pagerank parameters - university of chicagolekheng/meetings/matho...40 60 80 100 120 40 60 80 mm...
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PAGERANK PARAMETERS
David F. GleichAmy N. Langville
American Institute of MathematicsWorkshop on Ranking
Palo Alto, CAAugust 17th, 2010
Gleich & Langville AIM 1 / 21
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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
1
2
3
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6
→
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
10 20 30 40 50 60 70
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
![Page 15: PAGERANK PARAMETERS - University of Chicagolekheng/meetings/matho...40 60 80 100 120 40 60 80 mm PAGERANK PARAMETERS David F. Gleich Amy N. Langville American Institute of Mathematics](https://reader034.vdocuments.mx/reader034/viewer/2022051604/5ff5b097d9616f437e2bed28/html5/thumbnails/15.jpg)
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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
Gleich & Langville Related methods AIM 17 / 21
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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.
Gleich & Langville Related methods AIM 18 / 21
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Open issuesSlide 19 of 21
Choosing alpha
Choosing personalization
Related methods
Open issues
![Page 21: PAGERANK PARAMETERS - University of Chicagolekheng/meetings/matho...40 60 80 100 120 40 60 80 mm PAGERANK PARAMETERS David F. Gleich Amy N. Langville American Institute of Mathematics](https://reader034.vdocuments.mx/reader034/viewer/2022051604/5ff5b097d9616f437e2bed28/html5/thumbnails/21.jpg)
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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?