Graph Mining: Overview of different graph models · Graph Mining: Overview of different graph models Graph Mining course Winter Semester 2016 Davide Mottin, Konstantina Lazaridou

Graph Mining: Overview of different graph

models

Graph Mining course Winter Semester 2016

DavideMottin,KonstantinaLazaridouHassoPlattnerInstitute

Lecture road

2

Anomalydetection(previouslecture)

RepresentativesofProbabilistic(Uncertain)graphs

IntroductiontoSignednetworks

Graph models▪ Graphsareeverywhere!▪ Variousinterestingmodelsthatwehaven’tanalyzedinthelecture..

• graphstreams• evolvinggraphs• attributedgraphs• probabilisticgraphs• signedgraphs• coloredgraphs• ...

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Definitions

▪ Graphstream• sequenceofunorderedpairse={u,v}whereu,v∈ [n],S=(e1,e2,...,emi)

▪ Timeevolvinggraph• sequenceofstaticgraphs{G1,G2,...,Gn},whereGt=(Vt,Et)isasnapshotoftheevolvinggraphattimestampt

▪ Attributedgraph• G=(V,E,A)whereVisthevertexset,Eistheedgeset,andAistheattributesetthatcontainsunaryattributeai(linkedtoeachnodeni)andbinaryattributeaij (linkedtoeachedgeek=(ni ,nj )∈ E),

▪ Coloredgraph• G=(V,E)inwhicheachvertexisassignedacolor.

⁃ properlycoloredgraph:colorassignmentsconformtothecoloringrulesappliedtothegraph

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Probabilistic graphs - Outline

▪ Uncertaintyindata▪ Introductiontouncertaingraphs• Modeldefinition• Applications• Problems

▪ Findingrepresentativesinprobabilisticgraphs• Problemdefinition• Algorithms

GRAPH MINING WS 2016 5

Uncertainty in data

▪ Noiseingeneration• sensors

▪ Noiseincollection• missinginstances

▪ Biologicaldata• protein-proteininteractionprobability

▪ Problem’snature• risk,trust,influence,status

▪ Anonymizeddata• privacy preservation of user generated data


What is an uncertain graph?

▪ Agraphwhereeachedgehasanassociatedprobabilityp:[0,1]toit

GRAPH MINING WS 2016

Figure 1: (left) An unweighted probabilistic graph G, (right) G with the expected vertex degrees (in Italics)associated to each node

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Possible applications and problems

▪ Modellingofprobabilitiesinprotein-proteininteractiongraphs▪ Modellingrelationshipsinsocialgraphs


▪ Problemsthatapplytodeterministicgraphs• algorithmsneedtoberedesignedtoincorporateuncertainty

▪ Dataanonymization• oneofthepossibleworldscorrespondstheoriginaldata

▪ Frequentsubgraphmining• frequencyisredefinedusingtheedgeprobabilities

▪ Queriesbasedonshortestpaths• returns paths with very low probabilities

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Graph model definition

▪ AprobabilisticgraphisrepresentedasG =(V, E,W,p),whereV isthesetofvertices,E isthesetofedges,forweightedgraphsW:V х V →Rdenotestheweightsassociatedwitheveryedgeandpmapseverypairofnodestoarealnumber in[0,1]▪ puv representstheprobabilitythatedge(u,v)existsintheuncertainnetwork▪ ForaprobabilisticgraphG,2 " deterministicgraphscanbegenerated

• thesegraphsarecalledpossibleworlds


Possible world semantics [1]▪ Oftenintheliteratureitisassumedthattheedgeprobabilities

areindependent• isthisalwaysthecase?


[1] S. Abiteboul, P. Kanellakis, and G. Grahne. On the representation and querying of sets of possible worlds, SIGMOD 1987

▪ Forsimplicity,variousapproachestreattheprobabilitiesoftheedgesasweights

▪ Othersonlyconsidertheedgeshavingaprobabilityp>t• notvalidassumptionsinmanyscenarios!

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Sampling

▪ TheprobabilitythatacertaingraphG=(V,E)willbesampledfromGiscomputedasfollows:•P[G]=Π(u,v)ϵEPuv*Π(u,v)ϵ(VxV)\E(1– Puv)

▪ GivenGandthevertexdegrees,wecanalsocalculatedthevertexdiscrepancies•disu(G)=degu(G)– degu(G),whereuisanodeinG•G’sdiscrepancyisdefinedasthesumofallnodediscrepancies


Figure 2: (left) G with the expected vertex degrees associated to each node, (right) a certain instance G of Gwith the vertex discrepancies

G=argminG:worldofG Δ(G)

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What if we could work on a deterministic graph instead? How do we benefit?

▪ Computationalcomplexitywouldbemuchlower!▪ Traditionaldataminingalgorithmscouldbeapplied▪ Whichcharacteristicsshouldthiscertaingraphmaintainfromtheuncertainone?• samenumberofvertices..•whichedgesshouldbeincluded?


Outline - Probabilistic graphs

▪ Uncertaintyindata▪ Introductiontouncertaingraphs• Modeldefinition• Applications• Problems

▪ Findingrepresentativesinprobabilisticgraphs• Problemdefinition• Algorithms


Finding representatives in probabilistic graphs [2]▪ ArepresentativeGofaprobabilisticgraphG isadeterministicgraphthatitsverticeswillpresenttheleastpossiblediscrepancy

▪ Moreformally• Given an undirected uncertain graph G = (V, E, W, p), the representative isan exact instance G of G (possible world), such that each vertex degreewill have the minimum deviation from its expected value


[2]ThePursuitofaGoodPossibleWorld:ExtractingRepresentativeInstancesofUncertainGraphs,PanosParchaset.al,ACMSIGMOD2014

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Introduced algorithms▪ Baseline1:Greedyprobability

• eachedgee=(u,v)belongstoG,ifitdecreasesthetotaldiscrepancy

▪ Baseline2:Mostprobable• eachedgee=(u,v)belongstoG,ifpe ≥0.5holds

▪ ADR(averagedegreerewiring)• aimsatpreservingtheexpectedaveragedegreeofG

▪ ABM(approximateb-matching)• preserves the expected vertex degrees


ADR: average degree rewiring

▪ Whatistheexpectedaveragedegree?• degavg(G)=2*P/|V|,wherePisthesumofalledgeprobabilitiesinG

▪ Inordertopreserveit,GshouldcontainexactlyPedges▪ MainstepsofADR

•ConstructaseedsetE0oftheedgesinG• Foragivennumberoftimesk⁃SwaptheedgesinE0withedgesinE\E0,sothattheoveralldiscrepancyoftherepresentativedecreases


Pseudocode

▪ Initialization,computationofP,sortEindecreasingorderbytheedgeprobabilities▪ ForeacheinE

• ifrandomx<=pe:insertintoE0,updateG

▪ C=E\E0▪ Forktimes

• ForeachnodeuinG⁃I=incidentedgesofu⁃chooserandomlye1inIande2inCtoswap⁃computetheoveralldiscrepancybeforeandafterthepotentialswap⁃ifimprovement:swape1withe2inE,Crespectively,updatediscrepancies


ADR example: edge probabilities


ADR: a possible world and the discrepancies


ADR: first iterations


d1+d2 < 0 explanation▪ Forreplacing(u,v)with(x,y)

▪ d1=|disu(G)- 1|+|disv(G)– 1|- (|disu(G)|+|disv(G)|)

▪ d2=|disx(G)+1|+|disy(G)+1|- (|disx(G)|+|disy(G)|)

▪ Sumuv_bef=|disu(G)|+|disv(G)|

▪ Sumuv_after=|disu(G)– 1|+|disv(G)– 1|

▪ Sumxy_bef=|disx(G)|+|disy(G)|

▪ Sumxy_after=|disx(G)+1|+|disy(G)+1|

▪ d1=Sumuv_after– Sumuv_bef

▪ d2=Sumxy_after– Sumxy_bef

▪ Ifd1andd2arepositive,then• Sumuv_after>Sumuv_bef

• Sumxy_after>Sumxy_bef⁃ noneoftheunderlyingnodesbenefitsfromtheswap...


References

▪ Uncertaindata•Ontherepresentationandqueryingofsetsofpossibleworlds•Asurveyofuncertaindataalgorithmsandapplications

▪ Uncertaingraphs• Thepursuitofagoodpossibleworld:extractingrepresentativeinstancesofuncertaingraphs

•Uncertaingraphsparsification•Uncertaingraphprocessingthroughrepresentativeinstances• Triangle-basedrepresentativepossibleworldsofuncertaingraphs•Clusteringlargeprobabilisticgraphs•Algorithmsformininguncertaingraphdata•K-nearestneighborsinuncertaingraphs


Lecture road

23

Anomalydetection

RepresentativesofProbabilistic(Uncertain)graphs

Introductiontosignednetworks

What is a signed network?

▪ ItisagraphG=(V,E),whereeachedgeismappedtoasign▪ Asigncanbepositiveornegative▪ Thesignofapathistheproductofthesignsofitsedges▪ Typicallyasignednetworkisdenotedby:

• Σ=G(V,E,σ),whereσ,orthesignatureofthegraph,isthefunctionσ:E->(+,-)


u v

k

+/-

+/-+/-

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What is balance?

▪ History..• FritzHeider (psychologist)andFrankHarary (mathematician)laythefoundationsofthesignedgraphsandthebalancetheory

• OriginalideaofP-O-Xmodel⁃ howaresocialrelationsmodeled?aretheybalanced?


“Theenemyofmyenemyismyfriend”!

P O

X

+

+-

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Example of the P-O-X model

▪ ImaginethatyouarepersonPandthatOissomeone,whomyouthinkhighlyof,nowimagineXisapresidentialcandidateyoudislike,butXvehementlyendorseesO.▪ Whatdoyoususpectwouldhappen?


+

+-

thesituationisunbalanced...

PneedstoagreewithhisfriendO,orneedstounfriendO!

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Balance theory

▪ Theorem1:Gisbalancedifeverypathpbetweenu,vhavethesamesign▪ Theorem2:AsignedgraphisbalancedifandonlyifVcanbebipartitioned,s.t.eachedgebetweenthepartsisnegativeandeachedgewithinapartispositive


Status theory [3]

▪ Thesignsinbalancetheoryareperceivedaslikes/dislikes▪ Cantheyalsoindicateanotherrelation?

• inthecontextofdirectedsocialnetworks,theintentionoftheusercreatingthelinkmatters..


P O

X

+

-

“I think O has a lower status than I do”

“I think O has a higher status than I do”

[3]SignedNetworksinSocialMedia,JureLeskovec,SIGCHI2010

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Some possible applications

▪ ModellinginteractionsinChemical/Biologicalnetworks▪ Socialnetworkanalysis▪ Politicalandeconomicalrelations


GraphAlgorithms,ApplicationsandImplementations,CharlesPhillips

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References

▪ Morematerial• Signedgraphs,MatthiasBeck•GraphAlgorithms,ApplicationsandImplementations,CharlesPhillips•Harary:Onthenotionofbalanceofasignedgraph•Networks,Crowds,andMarkets:ReasoningaboutaHighlyConnectedWorld,Chapter5:PositiveandNegativeRelationships

▪ Researchproblemsonsignedgraphs• Signedgraphsinsocialmedia•CommunityMininginSignedSocialNetworks–AnAutomatedApproach•PolarityRelatedInfluenceMaximizationinSignedSocialNetworks•NodeClassificationinSignedSocialNetworks•PredictingPositiveandNegativeLinksinOnlineSocialNetworks


In the next episodes …

3rdpresentationdate

CourseEvaluation

Examsandmaybemore…!

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Questions?

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References▪ Akoglu,L.,McGlohon,M.andFaloutsos,C..Oddball:Spottinganomaliesinweightedgraphs.PAKDD,2010.

▪ Tong,H.andLin,C.Y.Non-NegativeResidualMatrixFactorizationwithApplicationtoGraphAnomalyDetection. In SDM,2011.

▪ Xing,E.P.,Ng,A.Y.,Jordan,M.I.andRussell,S.Distancemetriclearningwithapplicationtoclusteringwithside-information.In NIPS,2002.

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Documents

Graph Mining: Overview of different graph models · Graph Mining: Overview of different graph models Graph Mining course Winter Semester 2016 Davide Mottin, Konstantina Lazaridou