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