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PAGERANK PARAMETERS

David F. GleichAmy N. Langville

American Institute of MathematicsWorkshop on Ranking

Palo Alto, CAAugust 17th, 2010

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The most important page on the web

Gleich & Langville Recap AIM 2 / 21

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The most important page on the web

Gleich & Langville Recap AIM 2 / 21

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PageRank details

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→

1/6 1/2 0 0 0 01/6 0 0 1/3 0 01/6 1/2 0 1/3 0 01/6 0 1/2 0 0 01/6 0 1/2 1/3 0 11/6 0 0 0 1 0

︸ ︷︷ ︸

P

Pj≥0eTP=eT

“jump” → v = [ 1n ... 1n ]T ≥0

eTv=1

Markov chain�

αP+ (1− α)veT�

x = xunique x ⇒ j ≥ 0, eTx = 1.

Linear system (− αP)x = (1− α)vIgnored dangling nodes patched back to v

algorithms laterGleich & Langville Recap AIM 3 / 21

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Other uses for PageRankWhat else people use PageRank to do

GeneRankMorrison et al. GeneRank, 2005

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NM_003748NM_003862Contig32125_RCU82987AB037863NM_020974Contig55377_RCNM_003882NM_000849Contig48328_RCContig46223_RCNM_006117NM_003239NM_018401AF257175AF201951NM_001282Contig63102_RCNM_000286Contig34634_RCNM_000320AB033007AL355708NM_000017NM_006763AF148505Contig57595NM_001280AJ224741U45975Contig49670_RCContig753_RCContig25055_RCContig53646_RCContig42421_RCContig51749_RCAL137514NM_004911NM_000224NM_013262Contig41887_RCNM_004163AB020689NM_015416Contig43747_RCNM_012429AB033043AL133619NM_016569NM_004480NM_004798Contig37063_RCNM_000507AB037745Contig50802_RCNM_001007Contig53742_RCNM_018104Contig51963Contig53268_RCNM_012261NM_020244Contig55813_RCContig27312_RCContig44064_RCNM_002570NM_002900AL050090NM_015417Contig47405_RCNM_016337Contig55829_RCContig37598Contig45347_RCNM_020675NM_003234AL080110AL137295Contig17359_RCNM_013296NM_019013AF052159Contig55313_RCNM_002358NM_004358Contig50106_RCNM_005342NM_014754U58033Contig64688NM_001827Contig3902_RCContig41413_RCNM_015434NM_014078NM_018120NM_001124L27560Contig45816_RCAL050021NM_006115NM_001333NM_005496Contig51519_RCContig1778_RCNM_014363NM_001905NM_018454NM_002811NM_004603AB032973NM_006096D25328Contig46802_RCX94232NM_018004Contig8581_RCContig55188_RCContig50410Contig53226_RCNM_012214NM_006201NM_006372Contig13480_RCAL137502Contig40128_RCNM_003676NM_013437Contig2504_RCAL133603NM_012177R70506_RCNM_003662NM_018136NM_000158NM_018410Contig21812_RCNM_004052Contig4595Contig60864_RCNM_003878U96131NM_005563NM_018455Contig44799_RCNM_003258NM_004456NM_003158NM_014750Contig25343_RCNM_005196Contig57864_RCNM_014109NM_002808Contig58368_RCContig46653_RCNM_004504M21551NM_014875NM_001168NM_003376NM_018098AF161553NM_020166NM_017779NM_018265AF155117NM_004701NM_006281Contig44289_RCNM_004336Contig33814_RCNM_003600NM_006265NM_000291NM_000096NM_001673NM_001216NM_014968NM_018354NM_007036NM_004702Contig2399_RCNM_001809Contig20217_RCNM_003981NM_007203NM_006681AF055033NM_014889NM_020386NM_000599Contig56457_RCNM_005915Contig24252_RCContig55725_RCNM_002916NM_014321NM_006931AL080079Contig51464_RCNM_000788NM_016448X05610NM_014791Contig40831_RCAK000745NM_015984NM_016577Contig32185_RCAF052162AF073519NM_003607NM_006101NM_003875Contig25991Contig35251_RCNM_004994NM_000436NM_002073NM_002019NM_000127NM_020188AL137718Contig28552_RCContig38288_RCAA555029_RCNM_016359Contig46218_RCContig63649_RCAL080059

Use (− αGD−1)x =w tofind “nearby” importantgenes.

ProteinRankObjectRankEventRank

IsoRankClustering

Sports rankingFood websCentrality

Reverse PageRankFutureRank

SocialPageRankBookRank

ArticleRankItemRankSimRank

DiffusionRankTrustRankTweetRank

Note New paper LabRank with a random scientist?

Gleich & Langville Recap AIM 4 / 21

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Ulam NetworksChirikov mapyt+1 = ηyt+k sin(t+θt)t+1 = t + yt+1

Ulam network1. divide phase space into uniform cells2. form P based on trajectories.

log(E [x(A)]) log(Std [x(A)]))/ log(E [x(A)])

A ∼ Bet(2,16)Note White is larger, black is smaller

Google matrix, dynamical attractors, and Ulam networks, Shepelyansky and Zhirov, arXivGleich & Langville Recap AIM 5 / 21

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Choosing alphaSlide 6 of 21

Choosing alpha

Choosing personalization

Related methods

Open issues

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What is alpha? There’s no single answer.Ask yourself, why am I computing PageRank? Then use the bestvalue for your application.

web-search → tune α for the best featurevector

node centrality → understand what randomjumps mean in your graph

find important nodesin a web-graph → use the random surfer inter-

pretation

Author αBrin and Page (1998) 0.85Najork et al. (2007) 0.85Litvak et al. (2006) 0.5Pan el al. (2004) 0.15Algorithms (...) ≥ 0.85Experiment ???

Gleich & Langville Choosing alpha AIM 7 / 21

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The PageRank limit valueSingular? (− αP)x = (1− α)v

P = X�

00 J1

�

X−1

�

− αX�

00 J1

�

X−1�

x = (1− α)v

X

�

− α�

00 J1

��

X−1x = (1− α)v�

− α�

00 J1

��

y = (1− α)z

(1− α)y1 = (1− α)z1(− αJ2)y2 = (1− α)z2

Boldi et al. 2003: PageRank as a function of the damping parameter

Gleich & Langville Choosing alpha AIM 8 / 21

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TotalRank

t =∫ 1

0x(α)dα

Proposed by Boldi et al. (2005) as a parameter free PageRank.

Gleich & Langville Choosing alpha AIM 9 / 21

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Generalized PageRank

PageRank (− αP)x = (1− α)vx =

∑∞=0(1− α)(α

)Pv

Generalized PageRank y =∑∞

=0 ƒ ()Pv

∑

ƒ () <∞

TotalRank ƒ () = 1+1 −

1+2

LinearRank ...HyperRank ...

Baeza-Yates et al. 2006

Gleich & Langville Choosing alpha AIM 10 / 21

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Pick a distributionMultiple surfers should have an impact!

Each person picks α from distribution A

↓x(E [A])

...

↓E [x(A)]

↘ ↙x(E [A]) 6= E [x(A)]

TotalRank : E [x(A)] : A ∼ U[0,1]Constantine & Gleich, Internet Mathematics, in press.

Gleich & Langville Choosing alpha AIM 11 / 21

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From users

Raw α

density

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Sample mean μ̄ = 0.631.Gleich et al., WWW2010Note 257,664 users from Microsoft toolbar data

Gleich & Langville Choosing alpha AIM 12 / 21

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ChoosingpersonalizationSlide 13 of 21

Choosing alpha

Choosing personalization

Related methods

Open issues

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Personalization choices

Application specific

É GeneRank : v = normalized microarray weightsÉ TopicRank: v = pages on the same topicÉ TrustRank: v = only pages known to be goodÉ BadRank: v = only pages known to be bad (an reverse the

graph)

Super-personalized

É Set v to have only a single non-zero : v = e.

Gleich & Langville Choosing personalization AIM 14 / 21

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Personalized PageRank

B = (1− α)(− αP)−1

Bj = “personalized score of page when jumping to page ”

Gleich & Langville Choosing personalization AIM 15 / 21

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Related methodsSlide 16 of 21

Choosing alpha

Choosing personalization

Related methods

Open issues

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PageRank history

See Vigna 2010: Spectral Ranking andFranceschet 2010: PageRank: Standing on the shoulder of giants.

Let A be the adjacency matrix of a graph.PageRank (− αP)x = (1− α)v (αP+ (1− α)veT )x = x

Seeley 1949 Px = x

Wei 1952 ATx = x

Katz 1953 (− αA)x = e

Hubbell 1965 ATx = x+ v

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Graph centrality

For a graph G, a score assigned to each vertex ∈ V is acentrality score if larger scores are “more central” vertices andthe score is independent of the labeling on the vertices.

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Open issuesSlide 19 of 21

Choosing alpha

Choosing personalization

Related methods

Open issues

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From

Vigna, A history of spectral ranking, MMDS2010

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Other issues

Gleich & Langville Open issues AIM 21 / 21

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QUESTIONS?