ExactlythisquestionisansweredbyincludingnodeannotationsintoSBMasitisdonebyHricetal.

(2016),whichisbasedonthenotionthatannotationsonnodesarejustmeta-informationonehasaboutthenetwork—there isnoprincipleddifferencebetweenthedataaboutconnectionsbetweentwonodes(links)andbetweenanodeandits annotations.Inliteraturedealingwiththecommunitydetectioninnetworksthisdistinctionbetweendataandannotations isoftenmadeexplicit,eitherbytreatingannotationsasasortof“groundtruth”forgroups(Yang&Leskovec,2012a,2012b, 2015),orasfeaturesthatneedtobelearnedbythemodel(Newman&Clauset,2016).Hereinstead,annotationsaretreatedas nodesofabipartitenetworkconsistingof“data”nodes(journalsinthecitationnetwork)and“annotation”nodes(subfields orfieldsofthejournals),andconnectionexistsbetweendatanodeandallofitsannotations(therecanbeanynumberof them,includingzero).

Fig.6illustratestheresultingcombinednetworkthatconsistsoftwokindsofnodes(journalsandannotations)andtwo kindsoflinks(citationsandjournal-annotationassignment).SBMisthenfittedwithaconstraintthateachinferredblock mustcontainonlyonekindofnodes.Thebenefitsofthisprocedureisthatthenodeannotationscontributetotheinferred nodeblocks,andannotationsarealsogroupedintoblocksof“equivalence”.

Workinginthisframework,thequestionfromthebeginningofthissectioncanbeformulatedas:howmuchinformation gaindoesonegetaboutlinksofasinglenode,afterlearningthenode’sannotations?Toansweritthefollowingprocedureis used.Asmallfractionofnodesisremovedfromthenetwork(5%or100,whicheverissmaller),turningtheminto“extra nodes”—thenodeswearemissingtheinformationon,andwouldliketoknowourchancesincorrectlyguessingwhere theirlinksconnectto.Then,theblocksareinferredforboththeoriginalnetwork(withoutannotations),andonedescribed above(withannotationsincludedinanadditionallayer).Theprobabilitiesfornode’slinkstoconnecttonodesthatbelong toexistingblocksaredefinedonlybytheblockthenodebelongsto.Withoutannotationstotelluswhichblockdoesthe extranodebelongto,theonlythingweknowaboutthenodeisitsdegree,andthusourbestguessfortheprobabilityofthis nodetobelongtoablockistousethesizedistributionoftheblocks,asitistheonlyinformationwehave.Incasewedohave thenode’sannotations,weknowitslinksintheannotationslayerwhichnarrowsourchoiceofblocksitcanbelongtoand thusraisestheprobabilitiesofguessingthecorrectlinks.Ifwedenotetheprobabilityforguessingalllinksofnodeiwithout knowingannotationswithP_iandthesamebyusingannotationsasP_i(ann)wecanquantifytherelativeimprovementwith thepredictivelikelihoodratio_i:

_i= P_i(ann)

P_i+P_i(ann) . (3)

Fig.7.Nodepredictionperformance,measuredbytheaveragepredictivelikelihoodratioforsubfieldsandfields[seeEq.(3)].Thevaluesare calculatedforsimplifiednetworks(seeSection2.2)for14timeslicesusedpreviously,withfivethresholdvalues˛:0.05,...,0.25.Eachbarcorresponds toasingletimewindow,withtheheightofthebarbeingtheaverageover˛valuesshownasdotsontopofeachbar.Foreach˛value,theaverageand standarddeviationovertensamplesisshown.Eachsampleisformedbyrandomlyremoving5%or100nodes,whicheverissmaller.

Thepredictivelikelihoodratio_itakesvaluesfrom[0,1]andisabove0.5ifannotationsimprovelinkpredictionpower, around0.5iftheydonotchangeit,andbelow0.5iftheydecreaseit.Theaverage_iofallsamplenodesistheaverage predictivelikelihoodratioforadataset.

Thismeasureisnotsensitivetothetotalnumberofblocksorannotations,asitdependsonlyonthe“power”ofthose annotationstopredicttheblockstructure.Theonlyimportantthingishowalignedtheannotationsaretothestructural blocks.

Weusetheaveragepredictivelikelihoodratiotomeasuretheabilityofsubfieldandfieldclassificationstopredict journals’citationsinthesimplifiednetworks(seeSection2.2).Predictingthelinksofthesimplifiednetworksisequivalent topredictingwhichjournalsarethemostimportantsourcesanddestinationsforthecitationsoftheextranodes,because thesenetworksonlyincludethemostimportantlinksforeachnodeanddonotincludetheactualcitationcountsforthe links.ThevaluesforarepresentedinFig.7foreachindividualtimeslice,andforfivethresholdlevels˛:0.05,0.1,0.15, 0.2,and0.25.

Overall,bothsubfieldandfieldclassificationcorrelatepositivelywiththecitationstructure.Theonlyexceptionisthe lowthresholdnetworksfor1900s,inwhichknowingthejournal’s(sub)fielddoesnothelpinpredictingitscitations.Low thresholdvaluesinthisalreadysmallnetworkcausedthelossoflargefractionoflinksandnodes,whichloweredthequality oftheapproximationbythesimplifiedandthresholdednetwork.Higherthresholdvaluesdonothavethisproblem.

Subfieldsaremorepredictivewithtime,althoughthereisaslightdeclineforthelast15years,withthepossiblereason beingasomesortofover-specializationofsubfields,whichdoesnotnecessarilycorrespondtothecitationpatternsofthe journalsbeingclassified.Fields,ontheotherhand,remainagoodproxyforlarge-scalecitationstructurethroughoutthe wholetimeperiod.

7.1. Predictabilityofindividualfields

Themethoddescribedintheprevioussectionanswersthequestionofhowmuchinformationwegainaboutjournal’s citationsifweknowwhatsubfieldsitisclassifiedinto.Wewillnextdividethisquestionintosmallerparts,andaskhow muchinformationdowegainbyknowingthatajournalbelongstoaspecificsubfield.

UsingthemodelfromHricetal.(2016)itispossibletocalculatehowmuchinformationgain(forguessingnode’slinks) doesasingleannotationprovide,incomparisontoacasewhereannotationsareassignedrandomly.Informationgainrelative totherandomcaseisdefinedaspredictivenessa,itisdefinedperannotationblocka,anditisnotaffectedbythetotal numberofblocksorannotations.FurtherdetailsandformulascanbefoundinHricetal.(2016).

Fig.8. Predictivenessesmeasuredbyofthetop16fields,forclassificationintofields,forallavailableyears.Thevaluesareten-yearslidingaveragesof averageforalltimewindowsusingthatyear.Eachpanelcontainsagroupoffourfieldsintheorderofdecreasingafter1960.Shadedregioninthe backgroundisthetotalspanofvalues.

Hereweagainconsidersubfieldsandfieldsasannotationsofjournals.AfterfittingtheSBMontothetwo-layerednetwork ofcitationsandannotations,thepredictivenessofeachblockofsubfields(orfields)iscalculated.Fieldsinthesameblock inheritpredictivenessoftheblock,whichfollowsfromtheSBMassumptionthatallannotationsinablockareequivalent.

Bycalculatingthefieldpredictivenessesforalltimewindows,ontopofbeingabletocomparefieldstoeachother,their changeintimecanalsobetrackedbothrelativelyandabsolutely.

BecausetheimplementationofSBMfittingalgorithmisprobabilistic,variationsinresultsaretobeexpectedwithsmall differencesinthenetworks(forinstancetwo10-yearwindowswith9-yearoverlap),andevenindifferentrunsoffitting functiononthesamenetwork.Thesevariationscauseavalues forthenetwork fromadjacenttimewindowstovary considerably,obscuringmoregeneraltrends.Takinganaverageaofalltimewindowsthatayearbelongsto,weareable toovercomethesefluctuations.Additionalclarityisachievedbyten-yearslidingaveragesofthesevalues.

Herewepresentthepredictivenessesofindividualfields,forthecasewhereclassificationintofieldswasused.Theresults forthecasewheresubfieldswereusedinstead,arepresentedinAppendixD.

BasedonfieldpredictivenessinFig.8,thetimecanbesplitintothreeperiods:before1940s,thetransitionperiod,and after1970s.Before1940sthefieldshaveonaveragehigherpredictiveness,butsincethedataisquitescarceforthisperiod, oneneedstobecarefulwhendrawingconclusions.Inthetransitionperiodallthefieldshaveverypoorpredictiveness, experiencingareboundafter1970forallbuthandfuloffieldsinthelastpanels(Engineering,Environmental,Medicine, Biology,Multidisciplinary,and Health).ThiscanbeasignofmajorchangessciencehasgonethroughaftertheWWII.

Mathematicshasthehighestpredictivenessinthethirdperiodbyavisiblemargin,whilebefore1940sthebestscoring fieldsareEngineeringandMultidisciplinary.Ithasrisensharplyfromthebottominthetransitionperiodtothetopinjust 25years,i.e.from1955to1980.ThismeansthatcitationpatternsofMathematicspapersbecamemorecharacteristicafter 1970s,whichispickedupbySBMandMathematicsjournalsendupinasmallnumberofexclusiveblocks.ForEngineering itistheopposite:itfaredveryhighbefore1940s,didnotsufferhardinthetransition,butneverrecovered.Thesamecanbe saidaboutMultidisciplinaryfield:ithadevensharperdropanddidnotreallyrecover.

Ontheotherside ofthespectrumareoftenlargefields(Engineering,Medicine,Biology),orrelatedtoalargefield (Environmental,Anthropology,andHealth).Theirlargesizemeansthattheycontainrichstructure withinthemselves, whichgetsdetectedbytheSBMaslargenumberofblocks.Henceknowingjustthefieldlabeltellslittleaboutthesmall blockswithinthefield.