Spectralanalysisofrankingalgorithms

SocialandTechnologicalNetworks

Rik Sarkar

UniversityofEdinburgh,2018.

Recap:HITSalgorithm

• Evaluatehubandauthorityscores• ApplyAuthorityupdatetoallnodes:– auth(p)=sumofallhub(q)whereq->pisalink

• ApplyHubupdatetoallnodes:– hub(p)=sumofallauth(r)wherep->risalink

• Repeatforkrounds

Adjacencymatrix

• Example

Hubsandauthorityscores

• Canbewrittenasvectorshanda

• Thedimension(numberofelements)ofthevectorsaren

Updaterules

• Arematrixmultiplications

• Hubrulefori :sumofa-valuesofnodesthatipointsto:

• Authorityrulefori :sumofh-valuesofnodesthatpointtoi:

Iterations

• Afteroneround:

• Overkrounds:

Convergence

• Rememberthathkeepsincreasing• Wewanttoshowthatthenormalizedvalue

• Convergestoavectoroffiniterealnumbersaskgoestoinfinity

• Ifconvergencehappens,thenthereisac:

Eigenvaluesandvectors

• Impliesthatformatrix• cisaneigen value,with• asthecorrespondingeigen vector

Proofofconvergencetoeigen vectors

• UsefulTheorem:Asymmetricmatrixhasorthogonaleigen vectors.– Theyformabasisofn-Dspace– Anyvectorcanbewrittenasalinearcombination

• issymmetric

• FormatrixPwithallpositivevalues,Perron’stheoremsays:– Auniquepositiverealvaluedlargesteigen valuecexists

– Correspondingeigen vectoryisuniqueandhaspositiverealcoordinates

– Ifc=1,thenconvergestoy

Nowtoproveconvergence:• Supposesortedeigen valuesare:

• Correspondingeigen vectorsare:

• Wecanwriteanyvectorxas

• So:

• Afterkiterations:

• Forhubs:

• So:• If,onlythefirsttermremains.

• So,convergesto

Properties

• Thevectorq1z1 isasimplemultipleofz1– Avectoressentiallysimilartothefirsteigen vector– Thereforeindependentofstartingvaluesofh

• q1canbeshowntobenon-zeroalways,sothescoresarenotzero

• Authorityscoreanalysisisanalogous

Pagerank Updateruleasamatrixderivedfromadjacency

• w

• Scaledpagerank:

• Overkiterations:

• Pagerank doesnotneednormalization.

• Wearelookingforaneigen vectorwitheigenvalue=1

Randomwalks

• Arandomwalkerismovingalongrandomdirectededges

• Supposevectorbshowstheprobabilitiesofwalkercurrentlybeingatdifferentnodes

• Thenvectorgivestheprobabilitiesforthenextstep

Randomwalks

• Thus,pagerank valuesofnodesafterkiterationsisequivalentto:– Theprobabilitiesofthewalkerbeingatthenodesafterksteps

• Thefinalvaluesgivenbytheeigen vectorarethesteadystateprobabilities– Notethatthesedependonlyonthenetworkandareindependentofthestartingpoints

Historyofwebsearch

• YAHOO:Adirectory(hierarchiclist)ofwebsites– JerryYang,DavidFilo,Stanford1995

• 1998:Authoritativesourcesinhyperlinkedenvironment(HITS),symposiumondiscretealgorithms– JonKleinberg,Cornell

• 1998:Pagerank citationranking:Bringingordertotheweb– LarryPage,SergeyBrin,Rajeev Motwani,TerryWinograd,Stanfordtechreport