P.J.Mucha,T.Richardson,KevinMacon,M.A.Porter,&J.‐P.Onnela,arXiv:0911.1824(andanaddi=onalpaperin2010NonlinearScienceGallery,toappearinChaos)
CommunityStructurein“Mul1slice”Networks
(CommunityStructureinTime‐Dependent,Mul=scale,andMul=plexNetworks)
MasonA.PorterMathema=calIns=tute,UniversityofOxford
Goal• Extendquality‐op=miza=onmethodsofcommunitydetec=ontonetworkswiththefollowingfeatures:– Mul6scale:Considermul=pleresolu=onparametersatonce(withoutsweeping)
– Time‐Dependent:Nodesandedgescanchangein=me
– Mul6plex:Mul=pletypesofedges
Outline• Background:Communitystructureandcommunitydetec=on
• Mul=slicenetworks
• Examples• Conclusions
CommunityStructurebyhand?:BaseballSteroidsNetworks
ImagesfromA.Clauset,C.Moore,&M.E.J.Newman(Nature,2008)
Iden=fyingCommuni=esAlgorithmically
“Thiswouldn’tbeacommunitydetec=ontalkwithouttheZacharyKarateClub.”
Thispar==onop=mizesmodularity,whichmeasuresthenumberofintra‐community=es(rela=vetorandomness)
“NetworkScience”Coauthorship
FacebookFriendshipNetworks
A.L.Traud,E.D.Kelsic,P.J.Mucha,&M.A.Porter,arXiv:0809.0960
CongressionalCommi_ees(MAP,PJM,M.E.J.Newman,C.M.Warmbrand,&A.J.Friend)
Preliminaries• “Hard/rigid”versus“soc/fuzzy/overlapping”clustering
• Acommunityshoulddescribea“cohesivegroup”ofnodes– Tonsofalgorithmsavailable
• Unifyingno=on:moreintra‐communityedgesthanonewouldexpectatrandom– Butwhatdoes“atrandom”mean?
• Reviewar=cles– “Communi=esinNetworks,”M.A.Porter,J.‐P.Onnela&P.J.Mucha,No6cesoftheAmericanMathema6calSociety56,1082‐97&1164‐6(2009).
– “CommunityDetec=oninGraphs,”S.Fortunato,PhysicsReports486,75‐174(2010).
Quality/Modularity
• Popularapproach:Usea“modularity”qualityfunc=on
• GOAL:Assignnodestocommuni=estomaximizemodularity.
CommunityDetec=on:NullModels
• Erdös‐Rényi(Bernoulli) • Newman‐Girvan*
• Leicht‐Newman*(directed) • Barber*(bipar=te)
*withresolu6onparametersγ
CommunityDetec=on:Computa=onalHeuris=cs
• Cannotguaranteeop=malqualitywithoutfullenumera=onofpossiblepar==ons– NP‐completeproblem– Manyalgorithmsavailable(simulatedannealing,etc.)– Needtopicknullmodelappropriatetoproblem– Extremedegeneraciesingoodlocalop=maofQ
• (B.H.Good,Y.‐A.deMontjoye,&A.Clauset,toappearinPRE;seeAaron’stalkonWednesday)
Mul1sliceNetworks
• Typicalformula=onforstudyingnetworks:Sta=cnetworks,withasinglekindof=e,par==onedatasinglespa=alresolu=on– Alsopoten=allysweepovermul=pleresolu=ons(oroccasionally
mul=ple=mes)butinanadhocfashion• Mul=sliceframework:dynamic,mul=plex,andwith
communi=esatmul=plescales• Simpleidea:Gluecommonindividualsacross“slices”
Whatistheappropriatenullmodel?
• Eachsliceisanetwork(sta=c,singletype)withaspecifiedspa=alresolu=onofinterest
• Differentslicescanmean:differentvalueofresolu=onparameter,different=mesnapshot,differenttypeofconnec=on
• Havebothintra‐sliceedges&inter‐sliceedges• Howtochooseanullmodel?
QualityofPar==onvia“Stability”
• Idea:useadynamicalprocessonanetworktolearnaboutnetworkstructure– WebuildonworkofR.Lambio_e,J.‐C.Delvenne,&M.
Barahona[arXiv:0812.1770]• Qualityofanetworkpar==onexpressedintermsofits
“stability”(autocovariancefunc=onofanergodicMarkovprocessonthenetwork):
– P(C,t)=probability,foragivencommunityC,forarandomwalkertobeinthatcommunitybothini=allyandat=met
• Stabilitymeasuresthequalityofapar==onintermsofthepersistenceofthedynamicsbygivingaposi=vecontribu=ontocommuni=esfromwhicharandomwalkerisunlikelytoescapewithagiven=met
LaplacianDynamics(i.e.,randomwalks)
• Lambio_e,Delvenne,&Barahona[arXiv:0812.1770]derivedmodularityfromnormalizedLaplaciandynamics
Expansionofmatrixexponen=altofirst‐orderintrecoversNewman‐Girvanmodularitywithresolu=onγ=1/t.
Ques6on:Howdoweapplythisideatomul6slicenetworks?
GeneralizedLaplacianDynamics
a) Calculate(tofirstorderint)theprobabilityofobservinga=ebetweennodesiandj,condi6onalonthetypeofconnec6onnecessarytomoveji
b) Generalizedynamicstoincludemo=onalongdifferenttypesofedges
c) Differentspreadingweightsondifferenttypesofedges
Mul=sliceNetworks
SpecialCase:Bipar=teNetworks
• RecoverBarbernullmodelwithresolu=onparameter:
SpecialCase:DirectedNetworks
• RecoverLeicht‐Newmannullmodelwitharesolu=onparameter:
SpecialCase:SignedNetworks
• RecovernullmodelsofTraagetal.&Gomezetal.:
Examples• (Revengeofthe)ZacharyKarateClub– Mul=pleresolu=onparametervaluesatonce(“mul=scale”)
• Tastes,Ties,&Time– Mul=plex(mul=pleedgetypes)
• 200yearsofrollcallvotesinU.S.Senate– Time‐dependent
ZacharyKarateClub
Tastes,Ties,&Time
• DatafromLewisetal.2008• “not‐Harvarddataset”
• Firstwaveofprivatenortheasternschool
• Edgetypes:• Facebookfriends
• Picturefriends
• Roommates
• HousingGroups
RollCallVo=ngNetworks
• A.S.Waugh,L.Pei,J.H.Fowler,P.J.Mucha,&M.A.Porter,arXiv:0907.3509(withoutmul=sliceformula=on)
• Modularityasameasureofpolariza=on• Modularityasapredictorofmajorityturnover(the“par=al
polariza=onhypothesis”)• Onenetworksliceforeachtwo‐yearCongress
110Senates(220years)
110Senates
Grayareas:3communi=esexistatthesame=me(9communi=esintotal;ω=0.5)
Arrangedbystate…
Conclusions• “Mul=slice”frameworkgeneralizestheinves=ga=onofcommmunitystructuretomorecomplicated,morerealis=c,andmuchmoreinteres=ngsitua=ons:dynamic/longitudinaldata,mul.plex=es,andcommuni=esacrossmul.plescales.
• Visualiza=ontoolsforgraphsthatincorporatecommunitystructure:h_p://netwiki.amath.unc.edu/VisComms/VisComms
• Currentefforts:Applymul=slicecommunitydetec=ontoepidemics,poli=cians,andbrains– Otherideas:newnullmodels,choiceofinter‐slicecoupling,
generaliza=onsofclusteringcoefficientsandotherquan==estomul=slicenetworks,…
BinaryTrees