Ties That Bind - Characterizing Classes by Attributes and Social Ties

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by  Aria Rezaei

Ties That Bind - Characterizing Classes by Attributes and Social
Ties
AriaRezaei BryanPerozzi∗ LemanAkoglu
StonyBrookUniversity GoogleResearch H.JohnHeinzIIICollege
StonyBrook,NY NewYork,NY CarnegieMellonUniversity
[email protected] [email protected] [email protected]
ABSTRACT node.Biologicaldatacanalsoberepresentedasattributedgraphs;
Givenasetofattributedsubgraphs knowntobefromdifferent protein-proteininteraction(PPI)networkscanhavegeneencodings
7 classes, howcanwediscovertheirdifferences? Therearemany ofproteinsasattributes,orgeneinteractionnetworksmaycontain
1 caseswherecollectionsofsubgraphsmaybecontrastedagainst geneontologypropertiesasattributes[14,22].
0 eachother.Forexample,theymaybeassignedgroundtruthlabels Weconsiderthefollowingquestion: Givenacollectionofat-
2 (spam/not-spam), or it may be desired to directly compare the tributedsubgraphsfromdifferentclasses,howcanwediscoverthe
n biologicalnetworksofdifferentspeciesorcompoundnetworksof attributesthatcharacterizetheirdifferences?Thisisageneralques-
a differentchemicals. tion, whichfindsapplicationsinvarioussettingsdependingon
J Inthisworkweintroducetheproblemofcharacterizingthe how‘subgraphs’and‘classes’aredefinedandinterpreted.Insocial
1 differencesbetweenattributedsubgraphsthatbelongtodifferent networks,subgraphscouldbethelocalcommunitiesaroundeach
3 classes. Wedefinethischaracterizationproblemasoneofparti- individual.Thatisbecauseone’sacquaintancescarryalotofinfor-
tioningtheattributesintoasmanygroupsasthenumberofclasses, mationaboutthemduetothefactorsofhomophily(phenomenon
] whilemaximizingthetotalattributedqualityscoreofallthegiven that“birdsofafeatherflocktogether”)andinfluence(phenomenon
SI subgraphs. thatourattitudesandbehaviorsareshapedbyourpeers)[7].One
Weshowthatourattribute-to-classassignmentproblemisNP- canalsoconsiderthesubgraphsextractedbyacommunitydetec-
.
s hardandanoptimal(1−1/e)-approximationalgorithmexists.We tionalgorithm,thesocialcirclesasdefinedbyindividuals,orany
c alsoproposetwodifferentfasterheuristicsthatarelinear-timein collectionofsmallgraphletsthatcomefromanapplication(e.g.
[
thenumberofattributesandsubgraphs. Unlikepreviouswork PPInetworksofacollectionofflyspecies).Ontheotherhand,a

1 whereonlyattributesweretakenintoaccountforcharacterization, ‘class’correspondstoabroadcategorizationofsubjects.Insocial
v hereweexploitbothattributesandsocialties(i.e.graphstructure). networkanalysis,onemaytrytounderstandthedifferencesbe-
9 Throughextensiveexperiments,wecompareourproposedalgo- tweenindividualslivingindifferentcountries(e.g.U.S.vs.China),
3 rithms,showfindingsthatagreewithhumanintuitionondatasets orhavingdifferentdemographics(e.g.elderlyvs. teenagers). In
0 fromAmazonco-purchases,Congressionalbillsponsorshipsand biology,onemaywanttoanalyzePPInetworksofhealthyversus
9 DBLPco-authorships.Wealsoshowthatourapproachofcharac- sickindividualsorofmiceversushumans.1
.0 terizingsubgraphsisbettersuitedforsense-makingthandiscrimi- Inthiswork,weproposetocharacterizethedifferentclasses
1 natingclassificationapproaches. throughtheattributesthattheirsubgraphsfocuson.Intuitively,we
0 assumethenodesineachsubgraphshareasubsetofattributesin
7 CCSCONCEPTS common(e.g.acircleoffriendswhogotothesameschoolandplay
1 baseball).Thatis,membersofasubgraph“click”togetherthrougha
v: c•Iennftoerrmedatcioomnpsyusttienmgs→→SoDcaitaalmneitnwinogrk;Saoncaialyltsaisg;ging;•Human- sharedcharacterizingattributesubspace,calledthefocusattributes
[28].Itisexpectedthatoutofalargenumberofattributes,onlya
i
X ACMReferenceformat: fewofthemwouldberelevantforeachsubgraph.
r AriaRezaei,BryanPerozzi,andLemanAkoglu..TiesThatBind-Charac- Ourmaininsightforcomparingsubgraphsisthenthatthesub-
a terizingClassesbyAttributesandSocialTies.InProceedingsofDraft,Perth, graphsfromdifferentclasseswouldexhibitdifferentfocusattributes.
Australia,April2017(WWW’17),9pages. Inotherwords,theattributesthatcharacterizethesubgraphsof
DOI:
oneclassaredifferentfromtheattributesaroundwhichthesub-
graphsfromanotherclasscenterupon.Astereotypicalexampleto
1 INTRODUCTION thisinsightisteenagersfocusingonattributessuchas‘selfies’and
Besides connectivity, many graphs contain a state (or content) ‘partying’whereaselderlybeingcharacterizedby‘knitting’and
vectorforeachnode.Thistypeofgraphisknownasanattributed ‘teapartying’.Notethateventhoughclassesmightsharecommon
graph, and is a natural abstraction for many applications. For attributes,weaimtoidentifythosethatareexclusiveandnotthe
example,inasocialnetworktheprofileinformationofindividuals overlappingones;asthosebesthelpcharacterizethedifferences.
(e.g.age,occupation,etc.)constitutetheattributevectorforeach Figure1presentsacompletesketchofourproblem.
Avastbodyofmethodsforcommunitydetectionhasbeenpro-
∗WorkperformedwhileatStonyBrookUniversity posedforbothsimple[4,8,16,26,36]aswellasattributedgraphs
WWW’17,Perth,Australia
2017.978-x-xxxx-xxxx-x/YY/MM. 1Herewefocusontwoclassesforsimplicityhoweverourmethodseasily
DOI: generalizetosubgraphsfromanynumberofclasses.
WWW’17,April2017,Perth,Australia AriaRezaei,BryanPerozzi,andLemanAkoglu
class A subgraphs a a a a a a
G g g g g 1 2 3 4 5 6 class A class B
A1 A2 A3 A1
gA2 aa3 aa5
g 4 6
A3 a
class B subgraphs gB1 a1
g g 2
B1 B2 g
B2
assignment
attributed graph characterizing subspaces & ranking
(a) (b) (c) (d)
Figure1:Problemsketchontoydata. Given(b)node-attributedsubgraphs(or(a)nodesaroundwhichweextractsubgraphs)
fromdifferentclasses(AandB),wefind(c)thecharacterizingsubspace(i.e.,thefocusattributesandrespectiveweights)for
eachsubgraph,and(d)splitandranktheattributesforcharacterizingandcomparingtheclasses.
[2,12,13,15,23,25,28,33,37].Thoseareprimarilyconcernedwith humanintuitiononreal-worldscenariosfrom3datasets,
extractingdisjointoroverlappinggroupsofnodes,whileoptimiz- anddemonstratethatourcharacterizationapproachisbet-
ingsomegraphclusteringobjective.Ourproblemisconsiderably tersuitedtosense-makingthandiscriminativeclassifica-
different. Unlikethem,ourgoalistounderstandthedifferences tionapproaches(§5).
betweendistinctclassesofsubgraphs(orcommunities)through
theattributesthatcharacterizethem,nottoextractbetterones. 2 PROBLEMDEFINITION
Similarstudieshavebeendoneincharacterizingandcomparing
Inthissection,weintroducethenotationusedthroughouttext
thesocialmediauseofdifferentclassesofsubjects.Forexample,
andpresenttheformalproblemdefinition. Anattributedgraph
featuresfromauser’ssocialmediainteractionshavebeenshownto
G=(V,E,A)isagraphwithnodesetV,undirectededgesE ⊆V×V,
predictdemographicinformationsuchasgender[5],age[29,31],
occupation[30],location[9,18],andincome[10].Morenuanced andasetofattributesA={a1,...ad}associatedwitheverynode,
traitshavealsobeenpredictedaboutindividuals,suchaspersonality whereai ∈(cid:60)d denotesthed-dimensionalattributevectorfornode
i. Inthisworkweconsiderreal-valuedandbinaryattributes. A
[32],ormentalillness[6]. However,thesestudiestendtofocus
categoricalattributecanbetransformedtobinarythroughone-hot
solelyontextattributesanddonotconsiderbroaderlevelsofsocial
encoding.
interactionsinanetwork.
Givenacollectionofattributedsubgraphsfromc classes,our
ArecentworkinthesamelineswithoursisbyDellaPostaet aimistosplittheattributesinAintoc disjointgroupssuchthat
al.[7],whichstudiednetworkeffectsforexplaininghowpolitical thetotalqualityscoreQ ofallthesubgraphsbasedonfunction
ideologybecomeslinkedtolifestyles,suchas“latteliberals”and
q(·)andtheirassignedattributesismaximized. Hereweusethe
“bird-huntingconservatives”.Theirsimulatedmodelsrevealstrong
normalitymeasure[27]forq(·),whichcanbereplacedwithany
indicationsforinfluencesoperatingbetweenindividualsinpolitical othermeasureofinterestthatcanutilizebothgraphstructureand
“echo chambers” rather than within individuals, demonstrating
attributesingeneral.
evidencetoward“oneisthecompanytheykeep”,i.e.,thatsocial
Ourproblemstatementisgivenfortwoclassesasfollowsfor
tiesmatter.
simplicity,whichcanbegeneralizedtomultipleclasses.
Inthiswork,weanalyzethedifferencesbetweenindividuals
fromdifferentclasses. Unlikepreviousworkwhichhasfocused Definition2.1(CharacterizationProblem).
primarilyontheindividual’sattributes(mostlytext),weuselo- Given
calcommunitiesaroundindividualnodesinadditiontoattributes • pattributedsubgraphsд1+,д2+,...,дp+fromclass1,
tocharacterizethem. Specifically,ourcontributionsincludethe • nattributedsubgraphsд1−,д2−,...,дn−fromclass2,
following:
fromgraphG,andattributevectora∈(cid:60)d foreachnode;
• We introduce the general characterization problem for
Find
agivencollectionofattributedsubgraphsfromdifferent • apartitioningofattributestoclassesasA+andA−,where
classes—whichleveragesboththestructureofsocialtiesas A+∪A−=AandA+∩A−=∅,
wellastheattributes.Ourformulationentailspartitioning • focusattributesA+ ⊆ A+(andrespectiveweightsw+)for
theattributesintoasmanygroupsasthenumberofclasses, eachsubgraphд+,i∀i,and i
whilemaximizingthetotalattributedqualityscoreofthe • focusattributesiA− ⊆ A−(andrespectiveweightsw−)for
inputsubgraphs(§3). eachsubgraphд−,j∀j; j
• Weshowthatourattribute-to-classassignmentproblem j
isNP-hardandanoptimal(1−1/e)-approximationalgo- suchthat
rithmexists(§4.1). Wealsoproposetwodifferentfaster • totalqualityQ ofallsubgraphsismaximized,whereQ =
heuristicsthatarelinear-timeinthenumberofattributes (cid:205)pi=1 q(дi+|A+) + (cid:205)nj=1 q(дj−|A−);
andsubgraphs(§4.2). RankattributeswithinA+andA−.
• Throughextensiveexperiments,wecomparetheperfor-
Theaboveproblemcontainsthreesubproblems,inparticular,
manceofthealgorithms,presentfindingsthatagreewith
(P1)howtomeasurethequalityofanattributedsubgraph,(P2)
TiesThatBind-CharacterizingClassesbyAttributesandSocialTies WWW’17,April2017,Perth,Australia
howtofindthefocusattributes(andtheirweights)ofagivensub- д.Thisobjectiveiswrittenas
graph,and(P3)howtoassignandranktheattributesfordifferent
classessoastomaximizetotalquality. Inpractice,classesfocus max. wgT ·(aI +aX)
wg
onasmallsetofattributes.Further,ourrankingoftheattributes
ensuresthoseirrelevanttobothclassesandthosecommonbetween s.t. (cid:107)wg(cid:107)p =1, wg(a)≥0, ∀a=1...d (2)
themarerankedlowerandonlyafewofthemostdifferentiating Notethatwgisnormalizedtoitsp-normtorestrainthesolution
attributesstandout.Figure1showsanexampleforourproblem space.Wealsointroducenon-negativityconstraintontheweights
for5subgraphsfrom2classes,where6attributesaresplitintotwo tofacilitatetheirinterpretation.Inthefollowingweletxˆ =(aI +
andrankedforcharacterizingandcomparingtheclasses. aX),wherexˆ(a)∈[−1,1].
Inthenextsection,weaddressthesubproblemsinthegiven Ifoneuses(cid:107)wg(cid:107)p=1,ortheL1norm,thesolutionpicksthesingle
orderabove,in§3.1through§3.3respectively,tobuildupasolution attributewiththelargestxˆ entryasthefocus.Thatis,wg(a)=1
forourmainproblemstatement. wheremax(xˆ) = xˆ(a)and0otherwise. Thiscanbeinterpreted
asthemostimportantattributethatcharacterizesthesubgraph.
Notethatxˆmaycontainonlynegativeentries,inwhichcasethe
3 FORMULATION
largestnegativeentryisselected,andthesubgraphisdeemedas
3.1 QuantifyingQuality
lowquality.
Toinferthecharacterizingsubspaceforagivensubgraph,weuse Iftherearemultipleattributesthatcanincreasenormality,we
ameasureofsubgraphquality. Theideaistofindtheattribute canalsoselectalltheattributeswithpositiveentriesinxˆ asthe
subspaceandrespectiveweightsthatmaximizethequalityofeach subgraphfocus.Theweightsoftheseattributes,however,should
subgraph.Inthiswork,weusethenormalitymeasure[27],which beproportionaltothemagnitudeoftheirxˆvalues.Thisisexactly
notonlyutilizesbothsubgraphstructureaswellasattributes,but what(cid:107)wg(cid:107)p=2,ortheL2normyields. Itisshown(see[27])that
alsoquantifiesbothinternalandexternalconnectivityofthesub- underp=2,
graph.
xˆ(a)
Foragivensubgraphд,itsnormalityN(д)isgivenasinEq. wg(a)= (cid:113) , (3)
(1),whereW istheadjacencymatrix,ki isnodei’sdegree,sim(·) (cid:205)xˆ(i)>0xˆ(i)2
itshethneusmimbielraroiftyedfugnesc,tiaonndoBf(aдtt)rdiebnuotetevsetchteornsowdeeisgahttethdebbyowungd,eariys wherexˆ(a)>0and0otherwise,suchthatwgisunit-normalized.
ofthesubgraph(forisolatedsubgraphs,B(д)isempty). Thetwo ThenormalityscoreofsubgraphдthenbecomesN(д)=wgT·xˆ =
tseurrmprsisinin(g1,)arnedspheicgthivlyelsyimquilaanrtciofynnдeicnttieornnsailnlysiadnedдeixntcerrenaaslelyin:tmerannayl, n(cid:205)oxˆr(ma)>o0fx√ˆi(cid:205)nxˆd(xˆiu)(>ac0)exˆd(io)2nxtˆh(ae)att=ri(cid:113)bu(cid:205)texˆs(i)w>i0thxˆp(io)s2it=ive(cid:107)xxˆˆ+e(cid:107)n2t,riie.es..,the2-
quality,whereasifsuchedgesareattheboundary,theydecrease
externalquality.Fortechnicaldetailsofnormality,see[27].
3.3 IdentifyingClassDifferences
3.3.1 Splittingattributesbetweenclasses. Inthislastpartwe
N(д)=I+X = (cid:213) (cid:0)Wij −k2ikej(cid:1)sim(ai,aj|wg) raetttruirbnutteosopuarcembaeitnwpereonbdleiffmersetanttecmlaesnset,swsohearsetowbeeseaebklettoosipdleitnttihfye
i∈д,j∈д
theirdifferences.Weaimtoobtainsuchanassignmentofattributes
− (cid:213) (cid:0)1−min(1,ki2keb)(cid:1)sim(ai,ab|wg) withagoaltomaximizethetotalquality(i.e.,normality)ofallthe
i∈д,b∈B(д) subgraphsfrombothclasses.Thisensuresthatthesubgraphsare
(i,b)∈E stillcharacterizedwell,evenundertheconstraintthattheattributes
= wgT · (aI +aX) (1) arenotsharedacrossclasses.
LetS+ = {д1+,...,дp+}andS− = {д1−,...,дn−}denotethesets
Onecanhandlehighlyheterogeneousattributessimplybychoosing ofallsubgraphsinclass1andclass2,respectively,whereeach
therightsim(·)function.AlsonotethataI andaX arevectorsthat subgraphisnowassociatedwithad-dimensionalnon-negative
canbedirectlycomputedfromdata.Attributeswithlargenon-zero vectorx.Thisisthesameasthexˆvectorintroducedin§3.2,except
weightsinwgarecalledthefocusattributesofsubgraphд. thatallthenegativeentriesaresettozero.Recallthattheentries
ofxˆdepictthecontributionofeachattributetothequalityofthe
subgraph.Therefore,wecandropthenegativeentries(recallthat
3.2 DiscoveringCharacterizingSubspaces
theoptimizationin(2)selectsonlythepositiveentries,ifany).2
ForasubgraphwecanuseEq.(1)tocomputeitsnormalitypro- ThegoalisthentofindtwodisjointattributegroupsA+andA−,
videdwg,theweightsforthe(focus)attributes.Howeverthefocus A+∪A− =AandA+∩A− = ∅,suchthatthetotalqualityofall
isoftenlatentandhardtoguesswithoutpriorknowledge,espe-
subgraphsismaximized(seeproblemstatementin§2).Givenaset
ciallyinhighdimensionswherenodesareassociatedwithalong
ofselectedattributesS,thequalityofasubgraphдcanbewritten
listofattributes. Evenifthefocusisknownapriori,itishardto
as
manuallyassignweights.
Instead,weinfertheattributeweightvectorforagivensubgraph, 2Theremaybesubgraphsforwhichxˆcontainsonlynegativeentries.We
soastomaximizeitsnormalityscore.Inotherwords,weleverage excludesuchsubgraphsfromthestudyofdiscoveringclassdifferences,as
normalityasanobjectivefunctiontoinferthebestwgforagiven theyaredeemedlowquality.
WWW’17,April2017,Perth,Australia AriaRezaei,BryanPerozzi,andLemanAkoglu
addingthesameattributetoitssmallersubsetS(cid:48); N(д|a∪S)−
(cid:115) N(д|S)≤N(д|a∪S(cid:48))−N(д|S(cid:48)),S(cid:48)⊆S ⊆A.
N(д|S)= (cid:213)x(a)2=(cid:107)x[S](cid:107)2 (4) Underthissetting,wefindthatourproblemin(5)canbestated
a∈S asaninstanceoftheSubmodularWelfareProblem(SWP),whichis
i.e.,the2-normofxinducedontheattributesubspace.Therefore, definedasfollows.
theoverallproblemcanbe(re)formulatedas
Definition 4.1 (Submodular Welfare Problem). Given d items
A+⊆mAa,Ax−.⊆A p1i(cid:213)∈S+(cid:107)xi[A+](cid:107)2+n1 j(cid:213)∈S−(cid:107)xj[A−](cid:107)2 wanid:2m[d]pl→aye(cid:60)rs+,hfianvdinagpmarotnitoiotonninegsoufbtmheodduiltaermustiinlittoymfudnicstjiooinnst
suchthat A+∩A−=∅ (5) setsI1,I2,...,Im inordertomaximize(cid:205)mi=1wi(Ii).
In our formulation items map to the attributes ford = |A|,
Notethatwenormalizethetermsbythenumberofsubgraphsin
whereasplayerscorrespondtotheclasses, inthesimplestcase
eachclasstohandleclassimbalance.Wealsoemphasizethatour
form=2.Inaddition,theutilityfunctioniswrittenforeachclass
objectivein(5)isdifferentfromaclassificationproblemintwo
c ∈{+,−}as
keyways.First,weworkwithxvectorsthatembedinformation
othnesourbiggirnaaplhacttornibnuetcetivveictytoarssaw’se.llSaescofoncdu,souattrroibbjuetcetsivreaethmebrrtahcaens wc(Ic)=N(S(c)|A(c))= n(1c) k∈(cid:213)S(c)(cid:107)xk[A(c)](cid:107)2 (7)
characterizationandaimstofindapartitioningofattributesthat whichistheaveragenormalityscoresofsubgraphsSbelonging
maximizestotalquality,whichisdifferentfromfindingadecision
toclassc.As(cid:107)·(cid:107)2isamonotoneandsubmodularfunction,sois
boundarythat minimizes classificationloss as in discriminative N(S(c))sincethesumofsubmodularfunctionsisalsosubmodular
approaches(See§5).
[21].Notethatalthoughwefocusontwoclassesinthiswork,the
3.3.2 Rankingattributes. Asolutionto(5)(nextsection)pro- SWPisdefinedmoregenerallyformplayers,i.e.,classes.Assuch,
vides a partitioning of the attributes into two groups. We can itiseasytogeneralizeourproblemtomoreclassesfollowingthe
analyzethespecificattributesassignedtoclassestocharacterize samesolutionsintroducedfortheSWP.
theirdifferences.Sincethisisanexploratorytask,analyzingalarge TheSWPisfirststudiedbyLehmannetal.[19],whoproposed
numberofattributeswouldbeinfeasible.Foreasierinterpretation, asimpleon-linegreedyalgorithmthatgivesa1/2-approximation
weneedarankingoftheattributes. forthisproblem.Later,Vondra´kproposedanimproved(1−1/e)-
eacOhnaettcriobuultdetah.iThnkisofhouwsienvger(cid:205)dio∈eSs(cn)oNt(rдeifl|aect∈thAe(cd)i)fffeorrenstcioartiinngg acapnpnrooxtibmeaatpiopnrosxoimluatitoendt[o35a].faKcthoortbeetttaerl.thshanow1e−d1t/hea,tutnhleesSsWPP=
power but only the importance ofa for classc. We want both NP[17].Mirroknietal.furtherprovedthatabetterthan(1−1/e)-
importantanddifferentiatingattributestorankhigherastheytruly approximationwouldrequireexponentiallymanyvaluequeries,
characterizethedifferencebetweensubgraphsofthetwoclasses. regardlessofP=NP[24].Assuch,Vondra´k’ssolutionistheopti-
Specifically,someattributesmayexhibitpositivexentriesfora malapproximationalgorithmfortheSWP,whichweusetosolve
particularclass,howeververysmallvalues,indicatingonlyslight ourproblemin(5).Thesolutionusesamultilinearextensiontore-
relevance. Wemayalsohavesomeattributesthatexhibitlarge laxthesubsetoptimizationintoanumericaloptimizationproblem
positivexentries,howeverforbothclasses.Whilerelevant,such suchthatadvancedoptimizationtechniques,inparticularacontin-
attributesarenon-differentiatingandwouldbeuninformativefor uousgreedyalgorithm,canbeapplied.Thecontinuoussolutionis
ourtask. thenroundedtoobtainanear-optimalsetwiththeaforementioned
To get rid of only slightly relevant or non-differentiating at- guarantee[35].
tributesandobtainasparsesolution,wedefinearelativecontribu-
4.2 FasterHeuristics
tionscorerc(·)foreachattributeaas
1 (cid:213) 1 (cid:213) 4.2.1 Pre-normalizedweights. Fortheformulationshownin(5),
rc(a)= xi(a)− xj(a) (6) weunit-normalizetheattributeweightsasinEq.(3),onlybasedon
p n
wagheicqhuiaslitthyeodfiffsuebregnracephbseiti∈wnSec+elansas’1scaonndtrcjil∈baSuss−tio2.nWaleontheetnorthanekavthere- athseelqeucatelidtysufubsnecttSio:nwNg((aд)|S=)=√(cid:205)(cid:112)a(cid:205)x∈(Saa)x∈(Sa)x2(.aTh)2,isanndorrmeqauliziraetsiotnhaytieSldiss
attributeswithineachclassbytheirrcvalues. given/known.Awaytosimplifythisfunctionistofixtheattribute
4 ALGORITHMS walelitghhets(kantowwgn(a))po=si√tiv(cid:205)eax∈a(Aatt)xr(iab)u2t,eis.e.i,ntoAnroatrhmearlitzheanthaem(unbaksneodwonn)
4.1 OptimalApproximation subset.Thiswaytheattributeweightscanbepre-computedanddo
notdependontheto-be-selectedattributesubsets.Thesimplified
ItiseasytoshowthatourqualityfunctionN(д|S) = (cid:107)x[S](cid:107)2 in versionofthemaximizationin(5)isthenwrittenas
Eq. (4)isamonotonesubmodularsetfunctionwithrespecttoS
fornon-negativex. Thatis,thequalityofasubgraphincreases max. 1 (cid:213) (cid:213) xi(a)2 + 1 (cid:213) (cid:213) xj(a)2
mfoollnowotsonthiceadlliymwiniitshhiinngcrreeatsuirnngsspertospiezretyS.kInnoawdndiitnioenc,otnhoeminiccsre,ai.see., A+⊆A,A−⊆A pi∈S+a∈A+ Di n j∈S−a∈A− Dj
addinganewattributeatoasetSincreasesthefunctionlessthan suchthat A+∩A−=∅ (8)
TiesThatBind-CharacterizingClassesbyAttributesandSocialTies WWW’17,April2017,Perth,Australia
1 1 1
mal Obj. Value00.9.95 mum Obj. Value000...789 SSTTToooWimpppAp---525li50fied mal Obj. Value00.9.95
Ratio to Opti00.8.85 SSTTooWimppAp--35lified Ratio to Maxi0000....3456 Ratio to Opti0.85 SSWimAplified
5 10 15 20 200 400 600 800 1000 0.95 0.80 0.55 0.30 0.05
# Attributes d # Attributes d P
Figure2:Ratioofobjectivevalueachievedbyeachtestalgorithm(left)tooptimalvaluefoundbybrute-force(d =3,4,...,20)
and(center)tomaximumachievedvalue(d =50,100,...,1000).(right)PerformanceofSimplifieddegradesasitassignsnearly
allattributestotheclasswithhigherexpectedxvalues,ignoringthediminishingreturnspropertyoftheobjectivefunction,
whereasSWAremainsnear-optimalunderallsettings.Allresultsareaveragedover10randomdatasets.
wherethedenominatorDi = (cid:112)(cid:205)a∈Axi(a)2 = (cid:107)xi(cid:107)2,whichcan these(separate)solutionsmayenduphavingcommonattributes,
nowbetreatedasconstantasitdoesnotdependonA+(samefor weresolvethesolutionsbyassigningeachcommonattributeonlyto
Dj). theclassforwhichitsaveragecontributionishigher(theindividual
ThesimplifiedfunctionN(д|S) = (cid:107)x[S](cid:107)22 isnowamonotone termsinEq.(6)).Thesearchrepeatsuntileachclassgetsassigned
(cid:107)x(cid:107)2 kuniqueattributes.Finally,kisnotacriticalparametertoset,but
modularfunctionwithrespecttoS.Thecontributionofaparticular
rathercanbechoseninteractively.
newattributetothequalityofasubgraphdoesnotanymoredepend
ontheotherattributesthatarealreadyintheselectedset.Thatis,
5 EVALUATION
N(д|a∪S)−N(д|S)=N(д|a∪S(cid:48))−N(д|S(cid:48))= x(a)2,∀S,S(cid:48)⊆A.
(cid:107)x(cid:107)2 We evaluate our approach to the characterization problem and
Asaresult,asimplelinear-timealgorithmcanbeemployedto
proposedalgorithmsonbothsyntheticandreal-worlddatasets.
solvetheobjectivein(8).Thealgorithmiteratesovertheattributes
Ourgoalistoanswerthefollowingquestions:
(orderdoesnotmatter),andassignseachattributeatotheclasscfor • Howdotheproposedalgorithmsperformandcompareto
whichtheaveragesubgraphqualityisimprovedmorethanothers,
eachother?Whatistheirscalabilityandruntime?
thaWtihs,ilaertghmeoabxjcecnt1(icv)e(cid:205)vaklu∈eSs(co)fxt(cid:107)hkx(eka(cid:107)s)22o,lubtrieoanksitnog(5ti)easnadrb(8it)raarreillyi.kely •• AHroewthdeoefisncdhinagrascotenrirzeaatli-ownocroldmdpaatraemtoeacnlaisnsgiffiucla?tion?
todifferduetothedifferenceincomputingtheattributeweights,
theweightnormalizationdoesnotchangetheorderoftheattributes 5.1 AnalysisonSyntheticDatasets
byimportancewithinagivenset.Therefore,weconjecturethatthe Throughsyntheticdataexperiments,ourgoalistocomparethe
twosolutionswillperformsimilarly,whichweinvestigatethrough algorithmicandcomputationalperformanceoftheproposedal-
experimentsin§5. gorithms,respectivelyintermsofobjectivevalueachievedand
runningtime.Specifically,wecompare:
4.2.2 Top-kattributesperclass. Forexploratorytasks,suchas
understanding the class differences via characterizing attribute • SWA(SubmodularWelfareAlgorithm,§4.1)
subspaces,itwouldbemostinterestingtolookatthetopkmost • Simplified(withpre-normalizedweights,§4.2.1)
importantattributesfromeachclass. Wealsoexpecteachsub- • Top-k(§4.2.2)
graphtoexhibitonlyahandfuloffocusattributes(experimentson Wegeneratethexvectorsforp=n=100subgraphseachfrom
real-worldgraphsconfirmthisintuition).Therefore,limitingthe c =2classes,whilevaryingthenumberofattributesd.Thex(a)
analysistoatopfewattributeswouldbesufficient. valuesofeachfeatureaforsubgraphsfromclasscaredrawnfroma
Foragiven(small)k,findingthetopkattributesA∗k thatmaxi- Normaldistributionwithadistinctµac andσac.Theµac’sthemselves
mizeN(д|A∗k)forasinglesubgraphдiseasy—thatwouldbethe aredrawnfromazero-meanunit-varianceNormal(notethatthose
kattributeswiththelargestvaluesinд’sx(seeEq.(4)).However, attributeswithnegativemeantendtobelessrelevantfortheclass).
wehaveamulti-criterionobjective,withagoaltofindthetopk Theσac’sarerandomlydrawnfroma[0,1]uniformdistribution.
attributesthatmaximizethetotalnormalityforallsubgraphsfrom Algorithmic performance. In the first experiment, we test
aclassatonceratherthanasingleone,thatisN(S(c)|A(c))(seeEq. theoptimalityofthealgorithmsbycomparingtheirachievedob-
∗k
(7)). jectivevaluetothatofbrute-forcewherewetryallpossiblepar-
Themulti-criterionoptimizationproblemisNP-hard[21].On titioningstoidentifytheoptimalsolution. Weexperimentwith
theotherhand,weknowthat (cid:107) · (cid:107)2 isamonotonesubmodular d ={3,4,...,20}asbrute-forceisnotcomputationallyfeasiblefor
function,andsoistheclassqualityfunctionN(·) = (cid:205)(cid:107)·(cid:107)2. As morethan20attributes,andk ={3,5}forTop-k.
such,wefindthetopkattributesforeachclassseparately,using InFigure2(left)wereporttheratiobetweeneachtestalgorithm’s
thelazygreedyhill-climbingalgorithmintroducedin[21].Since objectivevalueandtheoptimalasfoundbybrute-force(ratio1
WWW’17,April2017,Perth,Australia AriaRezaei,BryanPerozzi,andLemanAkoglu
meanstheyareequal)withvaryingattributesize.Theresultsare
8
averaged over 10 random realizations of the synthetic datasets
SWA
asdescribedabove. WenoticethatSWAachievesnear-optimal Simplified
performance throughout, which Simplified catches up with as 6 Top-5
Top-25
thenumberofattributesd increases. Top-k losesoptimalityas Top-50
kbecomessmallercomparedtod,wherethedeclineisfasterfor e (s)4
smallerk. m
Figure2(center)issimilar,wherewecompareperformances Ti
underlargerattributesizesd ={50,100,...,1000}.Asdislarger, 2
wealsouselargerk = {5,25,50}. Asbrute-forcecannotbecom-
puted in reasonable time, we report ratios w.r.t. the maximum
objectivevalueachievedamongthetestedalgorithms.Wefindthat 0
200 400 600 800 1000
Simplifiedachievesnear-identicalperformancetoSWA.Again,the # Attributes d
ratiosofTop-kmethodsdropastheattributespacegrows.Inter-
Figure3:Runningtime(seconds)withincreasingnumberof
estingly,Top-50(outof1000)attributesfromeachofbothclasses
attributes.Allalgorithmsscalenear-linearlywithd.
yield64.78%ofthemaximumobjectivevalue.
WhileitmayappearthatSimplifiedperformsaswellasSWA,
wecanshowthatundercertainconditionswherethediminish- Democrats Republicans
health government
ingreturnspropertyofsubmodularproblemsplaysamajorrole,
families taxation
itbecomesinferiortoSWA.Toshowsuchasetting, wedesign education law
anexperimentwherethex(a)valuesofanattributeaaredrawn commerce employment
uniformlyfrom[P,1]forclass1, andfrom[0,1−P]forclass2 housing public works
employment natural resources
aswedecreaseP from0.95to0.05. Notethatastheranges(and emergencies congress
hencethevariance)ofthevaluesincrease,theexpectedvalueof foreign trade finance
everyattributeremainshigherforclass1.Theresultsareshown environment commerce
criminal law immigration
inFigure2(right). ForlargeP,thevaluesforclass1aresignifi-
cantlylargerandbothalgorithmsassignallattributestoclass1.
Figure4: Top10attributesinrankedorderforDemocrats
Astherangesstartoverlappingandtheexpectedvaluesgetcloser,
andRepublicansinCongress. Characterizingattributesre-
Simplifiedcontinuestoassignallattributestoclass1(withhigher
vealthecontrastbetweenliberalandconservativeideas.
expectedvalue)eventhoughthemarginalincreasetotheobjective
valuedecreasessignificantlyaswegoonduetodiminishingreturns.
PARTY FOCUS ON
Asthevariancegetsevenlarger,Simplifiedagainperformssimilar ARMED FORCES War in
toSWAasitstartsassigningsomeattributestoclass2duetothe War in Iraq
Afghanistan
randomvariation.Arguably,itisunlikelytoencounterthissetting
Bombing
inreal-worlddatasets,wherethereexistmanysimilarly-distributed of Iraq
attributesforsufficientlydifferentclassesofsubgraphs.
Computational performance. Finally, we compare the pro-
Democrat
posedalgorithmsintermsoftheirrunningtimeandscalability,as Republican
thenumberofattributesgrows.Figure3showsruntimeinseconds
1993 1995 1997 1999 2001 2003 2005 2007
ford ={50,100,...,1000}.Wenotethatallthealgorithmsscale
near-linearly.SWAhasthelargestslopewithincreasingd,while Figure 5: Change of focus on attribute “National Security
finishingunder8secondsford =1000andp=n=100subgraphs andArmedForces”amongDemocratsandRepublicans.We
fromtwoclasses. ThescalabilityofTop-k dependsonk which observeincreasedinterestbyRepublicansintimeofwar.
decreaseswithincreasingk.Simplifiedheuristicliesinthebottom
andisreliablyoneofourfastestmethods.
Overall,SWAandSimplifiedworkbestonalldatasets. Sim- localcommunityextractionalgorithmofAndersenetal.[3].One
plifiedcanbeparallelizedeasily,aseachattributeisprocessed canalsouseego-networks,whereanodeisgroupedwithallits
independently. For massive datasets, one can also fall back to immediateneighbors.
Top-k,whichiscapableofidentifyingthefewkeyattributesfor Wereportthetop-10attributesbyrelativecontributionin(6)
characterization. perclasssidebysideforcomparison.Tobeprecise,werandomly
sample90%ofoursubgraphs100timesandpresenttheaverage
5.2 AnalysisonReal-worldDatasets relativecontribution(bars)andstandarddeviation(errorbars)so
Forreal-worlddataanalysisweconsiderattributedgraphswhere astoensurethatourresultsarenotanartifactofthesubgraphsat
nodesareassignedclasslabels.Westudytheclassdifferencesof hand.
nodesbythe“companythattheykeep”. Thatis,wecharacterize Weexperimentwith3real-worldattributednetworks: (i)bill
eachnodewithalocalcommunitysurroundingthem,usingthe co-sponsorshipsofCongressmen[11],(ii)co-purchasenetworkof
TiesThatBind-CharacterizingClassesbyAttributesandSocialTies WWW’17,April2017,Perth,Australia
Under 13 Over 13
Kids & Family Comedy
Animation Fitness
VHS Documentary
Clifford Classic Comedy
Action Performing Arts
Adventure Horror
Indiana Jones Musicals
Dinosaurs Bible
Cartoon Network Christian Video
Land Before Time Yoga
(a)Rankingbyrelativecontribution(Proposed) (b)Rankingbyrelativecontribution(Proposed)
Animation Classics Under 13 Over 13
Kids & Family Drama Videos for Babies Fitness
Educational Comedy Charlie Brown Infantil y familiar
Dr. Seuss Performing Arts Kids & Family Comedy
7-9 Years Action Magic School Bus Spanish Language
Holidays Westerns Mary-Kate & Ashley MGM Home
Dragon Tales Musicals For the Whole Family School Days
Infantil y familiar Mystery There Goes A... Military & War
Nickelodeon French Rugrats - All Grown Up Drama
Warner Video Sesame Street Little Bear Ballet & Dance
Franklin Puppets Scooby-Doo Eyewitness
(c)Rankingbycoefficients(LR) (d)Rankingbycoefficients(LR)
Figure6:Characterizationvs.Classification:LogisticRegression(LR)prefersinfrequentattributesthatdiscriminatewell.Our
proposedmethoddiscoverssubspacesthatcharacterizethedatainamorenaturalway.
Amazonvideos[20],and(iii)DBLPco-authorshipnetwork. We Amazon. Thisnetworkcontains4011nodes,9487edges,and903
describetheindividualdatasetsandpresentourfindingsnext. attributes. Nodesarevideosfromamazon.com,andedgesdepict
co-purchaserelationsbetweentwovideos,indicatingthattheyare
Congress. Weconsider8co-sponsorshipnetworksfromthe103rd
frequentlyboughttogether.Attributesrangefromdescribingthe
Congressto110th.Thenodesarecongressmen.Anedgedepicts
videogenresuchas“Comedy”and“Drama”,totheage-rangeof
co-sponsorshipofabillbytwocongressmen,andtheedgeweightis
theaudienceintendedforthevideossuchas“7-9Years”,popular
thenumberoftimestwonodessponsoredabilltogether.Eachbill
franchiseslike“SpongeBobSeries”,theformofthevideoslike
isassignedaphrasethatdescribesitssubject,withatotalof32such
“Animation”,andthedeviceitcomesinsuchas“VHS”or“DVD”.
phrases.Wemirrorthesebillsubjectstotheirsponsorstocreate
nodeattributes.Thenetworksarehighlydense,soweremovelow- WehaveexperimentedwithtwoscenariosonAmazontoshow-
casethestrengthofourmethodincharacterizingdifferentclasses.
weightededgessuchthatthesizeofthegiantconnectedcomponent
Wesetsemanticallydifferentattributesasclassesandusetherest
maintainsmorethan95%ofitsoriginalsize.
Figure4presentsthetop-rankingattributesamongtwoclasses, forcharacterization.Thespecificqueriesare(1)Animationvs.Clas-
DemocratsandRepublicans(averagedover8congressesinthe sicsand(2)VideosforUnder13yearsoldandVideosforOver13
dataset).Asexpected,Democratshavealiberalagendacentered yearsold. Under13classconsistsofthevideosexhibitingtheat-
tributes“Birth-2Years”,“3-6Years”,“7-9Years”and“10-12
uponsocialandenvironmentalprograms,whileRepublicansmainly
focusonregulatinggovernment,immigrationandfinancialissues. Years”.RestofthevideosbelongtotheOver13class.
Figure6aand6brespectivelyshowtop-10attributesperclass
SincetheCongressdatasetistemporal,wecanalsoexplorehow
asrankedbyourmethodonthesetwoscenarios. Wefindthat
thefocusofthetwopartieschangesonaparticularsubjectovertime.
“Kids&Family”andagegroups“3-12Years”arekeycharacterizing
Aclearexampleofthisisbillson“NationalSecurityandArmed
Forces”.Figure5showstheaveragecontributionofthisattributefor attributesforAnimation.“WarnerVideos”and“CartoonNetwork”
arealsoamongthetopattributes.Perhapssurprisingly,“Christian”
bothparties(individualtermsinEq.(6))overyears.Startingwith
videosand“Bible”storiesfollowtheabove. Ontheotherhand,
theUSconflictwithIraq,thisattributeseizesRepublicans’attention,
anditcontinuestogrowafterthe9/11attacksandthebeginningof wenotegenre-relatedattributesthattrulydefineClassics,suchas
“PerformingArts”,“Comedy”,and“Musicals”.
thewarinAfghanistan.Itreachesitspeakduringthestartofwar
Forthesecondscenario,weobserve“Kids&Family”and“Ani-
inIraqandthenstartstoloseattentiontowardsthelastyearswhen
theUStroopsarewithdrawnfromthemiddle-east.Thisabnormal mation”attributestomostlycharacterizetheUnder13videos.In
changeininterestinnationalsecurityandarmedforcesisespecially contrast,thecharacterizingattributesforOver13arethosethatcan-
notreallybeconsumedbychildren,including“Comedy”,“Fitness”,
interestingsincebeforeandaftertheyearsofinternationalcrisis,
and“Documentary”videos.
thisattributehasclosetozeroattention,evenachievesnegative
valuesduringthelastyearsasaDemocratattribute,whichindicates DBLP. Thisnetworkcontains134Knodes,1.478Medgesand2Kat-
itisnotcharacteristicofneitheroftheparties. tributes.Nodesarecomputerscientistsandlinksareco-authorship
WWW’17,April2017,Perth,Australia AriaRezaei,BryanPerozzi,andLemanAkoglu
WeseethisbehavioragaininFigure6d,whereLRranksrare
attributeshighly(suchas“CharlieBrown”and“Mary-Kate&Ash-
ley”)abovemorefrequentattributeswhicharequitediscriminative
(“Kids&Family”and“Animation”).IncontrasttoLR,ourproposed
methodranksattributesbytheircontributionacrosssubgraphs,
findingasubspaceofattributeswhichbettercharacterizestheinput
subgraphs.
Intuitively,whenanattributeispresentinmanyofthenodesthat
belongtoaclass,itisconsideredtobeacharacterizingattribute
Figure 7: Top 10 attributes in ranked order for ICC and
of that class. On the other hand, when observing an attribute
ICASSPconferencesinDBLP.
atanodeindicatesahighprobabilityofthenodebelongingto
a particular class, then the attribute is a discriminative one for
thatclass.Toquantifythese,weuseclasssupportandconfidence,
metricscommonlyusedinassociationrulemining[1].
relations. Attributesarecomputerscienceconferencesandjour-
nals, whereanodeexhibitsanattributeifs/hehasatleastone Let#(c,a)denotethenumberofnodesinclasscwithattributea,
publicationinthevenue. #(a)totalnumberofnodeswithattributea,and#(c)totalnumber
TheclassesareICC,aconferenceoncommunications,vs.ICASSP, ofnodesfromclassc,then:
aconferenceonspeechandsignalprocessing.Werandomlysample • Confidence(C):probabilityofbelongingtoclasscwhen
100nodesfromallnodesofeachclassandfindsubgraphsaround attributeaisobservedinsomenode:Cfd(c,a)=Pr(c|a)=
them,tomaintainamanageablesetofsubgraphs.Figure7shows #(c,a)
#(a)
top-10attributesforthetwoclasses.WefindthatattributesforICC • Class Confidence(CC): probability of belonging only
revolvearoundnetworking,communicationsandmobiletechnolo- to class c, when attribute a is observed in some node:
gies,including“INFOCOM”,“GLOBECOM”and“PMIRC”,while CC(c+,a)=Pr(c+|a)−Pr(c−|a)
attributesforICASSPareconferencesonspeech,videoandimage • Support(S): percentage of nodes in classc that exhibit
processingandlinguistics,including“INTERSPEECH”,“ICIP”and attributea:Sup(c,a)= #(c,a)
“EUSIPCO”. #(c)
• ClassSupport(CS):differenceofsupportforabetween
classes:CS(c+,a)=Sup(c+,a)−Sup(c−,a)
5.3 Characterizationvs.Classification Asweseekdistinctsubspaces,weonlyusetherelativemetrics,
Hereweexaminethedifferencesbetweencharacterizationanddis- i.e.CCandCS.Ideallyhavinganattributehighonbothmetricsis
criminativeclassification.Similartoourmethod,asparsesolution best,howeverthiscasehappensrarely.Anattributewithhighclass
fromaregularizedclassifierwillcontainasubspaceoftheattributes, supportcanbeconsideredasagoodrepresentativeofaclasswhile
whichcanberankedusinganattributeimportancescorefromthe anattributewithhighclassconfidencecanbeusedforclassification
model.Suchregularizedsparsemethodsarepopularapproaches purposes.Tomeasuretheaveragecharacterizationoftheattributes
forexploratorydataanalysisandformthefoundationsofmany assignedtoagivenclass,wehave:
intFeroprrtehtiasbcloemmpoadriesloinng,wmeeuthseodLso.gisticRegression(LR)withLASSO CS(c,A(c))= (cid:205)a∈A(c)waCS(c,a), (9)
regularization[34]tolearnasparsesolution. WefirsttrainaLR (cid:205)a∈A(c)wa
modelforbinaryclassificationbetweenclassesc+ andc− using whereCSistheweightedaverageofCSoverallattributesassigned
rawnodeattributesasinput. Afterclassification,weusetheLR toclassc. Weightwa hereisthemetricthatweuseforranking
coefficientstopartitiontheattributesbetweentheclasses,assign- attributesincorrespondingmethods,whichistherelativecontribu-
ingthosewithpositivecoefficientstoc+andnegativeonestoc−, tionforourproposedapproachandtheabsolutecoefficientvalues
andrankbytheirmagnitude.Toovercomeclassimbalanceinthe forLR.Likewise,tomeasurethetotaldiscriminationofasetof
dataset,weoversamplefromthesmallclasstomaketheclasssizes attributes,wehave:
etiqmuFeaisglutaornede6litmhcoeinmnapdtaeorttehhseethecffelaetscostpifio-rcfaasntuikocihnn.gsWaamtteprrilebinpugete.astothbitsaipnreodcewdiuthreo1u0r CC(c,A(c))= (cid:205)a∈(cid:205)A(ac)∈wA(ac)CwCa(c,a), (10)
methodtothosefoundthroughLR.WeseethatLRprefersinfre- whereagain,CC istheweightedaverageofCCforallattributes
quent,buthighlydiscriminatingattributes.Forexample,consider assignedtoclassc.
thedifferenceinattributesfoundbetweenourmethodandLRfor Figure8presentsbothmeasuresforthetworankingmethods
theAnimationvs.Classicsclassificationtask(Fig.6aand6c).Here, forthereal-worldscenarios.OurmethodoutperformsLRw.r.t.the
LRcompletelyfailstoassignhighweighttotwoattributes(“3-6 characterizationaspect(CS).Thisistobeexpected–asourmethod
Years”and“10-12Years”)thatarebothveryprevalentinthedataset. searchesforattributespresentinafocusedsubspaceacrossmany
Insteaditsthirdstrongestattributeis“Dr.Seuss”,whichisperfectly subgraphs of a class and ranks them accordingly. Surprisingly,
discriminative(allitemswiththisattributeareAnimation),butis innearlyallcases,thesubspaceswefindalsohaveacomparable
presentinonly4%ofthenodes.Thisisaclearexampleofsacrificing discriminativepower(CC)toLR.TheUnder13casewherewehave
characterizationfordiscrimination. lowCCiswhenmostdiscriminatingattributes(asrankedhighby
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