NeMa: Fast Graph Search with Label Similarity - Charu Aggarwal

12 Pages·2012·0.3 MB·English

NeMa: Fast Graph Search with Label Similarity
Arijit Khan Yinghui Wu Charu C. Aggarwal Xifeng Yan
ComputerScience IBMT.J.WatsonResearch
UniversityofCalifornia,SantaBarbara Hawthorne,NY
{arijitkhan,yinghui,xyan}@cs.ucsb.edu [email protected]
ABSTRACT andlanguagesareproposed—suchasSPARQLandXDDforthe
RDFandXMLdata—whichrequireastandardschemaofqueries
Itisincreasinglycommontofindreal-lifedatarepresentedasnet-
and target graphs. Nevertheless, the real-lifegraphs are complex
worksoflabeled,heterogeneousentities.Toquerythesenetworks,
andnoisy,andoftenlackstandardizedschemas[1].Indeed,(a)the
oneoftenneedstoidentifythematchesofagivenquerygraphina
nodes may be heterogeneous, referring to different entities (e.g.,
(typicallylarge)networkmodeledasatargetgraph. Duetonoise
persons, companies, documents) [14]. (b)Nodelabelsinagraph
and thelackoffixedschema inthetargetgraph, thequery graph
oftencarryrichsemantics,e.g.,id,urls,personalinformation,logs,
cansubstantiallydifferfromitsmatchesinthetargetgraphinboth
opinions [15]. (c) Worse still, the semantics of entities and their
structure and node labels, thus bringing challenges to the graph
interconnectionsinvariousdatasetsmaybedifferentandunknown
queryingtasks.Inthispaper,weproposeNeMa(NetworkMatch),
tousers[1]. Inthiscontext,amatchmaynotnecessarilybe(even
a neighborhood-based subgraph matching technique for querying
approximately)isomorphictothequerygraphintermsoflabeland
real-life networks. (1) To measure the quality of the match, we
topological equality. Thus, traditional graph querying techniques
proposeanovelsubgraphmatchingcostmetricthataggregatesthe
arenotabletocapturegoodqualitymatches. Considerthefollow-
costsofmatchingindividualnodes,andunifiesbothstructureand
ingexampleovertheIMDBmoviedataset.
nodelabelsimilarities. (2)Basedonthemetric,weformulatethe
minimumcost subgraph matchingproblem. Givenaquery graph
andatargetgraph,theproblemistoidentifythe(top-k)matchesof K.( AWcitnosr)let (Mo?vie) K.( AWcitnosr)let (Mo?v ie)
thequerygraphwithminimumcostsinthetargetgraph. Weshow Winslet, K. Titanic
(actor) (Movie)
thattheproblemisNP-hard,andalsohardtoapproximate. (3)We
proposeaheuristicalgorithmforsolvingtheproblembasedonan ? ?
inferencemodel. Inaddition,weproposeoptimizationtechniques (Director) (Director)
toimprovetheefficiencyofourmethod. (4)Weempiricallyverify Cameron, J. Lang, S.
thatNeMaisbotheffectiveandefficientcomparedtothekeyword S. Lang (Director) (I)
(Actor) (Actor)
searchandvariousstate-of-the-artgraphqueryingtechniques. S. Lang ?
(Actor) (Movie)
1. INTRODUCTION (a) Query Graph 1 (b) Query Graph 2 (c) Top-1 Match
Withtheadvent of theInternet, sourcesof datahaveincreased Figure1:AQueryandItsMatch(Example1.1)
dramatically, including the World-Wide Web, social networks,
Example 1.1. A user wants to find a movie of actress ‘Kate
genome databases, knowledge graphs, medical and government
Winslet’thatisdirectedbythesamedirectorwhoalsoworkedwith
records. Such dataare often represented as graphs, where nodes
actor‘Stephen Lang’. Eveniftheschemaandexact entitylabels
arelabeledentitiesandedgesrepresentrelationsamongtheseenti-
ofthetargetnetworkarenotavailable, theusercanstillcomeup
ties[14,43].Queryingandminingofgraphdataareessentialfora
withsomereasonablegraphrepresentation ofthequery[15,17],
widerangeofemergingapplications[1,15,30].
asillustratedinFigure1(a)and1(b).Observethatsuchgraphical
Toquerythesegraphs,oneoftenneedstoidentifythematchesof
representationmaynotbeunique,andtheremightnotbeanexact
agivenquerygraphina(typicallylarge)targetgraph. Traditional
matchofthequerygraphinthedataset. Indeed,theresultinFig-
graph querying models are usually defined in terms of subgraph
ure1(c)(astar-shapedgraph)isbynomeanssimilartothequery
isomorphismanditsextensions(e.g.,editdistance),whichidentify
graphs inFigure1(a)and (b)(bothchain-shaped graphs) under
subgraphs that are exactly or approximately isomorphic to query
traditionalgraphsimilaritydefinitions. Grapheditdistanceofthe
graphs [35, 33, 43]. In addition, a wide range of query models
resultgraphwithquerygraphs1and2are4and6, respectively.
Thesizeofthemaximumcommonsubgraphis3inbothcases.Nev-
Permission tomake digital orhardcopies ofall orpartofthis workfor
ertheless,‘Titanic’isthecorrectanswerofthequery; andhence,
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare
the result graph should be considered a good match for both the
notmadeordistributedforprofitorcommercialadvantageandthatcopies
bearthisnoticeandthefullcitationonthefirstpage. Tocopyotherwise,to querygraphsusingsomenovelgraphsimilaritymetric.
republish,topostonserversortoredistributetolists,requirespriorspecific This motivates us to investigate fast subgraph matching tech-
permissionand/orafee. Articlesfromthisvolumewereinvitedtopresent niques suitable for query answering, which can relax rigidstruc-
theirresultsatThe39thInternationalConferenceonVeryLargeDataBases, turalandlabelmatchingconstraintsofsubgraphisomorphismand
August26th-30th2013,RivadelGarda,Trento,Italy.
other traditional graph similarity measures. Our proposed graph
ProceedingsoftheVLDBEndowment,Vol. 6,No.3
Copyright2013VLDBEndowment2150-8097/13/01...$10.00. similarity metric is based on following observations: (a) if two
NeMa BLINKS1 IsoRank SAGA NESS1 gStore
Precision 0.91 0.52 0.63 0.75 Filter:0.17 0.59
(Node) Filter+Verify:0.80
Recall 0.91 0.52 0.63 0.75 Filter:0.83 0.59
(Node) Filter+Verify:0.80
Precision 0.88 0.50 0.40 0.69 Filter:0.39 0.55
(Graph) Filter+Verify:0.74
Recall 0.88 0.50 0.40 0.69 Filter:0.75 0.55
(Graph) Filter+Verify:0.74
Top-1Match 0.97 1.92 4882.0 15.95 Filter:0.59 0.92
FindingTime(sec) Filter+Verify:56.16
Table1: NeMavs. KeywordSearchandGraphQueryingMethods: ThequerygraphswereextractedfromtheIMDBgraph,andlatermodified
byadding 30%structural noiseand50%labelnoise. Wedetermined thetop-1matchforeach querygraphusingvarious methods, andmeasured
effectiveness atthelevel of(a)querynodes, and(b)querygraphs. Atthenodelevel, precisionisdefinedastheratioofcorrectlydiscoverednode
matchesoveralldiscoverednodematches,whilerecallismeasuredastheratioofcorrectlydiscoverednodematchesoverallcorrectnodematches.
Similarly, atthegraphlevel, precisionisdefinedastheratioofcorrectlydiscoveredgraphmatchesoveralldiscoveredgraphmatches,andrecallis
measuredastheratioofcorrectlydiscoveredgraphmatchesoverallcorrectgraphmatches. Agraphmatchisconsideredcorrectifatleast70%ofits
nodesarematchedcorrectly.Sinceweconsideronlythetop-1match,precisionandrecallhavethesamevalue.Inaddition,wealsoreportprecisionand
recallofNESSfilteringphase.Fordetailsaboutthequerygraphs,noise,andevaluationmetrics,seeSection7.
nodes are closein a query graph, the corresponding nodes in the toapplyaninferencealgorithmtoheuristicallyidentifytheoptimal
resultgraphmustalsobeclose. However, (b)theremaybesome matches.Ourmethodavoidscostlysubgraphisomorphismandedit
differencesinlabelsofthematchednodes. distancecomputations. Wefurtherproposeindexingandoptimiza-
While the need for such a graph similarity metric is evident tiontechniquesforourmethodinSection6.
(e.g.,SAGA [35], IsoRank [33]), there is little work on subgraph (4)WeempiricallyverifytheeffectivenessandefficiencyofNeMa.
matchinginlargenetworksconsideringboththecriteria.Recently, Ourexperimentalresultsonreal-worldnetworksinSection7show
NESS [20] isproposed for subgraph matching that considers the thatNeMafindsbetterqualityresultsquicklyascomparedtokey-
proximityamongnodes, butresortstostrictnodelabelmatching. wordsearch(e.g. BLINKS[16])andvariousgraphqueryingtech-
The NESS algorithm is based on a filtering-and-verification ap- niques(e.g.,IsoRank[33],SAGA[35],NESS[20],gStore[43]).
proach. Inthefilteringphase, thelesspromisingcandidatenodes
areprunediteratively,untilnomorecandidatescanbepruned.The 2. RELATED WORK
outputofthefilteringphaseisalimitednumberoffinalcandidates
SubgraphMatching. Ullmann’sbacktrackingmethod[38], VF2
for each query node. Then, it verifiesallpossible graph matches
[9],SwiftIndex[32]areusedforsubgraphisomorphismchecking.
formedbythesefinalcandidates, inorder tofindthetop-k graph
The subgraph matching problem identifies all the occurrences
matches. OnecanmodifyNESStoleveragefornodelabeldiffer-
of a query graph in the target network. In bioinformatics, exact
ences. However, thismodificationreducestheeffectivenessofits
and approximate subgraph matching have been extensively stud-
filteringphase,andresultsinalargenumberoffinalcandidatesfor
ied,e.g.,PathBlast[19],SAGA[35],NetAlign[22],IsoRank[33].
each query node (See Appendix for an example). Indeed, in our
Amongthem,SAGAisclosetooursintermsofproblemformula-
experiments, wefindaverylowprecisionscoreforNESS, atthe
tion.However,thesealgorithmstargetsmallerbiologicalnetworks.
endofitsfilteringphase(Table1).Hence,itbecomesquiteexpen-
Itisdifficulttoapplytheminlargeheterogeneousnetworks.
sivetodeterminethetop-kgraphmatchesfromtheselargenumber
Therehavebeensignificantstudiesoninexactsubgraphmatch-
of final candidates. In contrast, our proposed NeMa framework
inginlargegraphs.Tongetal.[37]proposedthebest-effortpattern
employsaninferencealgorithmthatiterativelybooststhescoreof
matching,whichaimstomaintaintheshapeofthequery. Incon-
morepromisingcandidatenodes,consideringbothlabelandstruc-
trast,weidentifytheoptimalmatchesintermsofproximityamong
turalsimilarity;andtherebydirectlyfindsthetop-kgraphmatches.
entitiesratherthantheshapeofthequerygraph. Tianetal. [36]
Contributions.Inthispaper,weproposeNeMa,anovelsubgraph proposed an approximate subgraph matching tool, called TALE,
matchingframeworkforqueryingheterogeneousnetworks. withefficient indexing. Mongiovi et. al. introduced a set-cover-
(1)Wedefinethequeryresultasthematchofagivenquerygraph based inexact subgraph matching technique, called SIGMA [26].
inatargetgraph,intermsofanotionofhomomorphism-basedsub- Boththesetechniquesuseedgemissestomeasurethequalityofa
graphmatching. Tomeasurethequalityofthematches,wefurther match; and therefore, cannot incorporate the notion of proximity
define a novel subgraph matching cost metric between the query amongentities. Thereareotherworksoninexactsubgraphmatch-
graph and its matches (Section 3). In contrast to strict subgraph ing. Anincompletelist(see[13]forsurveys)includeshomomor-
isomorphism,ourproposedmetricaggregatesthecostsofmatching phismbasedsubgraphmatching[12],beliefpropagationbasednet
individualquerynodes,whichinturndependsonthecostofmatch- alignment [4], edge-edit-distance based subgraph indexing tech-
ingnodelabelsandtheirneighborhoodswithinacertainhops. nique[41],subgraphmatchinginbillionnodegraphs[34],regular
(2) Basedon thecost metric, wepropose theminimumcost sub- expressionbasedgraphpatternmatching[3],schema[25]andun-
graph matching problem (Section 4), which is to identify the balanced ontology matching [42]. Among them, homomorphism
matchesofthequerygraphwithminimumcostsinthetargetgraph. basedsubgraphmatching[12]isclosetoours.However,insteadof
WeshowthattheproblemisNP-hardandalsohardtoapproximate. identifyingthetop-k matches, thepaperreportsallthesubgraphs
(3)Weproposeaheuristicmethodfortheminimumcostsubgraph wherethequeryedgescanbemappedtopathsofagivenmaximum
matching problem (Section 5). In a nutshell, NeMa converts the
underlying graph homomorphism problem into an equivalent in- 1In this paper, all experimental results with NESS and BLINKS
ference problem in graphical models [29], and thereby allows us correspondtotheirmodifiedversions,whereweallowtwonodesto
bematchediftheirlabeldifferenceiswithinapredefinedthreshold.
lengthandthelabeldifferencesarewithinacertainthreshold. Q(VQ,EQ,LQ) querygraph
There are several works on simulation and bisimulation-based G(V,E,L) targetgraph
graph pattern matching, e.g., [11, 23], which define subgraph φ:VQ→V subgraphmatchingfunction
matchingasarelationamongqueryandtargetnodes.Comparedto ∆L labeldifferencefunction
them,NeMa,ismorestrict,sincewedefinesubgraphmatchingas M(v) candidatesetofnodev
afunctionfromquerynodestotargetnodes. RG(u) neighborhoodvectorofnodeu
Nφ(v,u) neighborhoodmatchingcostbetweenvandu
Label and Concept Propagation. Label propagation has been Fφ(v,u) individualnodematchingcostbetweenvandu
widelyusedinsemi-supervisedlearning,e.g.,labelingofunlabeled C(φ) subgraphmatchingcostfunction
graph nodes [31]. Concept Propagation /Concept Vector, on the Table2:Notations:TargetGraphs,QueriesandSubgraphMatching
otherhand,wasoriginallyformulatedtomeasurethesemanticsim-
Given a target graph G = (V,E,L) and a query graph Q =
ilaritiesbetweenterms/conceptsinataxonomy[21]. Wenotethat
thespreadingactivationtheoryofmemory[2]usedasimilaridea (VQ,EQ,LQ),(1)anodeu ∈ V isacandidateforaquerynode
of activation propagation. CP/CV and spreading activation have v ∈ VQ if the difference in their labels (i.e., L(u) and LQ(v),
respectively),determinedbyagiven(polynomial-timecomputable)
beeneffectivelyusedin[7,20]forapproximatestructuralmatching
intrees,graphsandalsoforinformationretrievalfromassociated labeldifferencefunction∆L,islessthanorequaltoapredefined
thresholdǫ.WedenoteasM(v)thecandidatesetofthequerynode
networks[5]. Theseworksconsideronlystrictnodelabelmatch-
ing. However, subgraphmatchingwithoutnodelabelsisaharder v.(2)asubgraphmatchingisamany-to-onefunctionφ:VQ→V,
problemthansubgraphmatchingwithnodelabels[40].Therefore, suchthat,foreachquerynodev∈VQ,φ(v)∈M(v).
insteadofstrictnodelabelequality,whenoneallowsapproximate Remarks.(1)Thelabeldifferencefunction∆Lbetweentwonode
nodelabelmatching(e.g.,inourcurrentwork),itsignificantlyin- labels canbe defined by avariety of criteria, such asthe Jaccard
creasesthecomplexityofthesearchproblem. similarity,stringeditdistance,ormoresophisticatedsemanticmet-
Querying Semi-structured Data. Lorel and UnQl are among rics, e.g., ontology similarity [10]. In this work, we use Jaccard
thepreliminaryquerylanguagesdesignedforsemi-structureddata. similaritymeasuretodetermine∆L(Section7). (2)Incontrastto
Both of them model input data as labeled graphs, while permit- strictone-to-one mappingasintraditionalsubgraphisomorphism
ting users to write queries without detailed knowledge about the tests,weconsideramoregeneralmany-to-onesubgraphmatching
schema. Later, an underlying query processing system converts function. Indeed, twoquerynodesmayhavethesamematch[12,
thosequeriesintostandardSQLorstructuralrecursionqueries,re- 30]. (3)Inpractice,thenodesinthetargetandquerygraphsmay
spectively, for retrieving the correct answers. This idea of query be annotated with types (e.g., Figure 1 and [15]), where a query
rewriting has been explored in the context of both relational and nodecanonlybematchedwithtargetnodeshavingthesametype.
semi-structureddata,e.g.,[10,28,39,15].Observethatsuchquery Insuchcases,oursubgraphmatchingmodelcanbeeasilyadapted
rewritingtechniquesalleviateusersfromthecomplexityofunder- tocapturethetypeconstraintsbyrefiningcandidatesets.
standingtheschema;nevertheless,theunderlyingqueryprocessing 3.2 Subgraph Matching CostFunction
systemstillrequiresafixedschema.
IntherealmofRDF,SPARQLiswidelyusedasthequerypro- Therecanbemany validmatching functions for agivenquery
cessinglanguage. However, writingofaSPARQLqueryisoften graph and a target graph [13]. As stated earlier, our novel graph
toochallenging, becauseitrequirestheexactknowledgeofstruc- similaritymetricmustpreservetheproximityamongnodepairsin
ture,nodelabelsandtypes.gStore[43],whichisthefirststudythat thequerygraph,whilethelabelsofthematchednodesshouldalso
considersasubgraphmatching-basedqueryansweringtechniquein besimilar.Takingthisasourguideline,weintroducethesubgraph
RDFdata,allowsapproximatenodelabelmatching,butadheresto matchingcostfunctioninNeMaasametrictomeasurethegood-
strictstructuralmatches.Incontrast,ourNeMaframeworkpermits ness of a matching. The function adds up the costs of matching
bothstructuralandnodelabelmismatches. a query node with its candidate, thereby capturing the difference
Our work is different from the keyword search in graphs [16, betweenlabelsandneighborhoodstructuresofthetwonodes. We
18],asourquerieshavebothstructureandkeywords(nodelabels). firstintroducethenotionofaneighborhoodvector.
Neighborhoodvectorization. Givenanodeuinthetargetgraph
3. PRELIMINARIES
G,werepresenttheneighborhoodofuwithaneighborhoodvector
Westartwithafewdefinitions. RG(u) = {hu′,PG(u,u′)i},whereu′isanodewithinh-hopsof
3.1 Target Graphs, Queries andMatching u,andPG(u,u′)denotestheproximityofu′fromuinG.
Target graph. A target graph that represents a heterogeneous αd(u,u′) ifd(u,u′)≤h;
network dataset can be defined as a labeled, undirected graph PG(u,u′)= (1)
(0 otherwise.
G=(V,E,L),withthenodesetV,edgesetE,andalabelfunc-
tionL, where(1)eachtarget nodeu ∈ V representsanentityin
Here,d(u,u′)isthedistancebetweenuandu′.Thepropagation
thenetwork,(2)eachedgee∈Edenotestherelationshipbetween
factorαisaparameterbetween0and1;andh>0isthehopnum-
twoentities,and(3)Lisafunctionthatassignstoeachnodeua
ber(effectively,theradius)oftheneighborhoodforvectorization.
labelL(u)fromafinitealphabet. Inpractice,thenodelabelsmay
The neighborhood vector of node u encodes the proximity infor-
representtheattributesoftheentities,e.g.,name,value,etc.
mationfromutoitsh-hopneighbors. Itoftensufficestoconsider
Querygraph.AquerygraphQ=(VQ,EQ,LQ)isanundirected, smallvaluesofh(e.g.,h=2),sincetherelationshipbetweentwo
labeledgraph,withasetofquerynodesVQ,asetofqueryedges entitiesbecomesirrelevantastheirsocialdistanceincreases[6].
EQ, and a label function LQ, which assigns to each query node Basedon neighborhood vectors, wenow proceed tomodel the
v∈VQalabelLQ(v)fromafinitealphabet. matching cost of theneighborhoods of aquery node and atarget
We next define the subgraph matching of a (connected) query node.Letusdenotethesetofneighboringnodeswithinh-hopsofv
graphinalargetargetnetwork. asN(v).Givenamatchingfunctionφ,theneighborhoodmatching
u
2
costbetweenvandu=φ(v),denotedbyNφ(v,u),isdefinedas: v2 c c v4 c
N (v,u)= v′∈N(v)∆+(PQ(v,v′),PG(u,φ(v′))) (2)
φ P v′∈N(v)PQ(v,v′)
where∆+(x,y)isafunPctiondefinedas v1 b a v3 u1 b a u3
Query Graph Q Match
x−y, ifx>y;
∆ (x,y)= (3) Figure2:ExampleofFalseExactMatchinNeMa
+ (0 otherwise.
Property2. Ifthematchidentifiedbyφisnotisomorphictothe
Intuitively, N (v,u) measures thematching cost of the neigh-
φ querygraphQ,andφisaone-to-onefunction,thenC(φ)>0.
borhood vectorsof v andu. Notethat (i)theuser issuesaquery
basedonhervaguenotionofhowtheentitiesareconnectedinthe PROOF. SinceQ is connected and φ is a one-to-one function,
target graph. Hence, ∆+ avoids penalizing the cases when two if the match identified by φ is not isomorphic to Q, one of the
nodes are closer in the target graph, as compared to their corre- following must hold. (1) There exists some node v ∈ VQ, s.t.,
sponding nodes in the query graph. (ii) We normalize Nφ(v,u) ∆L LQ(v),L(φ(v)) >0. Then,C(φ)>0,assumingλ6=0in
overtheneighborhoodofvthatincursmorecostwhensamenum- Eq.4.(2)Thereexistsanedge(v,v′)inEQ;butthecorresponding
berofnodemissesoccursinasmallerneighborhood. edge(cid:0)(u,u′)isnotin(cid:1)graphG. φ(v) = uandφ(v′) = u′. This
Recallthatweassumetheexistenceofthelabeldifferencefunc- impliesPQ(v,v′)=α,butPG(u,u′)<α,whichinturnimplies
tion 0 ≤ ∆L ≤ 1. Now, the individual node matching cost for Nφ(v,u)>0.Assumingλ6=1inEq.4,wegetC(φ)>0.
matchingfunctionφisdefinedasalinearcombinationofthelabel
differencefunctionandtheneighborhoodmatchingcostfunction.
4. PROBLEM FORMULATION
Fφ(v,u)=λ·∆L(LQ(v),L(u))+(1−λ)·Nφ(v,u), (4) The subgraph matching cost function favors matches with low
whereu=φ(v). matchingcosts.Basedonthematchingcostfunction,weintroduce
Thisnodematchingcostcombinesbothlabelmatchingcostand theminimumcostsubgraphmatchingproblemasfollows.
neighborhood matching cost viaaparameter 0 < λ < 1, whose
Problem Statement 1. Minimum Cost Subgraph Matching.
optimalvalueliesbetween0.3∼0.5empirically(Section7).
Given a target graph G, a query graph Q, and the label noise
Wearenowreadytodefineoursubgraphmatchingcostfunction.
thresholdǫ,findtheminimumcostmatchingφ,
Givenamatchingφfromthequerynodesv ∈ VQtotargetnodes
φ(v)∈V,thesubgraphmatchingcostfunctionisdefinedas: argminC(φ), (6)
φ
C(φ)= F (v,φ(v)) (5)
φ
vX∈VQ s.t. ∆L(LQ(v),L(u))≤ǫ,∀v∈VQ,u=φ(v) (7)
Intuitively, C(φ) is the matching cost of φ between the query Intuitively,insteadofcheckingsubgraphisomorphism,ourprob-
graphQandthetargetgraphG,andtheproblemistofindamatch- lemformulationidentifiestheoptimalmatchbyminimizingnode
ing function φ that minimizes C(φ). Note that, assuming ∆L label differences as well as node pair distances. The identified
is non-negative, F (v,φ(v)) and therefore, C(φ) are both non- matchesserveasanswerstothequerygraph.
φ
negative,sotheminimumvaluethatC(φ)cantakeis0. The problem is, however, nontrivial. The following theorem
showsthatthedecisionversionoftheproblemisintractable,even
3.3 CostFunction Properties
whenthesubgraphmatchingfunctionφisnotinjective.
Thefollowingpropertiesofoursubgraphmatchingcostfunction
illustratesitsconnectionwithsubgraphisomorphism. Theorem1. GivenatargetnetworkG,aquerygraphQ,itisNP-
completetodeterminewhetherthereexistsamatchφwithNeMa
Property1. IfthequerygraphQissubgraphisomorphic(interms subgraphmatchingcostC(φ)=0.
ofstructureandnodelabelsequality)tothetargetgraphG,then
thereexistsaminimumcostmatchingfunctionφwithC(φ)=0. PROOF. TheproblemisNP,sincethereisanondeterministical-
gorithmwhichguessesamatchingfunctionφ,andverifieswhether
Property1ensuresthatallthematchingfunctionsφ,whichiden-
itscostC(φ)= 0,inpolynomialtime. WeprovetheNP-hardness
tifiesexact (isomorphic) matchesfor Q, musthave cost 0. How-
byreductionfromthegraphhomomorphismproblem,whichisNP-
ever, a match φ of Q, where C(φ) = 0, may not necessarily be complete [8]. A homomorphism from a graph Q′ to a graph G′
isomorphictoQ.Werefertosuchmatchesasfalseexactmatches.
(bothunlabeled) isafunction thatpreservesnodeadjacency (i.e.,
Example 2.1. Consider aquery graph Q, atarget graph G(Fig- each edge inQ′ ismapped toanedge inG′). Given aninstance
ure 2), and a subgraph matching function φ, where φ(v )=u , ofthegraphhomomorphismproblem,weconstructaninstanceof
1 1
φ(v )= φ(v )=u , and φ(v )=u . Assuming h = 1 and α = theminimumcostsubgraphmatchingproblem,whereallnodesin
2 4 2 3 3
0.5,theneighborhoodvectorsinQare: RQ(v1)={hv2,0.5i,hv3, the target graph G and query graph Q have identical labels. We
0.5i}, RQ(v2)={hv1,0.5i}, RQ(v3)={hv1,0.5i, hv4,0.5i}, and alsoassume, w.l.o.g., thatthedepthofvectorizationh = 1. One
RQ(v4)={hv3,0.5i}. Similarly,wehavethefollowingneighbor- mayverifythatifthereexistsahomomorphismφ′fromQ′toG′,
hood vectors in G: RG(u1) = {hu2, 0.5i, hu3, 0.5i}, RG(u2) then there exists a corresponding matching φ from Q to G, s.t.
={hu1, 0.5i, hu3, 0.5i}, and RG(u3) = {hu1,0.5i, hu2, 0.5i}. C(φ) = 0. Conversely, if φ′ is not a homomorphic matching,
Therefore, the individual node matching costs Fφ is 0 for all thenthereexistsanedge(v,v′)inEQ,butthecorrespondingedge
v ∈ VQ,andthesubgraphmatchingcostC(φ)is0. Observethat (φ(v),φ(v′)) is not in G. Hence, C(φ) > 0 (λ 6= 1 in Eq. 4).
thematchidentifiedbyφisnotisomorphictoQ. Therefore,thereexistsamatchingfunctionφfromQtoG,where
However,ifthematchingfunctionφisone-to-one,thefollowing C(φ)=0,ifandonlyifthereisahomomorphicmatchingφ′from
propertyshowsthatthefalseexactmatchescanbeavoided,. Q′toG′.Thiscompletestheproof.
One may want to find a polynomial time approximation algo-
AlgorithmNemaInfer
rithm.However,theproblemisalsohardtoapproximate.
Input:TargetgraphG(V,E,L),QueryGraphQ(VQ,EQ,LQ).
Output:MinimumcostmatchingofQinG.
Theorem2. TheminimumcostsubgraphmatchingisAPX-hard.
1. foreachnodev∈VQdocomputeM(v);
PROOF. Weshow that this optimization problem is APX-hard 2. i:=0;flag:=true;
by performing a reduction (f,g) from the Maximum Graph Ho- 3. InitiateiterativeinferencingwithEq.8;
4. whileflagdo
momorphism (MGH) problemwithout self loops, whichisAPX-
5. i:=i+1;
hard [27]. An MGH problem identifies a matching which max- 6. foreachv∈VQdo
imizes the number of edges of the query graph Q that can be 7. foreachu∈M(v)do
mapped to edges of the target graph G (both unlabeled). Given 8. computeUi(v,u)withTheorem3;
an instance I of MGH, we construct an instance I′ of the mini- 9. keeptrackofthecurrentmatchesofneighborsv′∈N(v);
mumcostsubgraphmatchingproblem,whereallnodesinthetar- 10. computeoptimalmatchOi(v)usingEq.10;
11. ifmorethanathresholdnumberof
agnedt neeqtwboerkthGe taontadlqnuuemrybegrraopfhnoQdehsavaendideedngtiecsa,lrleasbpeelcs.tivLeleyt,ninq 12. flquaegry:=nfoadlesse;vsatisfyOi(v)=Oi−1(v)then
Q. w.l.o.g., assume the depth of vectorization h = 1, and the 13. constructΦforallv∈VQ(withEq.11,12).
proportionalityconstantλ = 1− 1 . WedenotebyOPT(I)the 14. returnΦ;
nq
valueoftheoptimalsolutionofprobleminstanceI,andVAL(I,x)
Figure3:IterativeInferenceAlgorithmNemaInfer
the valueof afeasiblesolution xof theproblem instance I. As-
sume OPT(I) = eo and VAL(I,x) = e for some feasiblesolu- iterativelycomputes aninference cost for each query node v and
tion x of instance I. Clearly, eo ≥ 1. Hence, (1) OPT(I′) < itscandidates, andselectstheoptimalmatchofv asitscandidate
1 ≤ eo = OPT(I). Also, given some feasible solution y of uwiththeminimuminferencecost. NemaInferkeepstrackofthe
instance I′, one may verify that |OPT(I) − VAL(I,g(y))| =
optimalmatchesforeachquerynode. Theprocedurerepeatsuntil
eo − e, and |OPT(I′) −VAL(I′,y)| ≥ 2enoq−eeq. Therefore, (2) it reaches a fixpoint, where the optimal matches for more than a
|OPT(I)−VAL(I,g(y))|≤2nqeq|OPT(I′)−VAL(I′,y)|.Thus, threshold number of query nodes remain identical intwo succes-
thereexistsareduction(f,g)fromMGHtotheminimumcostsub- siveiterations(lines4-12).Finally,NemaInferrefinesthematches
graphmatchingproblem,andthetheoremfollows. ofeachquerynodeanditsneighborhoodthatit“memorizes”viaa
memoizationtechnique,andobtainsthebestmatch(line13). The
5. QUERYPROCESSINGALGORITHM constructedsubgraphmatchΦisthenreturned(line14).
WenextintroduceseveralproceduresofNemaInferindetail.
In this section, we propose a heuristic solution to identify the
minimum cost matchings. We start by introducing the max-sum Inference cost and optimal match (lines 3-12). The algorithm
inference problem in graphical models [29], and show how our NemaInferimprovesthequalityofthematchingineachiteration,
graph homomorphism problem underlying the NeMa framework basedonthenotionofaninferencecostandtheoptimalmatch.
isequivalenttoaninferenceproblemingraphicalmodels.
Inferencecost.AteachiterationiofNemaInfer,theinferencecost
Max-Sum Inference. In graphical models, the joint probability Ui(v,u)foreachv∈VQandu∈M(v)isdefinedasfollows.
distributionfunctionp(X)ofasetofvariablesX = {x ,x ,...
1 2
,xM} can be expressed as a product of the form p(X) =
U (v,u)= min F (v,u) (8)
ifi(Xi), where each Xi ⊆ X. Alternatively, logp(X) = 0 {φ:φ(v)=u} φ
Qilogfi(Xi).TheMax-Suminferenceproblemistofindtheval-
IuPnesotohfetrhweovradrsia,bwleeswxo1u,lxd2li,k.e.t.o,xmMaxtihmaitzreelsougltpi(nXm)atxhiamtucamnpb(eXde)-. Ui(v,u)= min Fφ(v,u)+ Ui−1 v′,u′ (9)
{φ:φ(v)=u}
composedasthesumofseveralfunctionsoftheformlogfi(Xi), (cid:2) v′X∈N(v) (cid:0) (cid:1)(cid:3)
eachofwhichdependsonlyonasubsetoftheoriginalvariables. Weassumei>0,andu′ =φ(v′)inEquation9.Intuitively,the
The objective of the max-sum inference problem is similar to inferencecostistheminimumsumoftheindividualnodematching
that of the minimum cost subgraph matching problem, which is costFφ(v,u)andthepreviousiteration’sinferencecostsUi−1 v′,
tominimizetheoverallsubgraphmatchingcostC(φ). Recallthat φ(v′) forallneighborsv′ ofv, over allpossiblematchingfunc-
(1) C(φ)isanaggregationoftheindividual nodematchingcosts tionsφ,withtheconstraintφ(v)=u. (cid:0)
Fφ(v,φ(v)) of all query nodes v, and (2) the individual node Not(cid:1)e that although we consider the minimization over all pos-
matching cost ofaquerynode v depends only onthematches of siblematchingfunctionsφ,s.t.,φ(v) = u, inEquation9, itonly
v and its neighbors in N(v). In light of this, we propose an it- dependsonthematchesoftheneighboringnodesinN(v). Asdis-
erativeinferencealgorithmsimilartotheloopybeliefpropagation cussedlater,inferencecostscanbecomputedinpolynomialtime.
algorithm[29],usedforinferencingingraphicalmodels.
Optimalmatch.Ineveryiteration,wealsodefinetheoptimalmatch
5.1 Iterative Inference Algorithm ofeachquerynode. Theoptimalmatchofaquerynodevatitera-
Inthissection,weintroduceourinferencealgorithm,denotedas tioni,denotedbyOi(v),isdefinedasfollows.
NemaInferandillustratedinFigure3.
Overview. Given a query graph Q and a target graph G, Oi(v)=aur∈gMm(vin) Ui(v,u); i≥0 (10)
NemaInfer first computes the candidate set for each query node
usingthenodelabelsimilarityfunction∆L (line1). Next, itini- Example4.1. WeillustratetheideaofoneiterationofNemaInfer
tializesaninferencecostU (v,u)byassigningittotheminimum using Figure 4. Assume we have already determined the candi-
0
possiblevalueofindividualnodematchingcostsF (v,u),overall date matches M(v) for every query node v using the label sim-
φ
possible matching functions φ, s.t., φ(v) = u (line2-3). It then ilarity function ∆L. For example, M(v2) = {u2,u5,u9} and
M(v4)= {u7,u10}inFigure4. Also,considerh= 1. Ati= 0, v1 u1 u4 u8
U0(v2,u5) = U0(v2,u9) = 0. Therefore,wecannotdistinguish a a a a
bteertwmeaetcnhuo5favn2d.uH9oiwnetvheer,inoibtisaelirzvaetitohnatrUou0n(dv,4a,suw10h)ic<hoUn0e(vis4a,ub7e)t-. v2 b c v3 ub uc u5 b c u6 uu9 b c u12
u10 isaneighbor of u9, whileu7 aneighbor of u5. Thus, itnot v4 d 2 3 ud 10 d
onlyinfluencestheoptimalmatchO0(v4)ofv4atiterationi=0, v5 e 7 u11 e c u13
butitalsomakesU (v ,u ) < U (v ,u )atiterationi = 1,via
1 2 9 1 2 5
Eq.9. Hence, weimprovethematchesineachiterationandpro- Query Graph Q Target Graph G
ceedtowardstheminimumcost(heuristic)subgraphmatch.
Figure4:OptimalSubgraphMatchFindingAlgorithm
Invariant. ThealgorithmNemaInferpossesthefollowinginvari-
NemaInfer,wecomputeapartialinferencecostforeachnodev′∈
athnetiinnfeearcehncoefcitossittearnadtitohne,swuhbigcrhapilhlumstaratctehsinthgeccoosntn(Secetcitoionnb3et)w. een N(v),whichisdenotedbyWi(v,u,v′),anddefinedbelow.
Invariant1. Ifthereexistsamatchingfunctionφfromthenodes
ofQtothenodesofG,suchthat,C(φ)=0,thenUi(v,φ(v))=0 Wi(v,u,v′)={φ:φm(vi)n=u} β(v)·∆+ PQ(v,v′),PG u,φ(v′)
forallv∈VQandi≥0. (cid:2)+Ui−1 v′(cid:0),φ(v′) (cid:0) (1(cid:1)3(cid:1))
However, theconverseisnotalwaystrue. Infact,basedonthe
properties of the loopy belief propagation algorithm, there is no Here, β(v) = [ v′∈N(v)P(cid:0)Q(v,v′)]−(cid:1)1(cid:3). To compute
guaranteethatouralgorithmwillconvergeforallquerynodesafter Wi(v,u,v′), we only nPeed to find the minimum value in Eq. 13
acertainnumberofiterations. Therefore,weterminatetheproce- over the candidates in M(v′). Hence, the partial inference cost
durewhenmorethanathresholdnumberofquerynodesvsatisfy Wi(v,u,v′)canbecomputedinpolynomialtime,foreachtriplet
theconditionOi(v) = Oi−1(v). WeempiricallyverifiedinSec- v, u, v′, where u ∈ M(v) and v′ ∈ N(v). Next, we show the
tion 7 that our method usually requires about 2 to 3 iterations to relationbetweenthepartialinferencecostandtheinferencecost.
terminate — around 95% of query nodes converge using IMDB
dataset,andalsoperformswellinreal-lifenetworks. Theorem3. TheinferencecostUi(v,u)iscomputableinpolyno-
mialtimeviathefollowingformula:
Matchingrefinement(line13). Theoptimalmatchofeachquery Ui(v,u)=∆L LQ(v),L(u) + Wi(v,u,v′) (14)
node at the final iteration might not correspond to the subgraph
matching function with the minimum (heuristic) aggregate cost (cid:0) (cid:1) v′X∈N(v)
PROOF.
[29]. Thiscanhappen iftherearemultiplegraphmatchingfunc-
tions that result in the minimum cost graph matches. Therefore, Ui(v,u)
we need to refine the optimal node matches from the final round
ofNemaInfertoidentifyonesuchminimumcostsubgraphmatch- = min ∆L(LQ(v),L(u))+β(v)· P ∆+(PQ(v,v′),PG(u,φ(v′)))
ing function, say Φ. Werefer tothe matches of the query nodes {φ:φ(v)=u} v′∈N(v)
(cid:2)
corresponding to Φ as the most probable matches. To find these
Fφ(v,u)
most probable matches, the standard memoization technique can
beused afterthetermination ofour iterativeinference algorithm. | + P Ui−1{(zv′,φ(v′)) }
v′∈N(v)
First,aquerynode,sayv,isselectedrandomly,anditsmostprob- (cid:3)
ablematch,denotedbyΦ(v)∈M(v),isdeterminedasfollows: ={φ:φm(vin)=u}v′∈PN(v) β(v)·∆+(PQ(v,v′),PG(u,φ(v′)))+Ui−1(v′,φ(v′))
(cid:2) (cid:3)
Φ(v)=argminUi′(v,u) (11) +∆L(LQ(v),L(u))
u∈M(v)
In Eq. 11, i = i′ denotes the final iteration. For the remaining =v′∈PN(v){φ:φm(vin)=u} β(v)·∆+(PQ(v,v′),PG(u,φ(v′)))+Ui−1(v′,φ(v′))
nodes, the most probable matches are determined by memoizing (cid:2) (cid:3)
recursively, i.e., wekeep trackof thematches of theneighboring Wi(v,u,v′)
nodesthatgiverisetothemostprobablematchofthecurrentnode. | +∆L(LQ{(vz),L(u)) }
Forexample,themostprobablematchesΦ(v′)ofallv′∈N(v)are =∆L(LQ(v),L(u))+ Wi(v,u,v′)
obtainedusingthemostprobablematchofvasfollows.
v′X∈N(v)
Hence,thetheorem.
φp = argmin Fφ(v,Φ(v))+ Ui′−1 v′,φ(v′)
{φ:φ(v)=Φ(v)}(cid:2) v′X∈N(v) (cid:0) (cid:1)(cid:3) It follows from Theorem 3 that the inference cost Ui(v,u) of
Φ(v′)=φp(v′) (12) nodesvanducanbeefficientlycomputedinpolynomialtime,by
(1)determiningthepartialinferencecostWi(v,u,v′)foreachv′∈
The aforementioned memoization technique is performed until N(v), and (2) aggregating these partial inference costs with ∆L
themostprobablematchesofallquerynodesarecomputed.
LQ(v),L(u) . The aforementioned technique also keeps track
ComputationofInferenceCosts. Astraightforwardapproachto ofwhichmatchesoftheneighboring querynodesgiverisetothe
(cid:0) (cid:1)
determine the inference cost Ui(v,u) for a query node v and its mostprobablematchofthecurrentquerynode. Thisinformation
candidateu(Eq.9)considersallpossiblecombinationofmatches isrequiredduringmatchingrefinement.
for all nodes v′ ∈ N(v), which has exponential timecomplexity Time complexity. We analyze the time complexity of the algo-
and might beveryexpensive. Inthissection, wepropose atech-
rithm NemaInfer. Let us denote the number of nodes in the tar-
niquetocomputetheinferencecostinpolynomialtime.
getgraphGandthequerygraphQas|V|and|VQ|,respectively.
Partialinferencecost. ToevaluatetheinferencecostUi(v,u)for (1)ItrequiresO(|VQ|·|V|)timetoidentifyM(v)foreachquery
aquery node v anditscandidate uatiterationiof thealgorithm node v ∈ VQ (line 1). (2) We denote the maximum number of
candidates per query node as mQ, and the maximum number of the performance of our inference algorithm NemaInfer. Wefirst
h-hop neighbors of each query node as dQ. The computation of introducethenotionofisolatedcandidates.
Othe(mop2Qti·mdaQl)mfaotlclhowOintg(vT)hpeeorreqmue3ry(lninoede10v).haTshetirmefeorceo,mthpeletximitye MIso(lva)t,eadnCoadneduid∈atMes(.vG)iivseannaisqoulaetreydncoadnedivdaatnedoiftsv,caifndidateset
requiredforeachiterationofNemaInferisO(|VQ|·m2Q·dQ). If
therearetotalI iterations,theoveralltimecomplexityisgivenby
O(|VQ|·|V|+I ·|VQ|·m2Q ·dQ). Observe that |VQ|, I, |dQ| {u′ :u′ ∈M(v′),v′ ∈N(v)}∩{u′′ :u′′ ∈N(u)}=∅ (15)
andmQaretypicallysmall.Indeed,asverifiedinourexperiments
Intuitively, thenodeuisanisolatedcandidateofaquerynode
(Section7),Iistypicallylessthan4andmQis35,forquerygraphs
vifnoneofthecandidateswithinh-hopneighborsofvareinthe
with5nodesandreallifegraphscontaining12Mnodes.
h-hopneighborhoodofu;otherwise,itisanon-isolatedcandidate
5.2 Generalized Queries
ofv.Thus,anisolatedcandidateuofvcannotbematchedwithv.
Inthissection,weextendNemaInferforthreegeneralizedcases, Toefficientlyfindthenon-isolatedcandidates,weproposeanop-
namely,Top-kmatches,unlabeledqueries,andlabelededges. timizationproblem,basedonverificationcostandcandidatecover.
Top-kMatches. Inmanyapplications,thequerygraphisnotsub- Verification Cost. The verification cost associated with a query
graph isomorphic to the target network; and hence, weare inter- nodev isdefinedasthetimecomplexitytoverifyallnodesinits
ested in identifying the top-k matches rather than only the best candidatesetM(v),whethertheyarenon-isolatedcandidates.Note
match. Given the target network G and the query graph Q, the thatthecomplexityofverifyingwhethersomenodeu∈M(v)isa
top-ksubgraphmatchingproblemistoidentifythetop-kmatches non-isolatedcandidateisO |N(u)|+ |M(v′)| .
v′∈N(v)
foraselectedquerynodev∈VQ. Candidate Cover. There exists dependencies between two non-
(cid:0) P (cid:1)
ThealgorithmNemaInfercanbereadilyadaptedforthisprob-
isolatedcandidates:ifuisanon-isolatedcandidateofv,thenthere
lem.(1)Thealgorithmcomputestheinferencecostsandterminates must exist a node u′ ∈ N(u), such that, u′ ∈ M(v′) for some
at line12. (2)Weidentifythetop-k mostprobable matchesof v v′ ∈ N(v). Clearly, u′ is a non-isolated candidate of v′. If we
(Eq.11). (3)Foreachofthesetop-kmostprobablematchesofv, verifiedallcandidates{u′ : u′ ∈ M(v′),v′ ∈ N(v)},thereisno
weapplytherecursivememoizingtechnique(Eq.12)todetermine needtoverifythecandidatesinM(v)again.Thus,onemayreduce
thecorrespondingmostprobablematchesforotherquerynodes.
redundantverificationsusinganotionofcandidatecover.
Matching Query with Unlabeled Nodes. A query graph may WedefinecandidatecoverC(Q)asasetofquerynodesv,such
have nodes with unknown labels, e.g., query graphs constructed that,forallv′∈VQ,eitherv′ ∈C(Q),orv′ ∈N(v).
fromRDFqueries.NeMacanbeadaptedtoevaluatesuchqueries.
First,weidentifyallthenodesfromthetargetnetworkthatcanbe
C(Q)={v:∀v′(v′ ∈C(Q)∨v′ ∈N(v))} (16)
matched withsome labeled query node based on label similarity.
Next, we find the subgraph induced by all those matched nodes Allnon-isolatedcandidatenodescanbeidentifiedbyverifying
fromthetargetnetworkalongwiththeirneighborswithinh-hops. onlythequerynodesinC(Q). Wedefinetheverificationcostofa
Allnodesinthissubgraphareconsideredasthecandidatesforthe candidatecoverasthesumoftheverificationcostsofitsconstituent
unlabeledquerynodes. ThealgorithmNemaInferistheninvoked querynodes.Next,weintroducethecandidatecoverproblem.
toidentifythematches. Inaddition, iftheunlabeledquerynodes
containtypeinformation,thecandidatesetscanfurtherberefined. ProblemStatement 2. CandidateCover. Givena query graph
Q,findthecandidatecoverwiththeminimumverificationcost.
NeMawithEdgeLabels.TheNeMacostfunctioncanbeadapted
toconsidertheedgelabels. Specifically,weconcatenatetheedge Thefollowingresultshowsthatthecandidatecoverproblemis
labels along the shortest path between a pair of nodes, and then intractable,butapproximablewithinafactor2inpolynomialtime.
updatetheneighborhoodmatchingcost(Equation2)asfollows.
Theorem4. Thecandidatecoverproblemis(1)NP-hard,and(2)
N (v,u)
φ 2-approximable.
v′∈N(v)[∆+(PQ(v,v′),PG(u,u′)))+∆L(s(v,v′),s(u,u′)))] PROOF. We show that this problem is NP-hard by reduction
= P [PQ(v,v′)+1] from NP-complete weighted minimum vertex cover problem [8].
v′∈N(v) Given a decision version of the weighted minimum vertex cover
P problem,weconstructaninstanceofthecandidatecoverproblem,
Here, u = φ(v),u′ = φ(v′), ands(v,v′)concatenates theedge wherethevertexweightsareconsideredasthecorrespondingver-
labelsalongtheshortestpathbetweenvandv′.Since,weconsider ification costs. Assuming h = 1, the minimum weighted vertex
theedgelabelsalongtheshortestpathbetweenapairofnodes,the coverwillbeourcandidatecover. Onecanapplylinearprogram-
asymptoticaltimecomplexityofNeMaInferremainsthesame. mingtosolvethisproblemwith2-approximation[8].
6. INDEXINGANDOPTIMIZATION 6.2 Indexing
Inthissection,wediscussindexingandoptimizationtechniques We introduce indexing technique to improve the efficiency of
toimprovetheefficiencyofournetworkmatchingalgorithm. the inference algorithm. (1) During the off-line indexing phase,
itcomputestheneighborhood vectorsR(u)forallnodesuinthe
6.1 Candidate Selection
targetnetworkG,andstoresthevectorsintheindex.(2)Duringthe
Thecandidatesetofaquerynodeisdefinedintermsofthelabel on-linenetworkmatchingtechnique,ifuisselectedasacandidate
similarity function (see Section 3), which may include candidate ofv,itappliesEq.15toverifywhetheruisanisolatedcandidate
nodesthatdonotmatchthequerynodeduetoneighborhoodmis- ofv.Ifso,uiseliminatedfromthecandidatesetofv.
match. We introduce optimization techniques to efficiently filter Our index structure has space and time complexity O(ndh),
G
suchcandidatenodesasmuchaspossible,andtherebyimproving where|V|=n,dG =averagenodedegreeinG,andh=depthof
IMDB INDEX 1 LABEL NOISE TH.=35% LABEL NOISE TH.=35%
YAGO MATCH (TOP-1) LABEL NOISE TH.=50% LABEL NOISE TH.=50%
1 DBpedia MATCH (TOP-3) 0.03
F1-MEASURE 000...789 TIME (SEC) 1 10 010000 1000 MATCH (TOP-5) F1-MEASURE 000...789 TIME (SEC) 00..0012
0.3
0.6 0 0.3 0.5 0.7 1 0 .00.21 0.6 0 35 50 0.005 0 35 50
λ IMDB YAGODBpedia LABEL NOISE (%) LABEL NOISE (%)
(a) Effectiveness (b) Efficiency (a) Effectiveness(DBpedia) (b) MatchTime(DBpedia)
Figure5:QueryPerformance Figure6:PerformanceagainstLabelNoise
Dataset #Nodes #Edges
vectorization.ForNeMawithedgelabels(Section5.2),theasymp- IMDB 2,932,657 11,040,263
totical time and space complexity of indexing remains the same, YAGO 12,811,149 18,282,215
sinceweconsideredgelabelsalongtheshortestpaths. DBpedia 5,177,018 20,835,327
Dynamicmaintenanceof theindex. Our indexing methodscan Table3:DatasetSizes
efficientlyaccommodatedynamicupdatesinthetargetnetwork. If
The nodes in YAGO and DBpedia are annotated with labels,
anodeu(andtheedgesattachedtoit)isaddedordeleted,onlythe
whilethenodesinIMDBareannotatedwithbothtypesandlabels.
indexesofu’sh-hopneighborsneedtobeupdated.Ifasingleedge
Hence,weusedthetypeinformationassociatedwiththenodes,in
(u,u′)isaddedordeleted,onlytheh−1hopneighborsofbothu
additiontotheirlabels,whilequeryingtheIMDBnetwork.
andu′areupdated,thusreducingtheredundantcomputation.
Querygraphs. Wegeneratedthequerygraphsbyextractingsub-
6.3 Optimizationfortop-k matching graphs from the target graphs, and then introduced noise to each
The inference algorithm NemaInfer can be adapted toidentify querygraph.Specifically,thequerygenerationwascontrolledby:
the top-k matches. For small values of k, it is possible to prune • nodenumberanddiameter,denotedby|VQ|andDQ,respec-
candidatenodesbysettingacostthreshold. Thecostthresholdǫc tively, where the query diameter is the maximum distance
isinitiallysettoasmallvalueǫ0. IfUi(v,u) > ǫc forsomeu ∈ betweenanytwonodesinthequerygraphQ.
M(v)atiterationioftheinferencealgorithm,thenuiseliminated
from the candidate set M(v) for the subsequent iterations. After • structuralnoise,theratioofthenumberofedgeupdates(ran-
dominsertionanddeletionofedges)inQtothenumberof
termination,ifthetop-kmatchescannotbeidentified,weincrease
edgesintheextractedsubgraph;and
ǫcbyasmallvalue,andrepeatthestepsabove. Thecorrectnessof
thismethodisensuredbyTheorem5. • labelnoise,measuredbytheJaccardsimilaritybetweenthe
labelsof nodesintheextractedsubgraph andtheirupdated
Theorem 5. If Ui(v,u) > ǫc at the i-th round of inference al- counterpartsinQ,wheretheupdatedlabelswereobtainedby
gorithm, then for all j > i, Uj(v,u) > ǫc at the j-th round of insertingrandomlygeneratedwordstothequerynodelabels.
inferencealgorithm.
We used Jaccard similarity as the label similarity measure.
PROOF. ItfollowsdirectlyfromEq.9andthefactthatUi(v,u) Specifically, given a query node v and a target node u, the label
≥0foralli≥0,overallpairsv,u,whereu∈M(v). difference ∆L(LQ(v),L(u)) is defined as 1− ||wwvv∩∪wwuu||, where
Hence,wecaneliminateufromM(v),wheneverUi(v,u)>ǫc wv andwuarethesetofwordsintheirlabelLQ(v)andL(u),re-
spectively. Recall that weallowed some noise inthe node labels
occursforthefirsttimeatsomeiterationiofNemaInfer.
byvaryingthelabelmatchingcostthresholdǫinourmatchingal-
gorithm (Problem Statement 1). A node u in the target network
7. EXPERIMENTAL RESULTS
isconsideredacandidatetomatchwithaquerynodeviftheirla-
We present three sets of empirical results over three real-life bel difference ∆L(LQ(v),L(u)) isless than the predefined cost
datasets to evaluate (1) the effectiveness and efficiency (Sec- thresholdǫ,referredtoasthelabelnoisethreshold.
tion 7.2), (2) scalability (Section 7.3), and (3) optimization tech-
Evaluationmetrics. Sincethequerygraphswereextractedfrom
niques(Section7.4)underlyingtheNeMaframework.
targetgraphs,onealreadyhasthecorrectnodematches. Now,the
7.1 Experimental Setup effectivenessofNeMaismeasuredasfollows. Precision(P)isthe
ratioofthecorrectlydiscoverednodematchesoveralldiscovered
GraphDataSets. Weusedthefollowingthreereallifedatasets, node matches. Recall (R) is the ratioof the correctly discovered
each represents a target graph. (1) IMDB Network3. TheInter- nodematchesoverallcorrectnodematches.F1-Measurecombines
netMovieDatabase(IMDB)consistsoftheentitiesofmovies,TV theresultsofprecisionandrecall,i.e.,
series, actors, directors, producers, amongothers, aswellastheir
2
relationships. (2) YAGOEntityRelationshipGraph4. YAGO is F1= (17)
(1/R+1/P)
aknowledgebasewithinformationharvestedfromtheWikipedia,
WordNetandGeoNames.Itcontainsabout20millionRDFtriples. Weconsideredthetop-1matchtoevaluateprecision,recall,and
(3) DBpedia Knowledge Base5. DBpedia extracts information F1-measure. Thus, weobtainedthesamevaluesforthem. How-
fromtheWikipedia. Weconsidered 22 millionRDFtriplesfrom ever,precisionandrecallwillbeusefulwhileanalyzingNESS.
DBpediaarticlecategories,infoboxproperties,andpersondata.
ComparingMethods. WecomparedNeMawithkeywordsearch
3http://www.imdb.com/interfaces#plain (BLINKS[16])andvariousgraphqueryingmethods: SAGA[35],
4http://www.mpi-inf.mpg.de/yago-naga/yago/ IsoRank[33],NESS[20],andNeMa -avariationofNeMafol-
gs
5http://dbpedia.org/About lowinggStore[43].AllthesemethodswereimplementedinC++.
F1-MEASURE 0000....9999 12468 nnnQQQ===357,,, DDDQQQ===123 MATCH TIME (SEC) 00000 0.....00001.124682 nnnQQQ===357,,, DDDQQQ===123 F1-MEASURE 0000....9999 12468 nnnQQQ===357,,, DDDQQQ===123 MATCH TIME (SEC) 0 .1235 nnnQQQ===357,,, DDDQQQ===123
0.9 0 0.9 0.3
0 10 20 30 40 0 10 20 30 40 0 25 35 50 80 0 25 35 50 80
STRUCTURAL NOISE (%) STRUCTURAL NOISE (%) LABEL NOISE THRESHOLD (%) LABEL NOISE THRESHOLD (%)
(a) (b) (c) (d)
Figure7:QueryPerformancevs.Noise(IMDB);nQ:QueryNodes,DQ:QueryDiameter
F1-MEASURE 0000 ....16789 nnnQQQ===573,,, DDDQQQ===231 MATCH TIME (SEC) 00 0..12.255 nnnQQQ===357,,, DDDQQQ===123 F1-MEASURE 0000 000....6789 ...17895555 nnnQQQ===357,,, DDDQQQ===123 MATCH TIME (SEC) 001 ...1473 nnnQQQ===357,,, DDDQQQ===123
0.6
0.5 0.1 0.55 0.1
0 10 20 30 40 0 10 20 30 40 0 25 35 50 80 0 25 35 50 80
STRUCTURAL NOISE (%) STRUCTURAL NOISE (%) LABEL NOISE THRESHOLD (%) LABEL NOISE THRESHOLD (%)
(a) (b) (c) (d)
Figure8:QueryPerformancevs.Noise(YAGO);nQ:QueryNodes,DQ:QueryDiameter
DE DE
DIDATE PER QUERY NO 1 12350 2500000 nnnQQQ===573,,, DDDQQQ===231 DIDATE PER QUERY NO 1 12352 0000250 nnnQQQ===573,,, DDDQQQ===231 F1-MEASURE 00 0..89 .1955 nnQQ==57,, DDQQ==23 F1-MEASURE 0000 ....16789 nnQQ==57,, DDQQ==23
N N
# CA 1 0 25 35 50 80 # CA 1 0 25 35 50 80 0.8 0 1 2 0.5 0 1 2
LABEL NOISE THRESHOLD (%) LABEL NOISE THRESHOLD (%) # UNLABELED QUERY NODE # UNLABELED QUERY NODE
(a) IMDB (b) YAGO (a) Effectiveness(IMDB) (b) Effectiveness(YAGO)
Figure9:#Candidatesvs.LabelNoise Figure11:EffectivenesswithUnlabeledQueryNodes
# ITERATIONS 223 ...2482 nnnQQQ===573,,, DDDQQQ===231 # ITERATIONS 22 ..224 nnnQQQ===573,,, DDDQQQ===231 F1-MEASURE 0000....9999 12468 W W/O/ EEDDGGEE LLAABBEELL MATCH TIME (SEC) 0000....34565555 W W/O/ EEDDGGEE LLAABBEELL
1.7 1.7 0.9 0.25
0 25 35 50 80 0 25 35 50 80 0 10 20 30 40 0 10 20 30 40
LABEL NOISE THRESHOLD (%) LABEL NOISE THRESHOLD (%) STRUCTURAL NOISE (%) STRUCTURAL NOISE (%)
(a) IMDB (b) YAGO (a) Effectiveness (b) Efficiency
Figure10:#Iterationsvs.LabelNoise Figure12:PerformancewithEdgeLabels(IMDB)
Inour experiments, (1)propagation factorαanddepthofvec- Figure5(b)reportstheefficiencyofNeMausingthesameset-
torization h (Section 3) were set as 0.5 and 2, respectively [20], tingasinFigure5(a),includingtherunningtimeofoff-lineindex
(2)theoptimalvaluesoftheproportionalityconstantλ(Eq.4)for construction (INDEX) and online query evaluation (MATCH). We
different datasetswereobtained empirically(Figure5(a)), (3) the observedthefollowing. (a)NeMaidentifiesthebestmatchinless
indexeswerestoredintheharddisk. Alltheexperimentswererun than 0.2 seconds, over all three datasets 6. (b) The top-k match
usingasinglecoreina100GB,2.5GHzXeonserver. findingtimedoesnotvarysignificantly,overdifferentk,sinceour
inferencingmethodisalwaysexecutedonce.(c)Thetimerequired
7.2 Effectiveness andEfficiency
for indexing ismodest (e.g., 9862 secfor theYAGO dataset with
7.2.1 PerformanceoverReal-lifeDataSets 13Mnodesand18Medges). (d)Theindexingandqueryingtimes
arelongeroverIMDB,duetoitshigherdensity.
Intheseexperiments,weevaluatedtheperformanceofofNeMa
7.2.2 PerformanceagainstNoise
over three real-lifegraphs, averaged over 100 queries (Figure5).
For each target graph, we randomly generated 100 query graphs Inthissetofexperiments,weinvestigatedtheimpactofvarying
with|VQ|=7andDQ =3. Wefixedthestructuralnoiseas30%, noises on the performance of NeMa. Three setsof query graphs
labelnoiseas50%,andlabelnoisethresholdas50%. weregeneratedbysetting(i)|VQ| = 3, DQ = 1, (ii)|VQ| = 5,
Figure 5(a) shows the effectiveness of NeMa over various DQ = 2,and(iii)|VQ|=7,DQ = 3. Undereachqueryset,100
datasets,andwithdifferentvaluesoftheproportionalityconstantλ. querygraphsweregenerated.
Forallthethreedatasets,ouralgorithmalwayscorrectlyidentifies
Varyinglabelnoise.Fixingthestructuralnoiseas30%,wevaried
more than76% of thequery nodes. Specifically, theF1-measure
thelabelnoisefrom0%to50%,andinvestigatedtheeffectiveness
is0.94 forIMDB,withλ = 0.3, even whenweintroduced 30%
ofNeMa,whenthelabelnoisethresholdwassetas35%and50%,
structural noise and 50% label noise. Theeffectiveness ishigher
overIMDBduetothetypeconstraintposedwiththequerynodes. 6our indexingandmatchingalgorithmcanbeparallelizedforev-
Besides, the optimal value of λ lies between 0.3 ∼ 0.5 in these ery node. Hence, one may implement NeMa ina PREGEL[24]
datasets. platform,forlargergraphs.
P, R 1 P, R 1 EC) 4000
URE, 00..68 URE, 00..68 ME (S 500
MEAS 00..24 MEAS 00..24 CH TI 1.5
F1- 0 0 20 30 40 F1- 0 0 25 35 50 80 MAT 00..14
STRUCTURAL NOISE (%) LABEL NOISE (%) 0 25 35 50 80
LABEL NOISE (%)
IsoRank (F1) NESS (F1) IsoRank (F1) NESS (F1)
SAGA (F1) NeMags (F1) SAGA (F1) NeMags (F1) IsoRank NeMags
NESS (P) BLINKS (F1) NESS (P) BLINKS (F1) SAGA BLINKS
NESS (R) NeMa (F1) NESS (R) NeMa (F1) NESS NeMa
(a) (b) (c)
Figure13:ComparisonResults(IMDB):NESS,BLINKSModifiedforApproximateLabelMatch.NESSResultsCorrespondtoitsFilteringPhase.
respectively. Asshown inFigure 6(a) over DBpedia, (1) the F1- nodes from 0 to 2. As shown in Figure 11, (1) the F1-measure
measuredecreasesasthelabelnoiseincreases,sincethecandidate decreases over both datasets while the number of the unlabeled
set of each query node may contain more candidates that arenot nodesincreases,becausetheunlabelednodesintroducemorecan-
truematches, whichinturnreduces theeffectiveness, (2)theF1- didates, whichinturnreducestheeffectiveness, (2)theeffective-
measure ishigher when the label noise threshold ishigher, since nessishigheroverIMDBduetothetypeconstraints,and(3)over
thecandidate setsare morelikelytoinclude thecorrect matches. allthecases,theF1-measureisalwaysabove0.50.
ObservethattheF1-measureisalwaysabove0.60. 7.2.4 PerformancewithPropagationDepth
Figure 6(b) shows that NeMa efficiency is insensitive to label
In these experiments, we analyzed the effect of propagation
noise,butmoresensitivetolabelnoisethreshold.Thisisbecauseit
depth h in our query performance. We randomly selected 100
takesNeMamoretimetoprocessthelargercandidatesetsforthe
querygraphasthelabelnoisethresholdincreases. query graphs from YAGO, with |VQ| = 7, DQ = 3, structural
noise 30%, label noise 50% and label noise threshold 50%. Ta-
Varyingstructuralnoise.Fixingthelabelnoisethresholdas35% ble4showsthattheefficiencyofNeMadecreaseswithincreasing
and label noise as 35%, we varied the structural noise from 0% h,especiallytheindextimeincreasesexponentiallywithh. How-
to 40% inFigure 7(a) and 7(b). It can be observed that (a) both ever,forh=2,weobtainedanacceptableF1-measureof0.86.
effectivenessandefficiencydecreaseasweincreasethestructural
noise,and(b)withtheincreaseofthequerysize,botheffectiveness h=1 h=2 h=3
andefficiencyincrease. Thereasonisthat(1)largerquerieshave IndexTime(sec) 265 9862 236553
moreconstraintsintheneighborhoodofaquerynode,whichbene- MatchTime(sec) 0.58 0.92 2.76
F1-Measure 0.61 0.86 0.87
fittheidentificationofcorrectmatches,and(2)ittakeslongertime
forNeMatocomparethematchingcostforlargerqueries. More-
Table4:QueryPerformancewithVaryingh(YAGO)
over,theF1-measureisalwaysabove0.93,andtherunningtimeis
alwayslessthan0.1seconds,evenwith40%structuralnoise. 7.2.5 PerformancewithEdgeLabels
Weverifiedthequeryperformanceinthepresenceofedgelabels
Varyinglabelnoisethreshold.Fixingthestructuralnoiseas30%
(Section5.2).Werandomlyselected100querygraphsfromIMDB,
and the label noise as35%, weinvestigated theeffect of varying
thelabelnoisethresholdonthequeryperformance. with with |VQ| = 7, DQ = 3, label noise 50% and label noise
threshold 50%. Wevaried the structural noise from 0% to 40%.
TheeffectivenessandefficiencyofNeMaoverIMDBareillus-
The labels of the newly inserted edges in the query graphs were
tratedinFigures7(c)and7(d),respectively. Weobservedthefol-
assignedbygeneratingrandomstrings. Figure12showsthat, (1)
lowing. (1)TheF1-measureinitiallyincreaseswhileweincrease
whennostructuralnoiseisadded,theF1-measureremainssamefor
thelabelnoisethreshold.Thisisbecausethequerynodelabelsare
boththecasesoflabeledandunlabelededges. (2)However, with
updated by adding random words. Hence, thehigher isthe label
theadditionofstructuralnoise,theF1-measuredecreasesslightly
noise threshold, there is more chance that the correct match of a
for the case of labeled edges, since there are more noises in the
querynodewillbeselectedinitscandidateset.(2)Whenthelabel
query graphs due to the labels of newly inserted edges. (3) The
noise threshold is more than 35%, the F1-measure does not im-
runningtimeishigherforthecaseoflabelededgesbecauseaddi-
provesignificantly. Therefore,theoptimalvalueofthelabelnoise
tionaltimeisrequiredtomeasureedgelabelsimilarities.
thresholdcanbedeterminedempiricallybasedonthequeryandtar-
getgraphs.Ontheotherhand,therunningtimeofNeMaincreases 7.2.6 ComparisonwithExistingAlgorithms
withtheincreaseofthelabelnoisethreshold. Thisisbecause(a)
We compared the performance of NeMa with IsoRank [33],
thecandidatematchesperquerynodeincreases(seeFigure9),and
SAGA [35], NESS [20], gStore [43], and BLINKS [16]. (1)
(b)thenumberofiterationsrequiredfortheconvergenceofournet-
IsoRank and SAGA find optimal graph matches in smaller bio-
workmatchingalgorithmalsoincreases(seeFigure10). Hence,it
logicalnetworksconsideringstructureandnodelabelsimilarities.
takesmoretimeforNeMatofindthematches.
(2) NESS finds the top-k graph matches from a large network,
7.2.3 EffectivenesswithUnlabeledQueryNode
but with strict node label equality. Hence, we modified NESS
WenextverifiedtheeffectivenessofNeMainthepresenceofun- by allowing two nodes to be matched if their label difference is
labeled nodesinquerygraphs. Theseexperiments simulateRDF withinthelabelnoisethreshold. (3)Weconsideredavariationof
query answering (see Section 5.2). For these experiments, we NeMa,namely,NeMa ,whichallowslabeldifferencebutresorts
gs
randomly selected two sets of 100 query graphs each, where (i) to strict isomorphic matching. Thus, NeMags essentially follows
|VQ| = 7, DQ = 3, and (ii) |VQ| = 5, DQ = 2, respectively. the same principle as gStore, which is a subgraph isomorphism-
Fixingthestructuralnoiseas30%, labelnoiseas35%, andlabel based SPARQL query evaluation method with node label differ-
noisethresholdas35%,wevariedthenumberofunlabeledquery ences. (4)BLINKS[16],akeywordsearchmethod,supportsonly