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MULTILAYERNETWORKS MultilayerNetworks StructureandFunction GinestraBianconi
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ToChristoph
Preface Multilayernetworksareformedbyseveralnetworksthatevolveandinteractwitheach other.Thesenetworksareubiquitousandincludesocialnetworks,financialmarkets, multimodaltransportationsystems,infrastructures,climatenetworks,ecologicalnetworks,molecularnetworksandthebrain.Themultilayerstructureofthesenetworks stronglyaffectsthepropertiesofdynamicalandstochasticprocessesdefinedonthem, whichcandisplayunexpectedcharacteristics.Forexample,interdependenciesbetween differentnetworksofamultilayerstructurecancausecascadesoffailureeventsthat candramaticallyincreasethefragilityofthesesystems;spreadingofdiseases,opinions andideasmighttakeadvantageofmultilayernetworktopologyandspreadevenwhen itssinglelayerscannotsustainanepidemicwhentakeninisolation;diffusionon multilayertransportationnetworkscansignificantlyspeedupwithrespecttodiffusionon singlelayers;finally,theinterplaybetweenmultiplexityandcontrollabilityofmultilayer networksisaproblemwithmajorconsequencesinfinancial,transportation,molecular biologyandbrainnetworks.
Inthelasttwentyyears,considerableattentionhasbeendevotedtothestudyofsingle networks.Ithasbeenfoundthatdespitetheirdifferentfunctions,manybiological,social ortechnologicalsystemscansharesimilarpropertieswhentheyarestudiedfromthe networkperspective.Recently,themultiplexityofmanynetworkshasbeenidentifiedas animportantaspectofnetworkedsystemsthatneedstobeaddressedtoimproveour understandingofbiologicalandman-madenetworks.Thesubjectiscurrentlyraising greatscientificattention,andseveralimportantnewresultshavebeenobtained.This bookwillpresentacomprehensiveaccountofthisemergingfield.
Thebookincludesthreeparts:
-PARTI:SINGLEANDMULTILAYERNETWORKS Thispart(chapter1)outlinesthemainresearchquestionsthathavebeendriving theresearchonmultilayernetworkstructureandfunction.
-PARTII:SINGLENETWORKS ThispartprovidesanintroductiontothemainresultsobtainedinNetworkScience forthecharacterizationofthestructure(chapter2)andthefunction(chapter3) ofsinglenetworks.Thispartconstitutesthereferencepointforappreciatingthe resultsthatholdformultilayernetworks.
-PARTIII:MULTILAYERNETWORKS Thispartconstitutesthecoreofthebookanddiscussesthemainpropertiesof multilayernetworkstructureandfunction.
Preface Threeinitialchapters(chapters4–6)setthestagefortherestofbook.They discusstherelevanceofthemultilayernetworksframeworkforavarietyofapplications(chapter4),providethemathematicaldefinitionsofmultilayernetworks (chapter5)andintroducetheirbasicstructuralproperties(chapter6).
Subsequently,severalchaptersaredevotedtothecharacterizationofthestructureofmultilayernetworksandextractionofrelevantinformationusingtheirbuiltincorrelations(chapter7),theirmesoscalecommunitystructure(chapter8)and structuralpropertiesdeterminingthenodes’andlayers’centralities(chapter9).
Bridgingbetweenthechaptersfocusingonmultilayernetworkstructures(chapters5–9)andthechaptersfocusingonmultilayernetworkdynamics(chapters 11–16),novelmodellingframeworksespeciallytailoredtomultilayernetworksare presentedinchapter10,togetherwithrandomizationalgorithms.
Theactiveresearchactivityonthedynamicsandfunctionofmultilayernetworksiscoveredinchapters11–16.Thesechaptersprovideageneralperspective onthemajordynamicalprocesses,including:percolationandavalanches(chapters11–12),epidemicspreading(chapter13),diffusion(chapter14),dynamical systemsandsynchronization(chapter15)andfinallyopiniondynamicsandgame theorymodels(chapter16).
-APPENDICES Aseriesofappendicesprovidingmoredetailedmathemeticaldiscussionofsome ofthemajorresultsinmultilayernetworkscomplementsthematerialpresentedin themainbodyofthebook.
Ouraimhasbeentoprovideanoverviewofthefieldwhichcouldguidethereaderin understandingtherecentliteratureonmultilayernetworks.Giventhefastpaceatwhich newresultsarecontinuouslypublishedonthesubject,ithasbecomeimpossibletocover entirelytherapidlygrowingliteratureinthefield.Ouraimistoprovideapedagogical presentationandanin-depthdiscussionofthemainresultsonmultilayernetworks, allowingstudentsandreseacherstobequicklyintroducedtothefield.Wehavetherefore madesomechoicesbasedonourperceptionofwhatismorerelevanttocoverinthebook. Thisdoesnotimplythattheworknotcoveredhereislessvaluableandweapologizein advancetotheAuthorsofthepapersnotcitedhere.
ThisbookwillbeofinterestforgraduatestudentsandresearchersinNetwork Scienceworkingattheinterfacebetweentwoormoredisciplinessuchas:physics, mathematics,statistics,economy,engineering,computerscience,neuroscienceandcell biology.Whilethebookwillprovideatheoreticalintroductiontothemainresultson multilayernetworks,atthesametimeitwillremainwidelyaccessibletothegeneral interdisciplinaryreader.
GinestraBianconi London,31October2017
Acknowledgements Thisbookistheresultofmultipleinteractionswiththenetworkscientistsworkingin multilayernetworksandwithmanyinterestedstudentstowhomIgavelecturesonthe topic.Iammostgratefultoallofthemfortheirsharedpassionformultilayernetwork structureandfunction.
Specialthanksgotomypreciousmultilayernetworkcollaborators:A.Arenas,A. Barrat,A.Baronchelli,M.Barthelémy,G.J.Baxter,S.Boccaletti,F.Battiston,D.Cellai, R.A.daCosta,L.Dall’Asta,R.Criado,C.I.DelGenio,S.N.Dorogovtsev,J.P. Gleeson,J.Gómez-Gardeñes,A.Halu,M.Karsai,J.Iacovacci,V.Latora,E.López, J.F.F.Mendes,G.Menichetti,R.Mondragón,S.Mukherjee,V.Nicosia,P.Panzarasa, M.A.Porter,F.Radicchi,C.Rahmede,D.Remondini,M.Romance,I.Sendina-Nadal, J.Stéhle,Z.Wang,Z.Wu,M.Zanin,K.Zhao,J.Zhouamongwhicharethecoauthors ofaninfluentialreviewarticleonmultilayernetworksthathasbeenthestartingpointfor thisbook:S.Boccaletti,R.Criado,C.I.DelGenio,J.Gómez-Gardeñes,M.Romance, I.Sendina-Nadal,Z.Wang,M.Zanin.
IthankthePhysicsDepartmentatSeoulNationalUniversity,theLondonMathematicalSociety,the2016NetSciSchool,theMathematicsDepartmentofthePolitecnicoof Torino,theComoSchoolforAdvancedStudies,andtheShortCourseonComplex NetworksatOxfordUniversityforhostingmycoursesandlecturesonmultilayer networksthathavebeenveryusefulinshapingthisbook.Theseeventswouldnothave beenpossiblewithoutthesupportofmygreatfriendsandcolleagues:A.Arenas,J.Coon, B.Kahng,S.Majid,Y.Moreno,M.A.Porter,F.Vaccarino.
InwritingthisbookIbenefitedfromgreatdiscussionswithS.Havlinontherobustness ofinterdependentnetworks,withA.ArenasandY.Morenoonepidemicspreadingand diffusionprocessesonmultilayernetworksandwithS.Boccalettionthesynchronization ofmultilayernetworks.
SönkeAdlugandAniaWronskifromOxfordUniversityPresshavebeeninvaluable too,fortheirexcellenteditorialworkthathasmotivatedmethroughoutthewritingof thebook.IalsothanktheSchoolofMathematicalSciencesatQueenMaryUniversity ofLondonthathasallowedmetomakethisbookareality.
Finally,thisbookwouldnothaveseenthelightwereitnotfortheenthusiasm,support andencouragementofmyhusbandChristoph.Iammostgratefultohimforbeingthe firstreaderofthisbookandgivingmemanyrelevantcommentsandsuggestions.
PartISingleandMultilayerNetworks
3.1Interplaybetweenstructureandfunction47
3.2Phasetransitionsandemergentphenomena48 3.3Robustnessandpercolation49
3.4Epidemicspreading58
4.6Molecularnetworksandtheinteractome92
4.7Brainnetworks95
4.8Ecologicalnetworks98
4.9Climatenetworks99
5TheMathematicalDefinition 100
5.1Generalmultilayernetworksandmorespecifictopologies100
5.2Themostgeneralmultilayernetwork100
5.3Multiplexnetworks102
5.4Multi-slicenetworks106
5.5Othertypesofmultilayernetwork110
5.6Tensorialformalismformultilayernetworks114
6BasicStructuralProperties 117
6.1Theeffectofmultiplexityonnetworkstructure117
6.2Degree117
6.3Clusteringcoefficient122
6.4Distance-dependentmeasures127
7StructuralCorrelationsofMultiplexNetworks 129
7.1Correlationsinmultiplexnetworks129
7.2Interlayerdegreecorrelationsinmultiplexnetworks130
7.3Overlap,multilinksandmultidegrees135
7.4Correlationsinweightedmultiplexnetworks141
7.5Theactivitiesofthenodesandpairwisemultiplexity143
8Communities 146
8.1Therelevanceofcommunitiesinmultilayernetworks146
8.2Multilayercommunitydetection146
8.3Correlationsinthecommunitystructureofmultiplexnetworks159
8.4Toaggregateortodisaggregate?166
9CentralityMeasures 170
9.1Centralitymeasuresandmultiplexity170
9.2MultiplexPageRank170
9.3MultiplexEigenvectorCentralities172
9.4FunctionalMultiplexPageRank173
9.5MultiRank180
9.6Versatility183
9.7MultilayerCommunicability186
9.8Centralityofmulti-slicenetworks188
10MultilayerNetworkModels 190
10.1Differentapproachestomultilayernetworkmodelling190
10.2Growingmultiplexnetworkmodels190
10.3Multiplexnetworkensembles199
10.4Randomizationalgorithms210
10.5Modelsofmulti-slicetemporalnetworks212
10.6Ensemblesofmoregeneralmultilayernetworks220
11InterdependentMultilayerNetworks 226
11.1Interdependenciesinmultilayernetworks226
11.2Percolationofinterdependentnetworks227
11.3Interdependentmultiplexnetworkswithoutlinkoverlap230
11.4Interdependentmultiplexnetworkswithlinkoverlap243
11.5Partialandredundantinterdependencies246
11.6Percolationoninterdependentmultilayernetworks254
12ClassicalPercolation,GeneralizedPercolationandCascades 260
12.1Robustnessofmultilayernetworks260 12.2Classicalpercolation261 12.3Directedpercolation270 12.4Antagonistpercolation274
13.1Epidemicsandmultiplexity282 13.2SISmodel283 13.3SIRmodel293
13.4Interplaybetweenawarenessandepidemicspreading301
13.5Competingepidemicspreadingonmultiplexnetworks305
13.6Epidemicspreadingonmulti-slicetemporalnetworks306 14Diffusion 309
14.1Therelevanceofdiffusiononmultilayernetworks309
14.2Diffusiononmultiplexnetworks310
14.3Randomwalksonmultiplexnetworks314
14.4Randomwalksonmulti-slicetemporalnetworks321
15.1Dynamicalsystemsinmultilayernetworks324 15.2Synchronization324 15.3Patternformation334
15.4Multiplexvisibilitygraphs336
15.5Controlofmultilayernetworks337 16OpinionDynamicsandGameTheory 343
16.1Modellingsocialnetworkdynamics343 16.2Votermodel343
AppendixATheBarabási–Albertmodel:theMasterEquation
PartI 1 ComplexSystemsasMultilayer Networks 1.1Whataremultilayernetworks? ThefundamentalideabehindNetworkScienceisthatimportantinformationabouta complexsystemcanbegainedbystudyingitsunderlyingnetworkstructure.Thissimple yetpowerfulpointofviewhasprovidedthetoolsforgainingunprecedentedknowledge ontherichinterplaybetweenthestructureandfunctionofcomplexsystems.
ThefieldofNetworkSciencehasbeenflourishinginthelastdecades,wherewehave witnessedaBigDataexplosioninsocialscience,biologyandengineering.Network Scienceisahighlyinterdisciplinaryfieldthatcombinestoolsandtechniquescoming fromphysics,mathematics,statistics,biology,engineeringandcomputerscience.Now almosttwentyyearssincethebeginningofthefield,wehavereachedunderstandingof complexnetworksandtheiruniversaltopologicalpropertiesandwehaverevealedthe richinterplaybetweenstructureanddynamicsincomplexnetworkarchitectures.
Inthelastfewyearsithasbeenpointedoutbyseveralresearchersthatourunderstandingofcomplexnetworkshassofarhadanimportantlimitation.Infact,rarely donetworksworkinisolation.Frominfrastructuresandtransportationsystemstocells andthebrain,mostnetworksaremultilayer,i.e.theyareformedbyseveralinteracting networks.Forexample,inmodernsocietydifferentinfrastructuresarerelatedbya complexwebofinterdependenciesandafailureinthepowergridcantriggerfailuresin theInternet,thefinancialmarketandtransportation.Whencommutingtotheworkplace, theinhabitantsoflargecitiesusuallytakemorethanonemeansoftransportation includingbus,metropolitantrainsandunderground.Inthecell,theprotein–protein interactionnetwork,signallingnetworks,metabolicnetworksandtranscriptionnetworks arenotisolatedbutinteracting,andthecellisnotaliveifanyoneofthesenetworks isnotfunctioning.Inthebrain,understandingtherelationoffunctionalandstructural networksformingamultilayernetworkisoffundamentalimportance.
Multilayernetworkshavebeenfirstintroducedinthecontextofsocialsciencesto describedifferenttypesofsocialties.Uptonow,socialnetworksremainoneofthetypical
examplesofmultilayernetworks.Nevertheless,multilayernetworkshaveattracteda significantinterdisciplinaryinterestonlyinthelastfewyears,becauseithasbecomeclear thatcharacterizingmultilayernetworksisfundamentaltounderstandingmostcomplex networksincludingcellularnetworks,thebrain,complexinfrastructuresandeconomical networksinadditiontosocialnetworks(chapter4).
Interestingly,theframeworkofmultilayernetworks(chapters5–6)canalsobeapplied totemporalnetworks,i.e.networksthatchangeovertime.Temporalnetworkscanbe usedtodescribealargevarietyofdata,rangingfromcontactsnetworksrecordingfaceto-facesocialinteractionstotime-resolvedcorrelationsbetweendifferentregionsofthe brain.Inthiscase,themultilayernetworkisformedbytemporalsliceseachdescribing theinteractionsoccurringinagiventemporalinterval.Itturnsoutthatthemultilayer approachforstudyingtemporalnetworkscanbeextremelyusefulforadvancingour understandingofthedynamicalprocessesoccurringinthem,suchasdiffusionand epidemicspreading.
1.2Informationgaininmultilayernetworks Amultilayernetworkisnottobeconfusedwithalargernetworkincludingallthe interactions.Asanetworkultimatelyisawaytoencodeinformationabouttheunderlying complexsystem,thereisasignificantdifferencebetweenconsideringalltheinteractions atthesamelevelandincludingtheinformationonthedifferentnaturesofthedifferent interactions.Inamultilayernetwork,eachinteractionhasadifferentconnotation,andthis propertyiscorrelatedwithotherstructuralcharacteristics,allowingnetworkscientists toextractsignificantlymoreinformationfromthecomplexsystemunderinvestigation (chapters7–9).
Amajorthemeofthisbookisthediscussionofthemajortypesofcorrelationthat arepresentinmultilayernetworkdatasets.Wewillshowhowthesecorrelationscanbe quantifiedandwewillpresentseveraltechniquesforextractingrelevantinformationfrom multilayernetworkdatasetsthatcannotbefoundbyconsideringnetworksinisolation. Theseincluderankingalgorithmsaimedatassessingtherelevanceofnodesinmultilayer networksandalgorithmsthataimtoextractthemesoscaleorganizationofmultilayer networksindifferentmultilayernetworkcommunities.
Thisfieldisexpectedtohavesignificantimpactinavarietyofcontexts,including mostnotablynetworkmedicineandbrainresearch.Inbrainresearch,theabilityto makesenseofthemainstructuralcharacteristicsofbraindataisessentialtoadvance ourunderstandingoftheinterplaybetweenthestructureoftheconnectome,describing themacroscopicwiringofthebrain,andfunctionalbrainnetworks,sheddinglighton braindynamics.Networkmedicineandpersonalizedmedicineaimatfindingthebest treatmentforaspecificpatientbyintegratingseveralmedicaldatasetsthatusuallytakethe formofmultilayernetworks.Theadvanceinourabilitytoextractrelevantinformation fromthesedatasetsisthereforeoffundamentalsignificanceforthewell-beingofsociety.
Formakingsenseofthelargesetofmultilayernetworksweneedtocombineinference algorithmsandtechniqueswithnullmodelsofmultilayernetworks(chapter10).Thenull
modelsdefinewell-controllednetworkstructuresthatconstitutethereferencepointto whichtheresultsobtainedbyinvestigatingrealdatasetscanbecompared.Additionally, nullmodelscanbetakenasbenchmarkstructuresoverwhichwecanrunsimulationsof dynamicalprocesses.Thiscanallowustotesttheeffectofmultilayernetworkstructures onthecharacteristicbehaviourofthedynamicstakingplaceonthem.
1.3Overviewofdynamicalprocessesonmultilayer networks Inmultilayernetworks,linksmightindicatedifferenttypesofinteractions.Thisproperty ofmultilayernetworkshasessentialconsequencesonthedynamicalprocesses(chapters 11–16)definedonsuchstructures.Frompercolationtodiffusionandgametheory, ingeneralthedynamicalinteractionsbetweennodesinamultilayernetworkwilltake adifferentfunctionalformdependingonthenatureofthelink.Forexample,ifwe consideramultilayertransportationnetworkformedbytheairportnetworkofflight connections,thetrainandtheroadtransportationnetworks,wewillobservethatthe rulesdeterminingthediffusionwithineachofthenetworksmightbedifferent,and thatchangingfromonemeansoftransportation(diffusionfromonelayertotheother ofthemultilayernetworks)mightagainfollowotherdynamicalrules.Thisscenario doesnotonlyapplytotransportationnetworksbutalsotothediffusionofideasand behavioursinsocialnetworks.Therefore,thefactthatdiffusiononmultilayernetworks ischaracterizedbydifferentratesdependingonthetypeoflinksignificantlychanges thepropertiesofthisdynamicalprocessandhasavarietyofpracticalconsequences. Similarly,thenodesofamultilayernetworkmightresponddifferentlytothedamageof nodesinthesamelayerorinanotherlayer.Forexample,takeanInternetrouter.This routermightstillbefunctionalevenifoneoftheconnectedInternetroutersisdamaged, butitmightnotbefunctionalanymoreifthepowerplantprovidingenergytotherouter isdamaged.Therefore,inamultilayernetworkwecandistinguishbetweenconnectivity linksprovidingconnectivitytothenodesofeachlayerandinterdependencylinksthat implytheimmediatedamageofonenodeiftheotherlinkednodeisdamaged.This property,commontomanyinterconnectedinfrastructures,makesthemmorefragilethan singlenetworks.Therefore,theimplicationsofinterdependenciesontherobustnessof multilayernetworksisessentialtobuildmorereliableandresilientglobalinfrastructures.
Inrecentyearsithasbeenshownthatconsideringthemultilayernatureofnetworks cansignificantlymodifytheconclusionsreachedbyconsideringsinglenetworks.Anumberofdynamicalprocesses,includingpercolation,diffusion,epidemicspreadingand gametheory,presentaphenomenologythatisunexpectedifoneconsidersthelayersin isolation.Moreover,ithasbeenshownthatthestructuralcorrelationsbuiltinmultilayer networkstructurescansignificantlychangethedynamicalpropertiesofthemultilayer network.Thisspectacularinterplaybetweenstructureanddynamicsisverylikelyto opennewscenariosforapplicationsandcontrolofmultilayernetworks,includingthe designofmoreresilientinfrastructuresandtransportationsystemsandthepossibilityof reprogrammingcancercells.
PartII SingleNetworks TheStructureofSingleNetworks 2.1Networks Networksareformedbyasetofnodesdescribingtheelementsofacomplexsystem connectedpairwisebyseverallinksdescribingtheircomplexwebofinteractions.Most networksreflectintheirstructurearichinterplaybetweenrandomnessandorder.For instance,insocialnetworkstheestablishmentofafriendshipmaydependonaseriesof contingentevents,whileinthebraintheconnectionsbetweenneuronsarenotalldeterminedbygenomicinformation.Ifstochasticityisubiquitousincomplexnetworks,these networksarenotmaximallyrandomeither;rather,theyobeyorganizationprinciplesthat makethemfunctional.NetworkSciencecharacterizesnetworkstructurestoincreaseour understandingofcomplexsystems,asitisassumedthattheunderlyingnetworkstructure ofacomplexsystemencodesinformationaboutitsfunction.Inthisrespect,theeffort madeinbiologytogatherreliableandcompleteinformationonbiologicalinteractions isnoticeable.Thisworkrangesfromthehigh-throughputexperimentsthataimto completetheinformationaboutthehumanproteininteractionnetworktothebigprojects thataimtomapthehumanconnectome.NetworkScienceincludesnetworkinference andcharacterizationofnetworkstructure,butalsogoesbeyondtopologyandaimsat identifyingtheeffectsthatnetworkshaveonsocial,technologicalandbiologicalprocesses andatpredictingthebehaviourofcomplexsystems.Inthischapterwewillfocusonthe majorresultsobtainedbystudyingnetworkstructure,whileinthesubsequentchapterwe willfocusonnetworkdynamics.Ourintentionisheretogivesomerelevantbackground onsinglenetworkswhichmightserveasareferencetothecoreofthebookonmultilayer networks.However,giventhespacelimitations,wewillnotbeableinanywaytogivea completeaccountofthelargeliteraturethatexistsonNetworkScience.Wesuggestto thenovicewantingtodeepenhisunderstandingtoreadtherelevantmonographieson singlenetworks[14,107,225,105,184].Conversely,theveryexperiencedreaderfamiliar withmostoftheresultsvalidforsinglenetworkscanusethematerialofchapter2and chapter3onlyasareferenceforthediscussionofmultilayernetworkspresentedinPart III(chapters4–16).
2.2Singlenetworktypes Agraph G = (V , E ) isformedbythepairofsets V and E where V isthesetofnodes (orvertices)and E isthesetoflinks(oredges).Networksaregraphsthatdescribe realinteractingsystemsasdiverseasthebrainortheInternet.Singlenetworkscome indifferenttypesdependingonseveralaspectscharacterizingtheirinteractions.
Singlenetworkscanbeclassifiedasundirectedordirectednetworks.
Undirectednetworksareformedbyundirectedinteractions,andinthesenetworksif node i islinkedtonode j thenautomaticallynode j islinkedtonode i .Forinstance, Facebookisanundirectednetwork,astheFacebookfriendshipindicatesanundirected interactionthathasbeenagreedtobythetwoinvolvedaccounts.Similarly,inbiology aproteininteractionnetworkisundirected,asanyproteininteractionindicateswhether twoproteinsbindtogethertoformaproteincomplex.
Onthecontrary,directednetworksarenetworksinwhichtheinteractionsaredirected, andifnode i pointstonode j itisnotautomaticallytruethatnode j pointstonode i . TheWorldWideWebisaclearexampleofadirectednetworkwherelinksfromone webpagetoanotherarenottypicallyreciprocated.Withinonlinesocialnetworks,Twitter isaclearexampleofadirectednetworkwhereaccountsdonotalwaysfolloweach other.
Singlenetworkscanalsobeclassifiedas unweighted or weighted
Weightednetworksarenetworkswherea weight isassociatedwitheachinteraction, describingtypicallyameasureofthe‘intensity’oftheinteraction.Forinstance,in theairportnetworkformedbyflightconnectionsbetweenairports,aweightcanbe associatedwiththelinksaccordingtothetraffic(intermsofnumberofpassengers) ofeachconnection.Innetworksgeneratedfromcorrelationsbetweentimeseriessuchas brainfunctionalnetworksorfinancialnetworksbetweenassets,weightscanbeassociated withlinkswherestrongerweightsindicatelargeandpositivecorrelations.
Unweightednetworks,onthecontrary,arenetworksinwhicheachinteractionis eitherpresentorabsent.Unweightednetworksmightcorrespondtonetworksinwhich theweightsaredisregardedornetworksinwhichtheweightsarethesameforevery interaction.
Themostfundamentaltypesofsinglenetworksare simplenetworks thatareundirected andunweighted,inwhichinteractionsexistonlybetweendifferentnodes.
Whiletheaboveclassificationofsinglenetworksdependsonthepropertiesofthe networkinteractions,itisalsopossibletoconsidernetworkshavingnodeswithdifferent properties.
Bipartitenetworks arenetworksformedbytwodistincttypesofnodesinwhich interactionsexistexclusivelybetweendifferenttypesofnodes.Bipartitenetworksinclude thenetworksbetweenactorsandmovieswhereeachactorisconnectedtoamovieif hehasactedinit,orthenetworkbetweenscientistsandpaperswhereeachscientistis connectedtoapaperifhehasauthoredit.
2.3Basicdefinitions 2.3.1Nodesandlinks
Themostbasicpropertiesofsinglenetworks G = (V , E ) arethetotalnumberofnodes N (alsocalledthe networksize)andthetotalnumberoflinks L with
wherethesymbol |X | indicatesthecardinalityoftheset X .Wewillindicatethelabelled nodesofthenetworkwith i = 1,2, ... , N .Therefore,thesetofnodes V isgivenby
Thelinkswillbeindicatedaspairsofnodelabels (i , j ) whereforundirectednetworksthe orderisirrelevant,whilefordirectednetworkstheorderindicatesthatnode i pointsto node j .Notethatforundirectednetworkseachundirectedlinkjoiningtwogivennodes ofthenetworkiscountedonce,whilefordirectednetworksalinkfromnode i tonode j iscountedindependentlyofthelinkwhichmighteventuallyconnectnode j tonode i Bipartitenetworks,wherenodescanbecastintotwodifferentsetsandinteractions onlyexistbetweennodesbelongingtodifferentsets,shouldbetreatedsomewhat differently.Infact,abipartitenetworkcomprisesthreesets: GB = (V , U , E ),wherethe sets V and U indicatetwodifferentgroupsofnodes(forinstance, V and U might indicateactorsandmovies).Thesetwosetsmighthavedifferentcardinality, |V |= NV and |U |= NU ,indicatinginourexamplethetotalnumberofactorsandthetotalnumber ofmoviesrespectively.Theelementsoftheset V willbeindicatedbyLatinletters i , j etc. Theelementsoftheset U willbeindicatedbyGreekletters μ, ν etc.Finally,theset E indicatesthesetoflinksconnectingnodesoftheset V onlytonodesoftheset U
2.3.2Adjacencymatrixandincidencematrix Anysinglenetwork G = (V , E ) isfullydeterminedbyitsadjacencymatrix.Theadjacency matrixisan N × N matrix a,whoseelements aij indicatewhethernode i islinkedto node j .Thespecificdefinitionoftheadjacencymatrixdependsonwhetherthenetwork isdirectedorundirected,weightedorunweighted.
Forunweightedandundirectednetworkstheadjacencymatrixelements aij are givenby aij = 1ifnode i islinkedtonode j , 0otherwise. (2.3)
