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A talk I gave at UTRC.

A talk I gave at UTRC.

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    PageRank PageRank Presentation Transcript

    • Adding uncertainty tothe PageRank randomsurferDAVID F. GLEICH, PURDUE UNIVERSITY, COMPUTER SCIENCEUTRC SEMINAR, 13 DECEMBER 2011 1/40 UTRC Seminar David Gleich, Purdue
    • + +Uncertainty Quantification 2/40 UTRC Seminar David Gleich, Purdue
    • are a great way to model and study problems in networkscience and physical science 3/40 UTRC Seminar David Gleich, Purdue
    • are a great way to model and study problems in network science and physical science I hope I’m preaching to the choir. 4/40 UTRC Seminar David Gleich, Purdue
    • A cartoon websearch primer1.  Crawl webpages2.  Analyze webpage text (information retrieval)3.  Analyze webpage links4.  Fit measures to human evaluations5.  Produce rankings6.  Continuously update 5/40 UTRC Seminar David Gleich, Purdue
    • 1 2 to 3 6/40UTRC Seminar David Gleich, Purdue
    • What is PageRank?PageRank by Google PageRank by Google 3 3 The Model 2 5 1.The Model uniformly with follow edges 2 4 5 1. follow edges uniformly with probability , and 4 2. randomly jump, with probability probability and 1 6 2. randomlyassume everywhere is 1 , we’ll jump with probability 1 6 equally, likely assume everywhere is 1 we’ll equally likely The places we find the The places we find the surfer most often are im- portant pages. often are im- surfer most portant pages. 7/40 David F. Gleich (Sandia) PageRank intro Purdue 5 / 36 David F. Gleich (Sandia) PageRank intro UTRC Seminar David Gleich, Purdue Purdue 5 / 36
    • The most important page on the web. 8/40 UTRC Seminar David Gleich, Purdue
    • PageRank via PageRank details PageRank by Google 3 3 2 5 The Model 0 0 0 3 2 1/ 6 1/ 2 0 2 5 6 1/ 6 0 0 1/ 3 0 0 7 1. follow edges uniformlyPwith j 0 ! 6 probability1/ 3, 0 0 7 eT P=eT 1/ 6 1/ 2 0 0 0 4 4 4 1/ 6 0 1/ 2 0 and 5 1/ 6 0 1/ 2 1/ 3 0 1 2. randomly jump 0 1/ 6 0 0 0 1 with probability 1 6 | {z } 1 6 1 , we’ll assume everywhere P equally likely T 0 “jump” ! v = [ 1 ... 1 ] n n eT v=1 î ó Markov chain P + (1 )ve T x=x The places we find the unique x ) j 0, eT x = 1. are im- surfer most often Linear system ( portant pages. P)x = (1 )v Ignored dangling nodes patched back to v 9/40 algorithms later David F. Gleich (Sandia) David F. Gleich (Sandia) PageRank intro PageRank intro Purdue 6 / Purdue 36 UTRC Seminar David Gleich, Purdue
    • ther uses for PageRankensitivity? else people use PageRank to do ProteinRank GeneRank ObjectRank NM_003748 NM_003862 Contig32125_RC U82987 AB037863 NM_020974 Contig55377_RC NM_003882 NM_000849 Contig48328_RC Contig46223_RC NM_006117 NM_003239 NM_018401 AF257175 AF201951 NM_001282 Contig63102_RC NM_000286 Contig34634_RC NM_000320 AB033007 AL355708 NM_000017 NM_006763 AF148505 Contig57595 NM_001280 AJ224741 U45975 Contig49670_RC Contig753_RC Contig25055_RC Contig53646_RC Contig42421_RC Contig51749_RC EventRank AL137514 NM_004911 NM_000224 NM_013262 Contig41887_RC NM_004163 AB020689 NM_015416 Contig43747_RC IsoRank NM_012429 AB033043 AL133619 NM_016569 NM_004480 NM_004798 Contig37063_RC NM_000507 AB037745 Contig50802_RC NM_001007 Contig53742_RC NM_018104 Contig51963 Contig53268_RC NM_012261 NM_020244 Contig55813_RC Contig27312_RC Contig44064_RC NM_002570 NM_002900 AL050090 NM_015417 Contig47405_RC NM_016337 Contig55829_RC Contig37598 Contig45347_RC NM_020675 NM_003234 AL080110 AL137295 Contig17359_RC NM_013296 NM_019013 AF052159 Contig55313_RC NM_002358 NM_004358 Contig50106_RC NM_005342 NM_014754 U58033 Contig64688 NM_001827 Contig3902_RC Contig41413_RC NM_015434 NM_014078 NM_018120 NM_001124 L27560 Contig45816_RC AL050021 NM_006115 NM_001333 NM_005496 Contig51519_RC Contig1778_RC NM_014363 NM_001905 NM_018454 NM_002811 Clustering NM_004603 AB032973 NM_006096 D25328 Contig46802_RC X94232 NM_018004 Contig8581_RC Contig55188_RC Contig50410 Contig53226_RC NM_012214 NM_006201 NM_006372 Contig13480_RC AL137502 Contig40128_RC NM_003676 NM_013437 Contig2504_RC AL133603 NM_012177 R70506_RC NM_003662 NM_018136 NM_000158 NM_018410 Contig21812_RC NM_004052 Contig4595 Contig60864_RC NM_003878 U96131 NM_005563 NM_018455 Contig44799_RC NM_003258 P)x = (1 NM_004456 NM_003158 NM_014750 Contig25343_RC NM_005196 Contig57864_RC NM_014109 NM_002808 Contig58368_RC Contig46653_RC ( )v NM_004504 M21551 NM_014875 NM_001168 NM_003376 NM_018098 AF161553 NM_020166 NM_017779 (graph partitioning) NM_018265 AF155117 NM_004701 NM_006281 Contig44289_RC NM_004336 Contig33814_RC NM_003600 NM_006265 NM_000291 NM_000096 NM_001673 NM_001216 NM_014968 NM_018354 NM_007036 NM_004702 Contig2399_RC NM_001809 Contig20217_RC NM_003981 NM_007203 NM_006681 AF055033 NM_014889 NM_020386 NM_000599 Contig56457_RC NM_005915 Contig24252_RC Contig55725_RC NM_002916 NM_014321 NM_006931 AL080079 Contig51464_RC NM_000788 NM_016448 X05610 NM_014791 Contig40831_RC AK000745 NM_015984 NM_016577 Contig32185_RC AF052162 AF073519 NM_003607 NM_006101 NM_003875 Contig25991 Contig35251_RC NM_004994 NM_000436 NM_002073 NM_002019 NM_000127 NM_020188 Sports ranking AL137718 Contig28552_RC Contig38288_RC AA555029_RC NM_016359 Contig46218_RC Contig63649_RC AL080059 10 20 30 40 50 60 70he (links : 1examined and understood se GD )x = w to Food webs nd “nearby” important Centrality enes. Teaching 10/40 Conjectured new papers: TweetRank (Done, WSDM 2010), WaveRank,he jump : examined, understood, and u Rank, PaperRank, UniversityRank, LabRank. I think theDavid Gleich, Purdue UTRC Seminar last one involves a
    • What else people use PageRank to do GeneRank NM_003748 NM_003862 Contig32125_RC U82987 AB037863 NM_020974 Contig55377_RC NM_003882 NM_000849 Contig48328_RC Contig46223_RC NM_006117 NM_003239 NM_018401 AF257175 AF201951 NM_001282 Contig63102_RC NM_000286 Contig34634_RC NM_000320 AB033007 AL355708 NM_000017 NM_006763 AF148505 Contig57595 NM_001280 AJ224741 U45975 Contig49670_RC Contig753_RC Contig25055_RC Contig53646_RC Contig42421_RC Contig51749_RC AL137514 NM_004911 NM_000224 NM_013262 Contig41887_RC NM_004163 AB020689 NM_015416 Contig43747_RC NM_012429 AB033043 AL133619 NM_016569 NM_004480 NM_004798 Contig37063_RC NM_000507 AB037745 Contig50802_RC NM_001007 Contig53742_RC NM_018104 Contig51963 Contig53268_RC NM_012261 NM_020244 Contig55813_RC Contig27312_RC Contig44064_RC NM_002570 NM_002900 AL050090 NM_015417 Contig47405_RC NM_016337 Contig55829_RC Contig37598 Contig45347_RC NM_020675 NM_003234 AL080110 AL137295 Contig17359_RC NM_013296 NM_019013 AF052159 Contig55313_RC NM_002358 NM_004358 Contig50106_RC NM_005342 NM_014754 U58033 Contig64688 NM_001827 Contig3902_RC Contig41413_RC NM_015434 NM_014078 NM_018120 NM_001124 L27560 Contig45816_RC AL050021 NM_006115 NM_001333 NM_005496 Contig51519_RC Contig1778_RC NM_014363 NM_001905 NM_018454 NM_002811 NM_004603 AB032973 NM_006096 D25328 Contig46802_RC X94232 NM_018004 Contig8581_RC Contig55188_RC Contig50410 Contig53226_RC NM_012214 NM_006201 NM_006372 Contig13480_RC AL137502 Contig40128_RC NM_003676 NM_013437 Contig2504_RC AL133603 NM_012177 R70506_RC NM_003662 NM_018136 NM_000158 NM_018410 Contig21812_RC NM_004052 Contig4595 Contig60864_RC NM_003878 U96131 NM_005563 NM_018455 Contig44799_RC NM_003258 NM_004456 NM_003158 NM_014750 Contig25343_RC NM_005196 Contig57864_RC NM_014109 NM_002808 Contig58368_RC Contig46653_RC NM_004504 M21551 NM_014875 NM_001168 NM_003376 NM_018098 AF161553 NM_020166 NM_017779 (g NM_018265 AF155117 NM_004701 NM_006281 Contig44289_RC NM_004336 Contig33814_RC NM_003600 NM_006265 NM_000291 NM_000096 NM_001673 NM_001216 NM_014968 NM_018354 NM_007036 NM_004702 Contig2399_RC NM_001809 Contig20217_RC NM_003981 NM_007203 NM_006681 AF055033 NM_014889 NM_020386 NM_000599 Contig56457_RC NM_005915 Contig24252_RC Contig55725_RC NM_002916 NM_014321 NM_006931 AL080079 Contig51464_RC NM_000788 NM_016448 X05610 NM_014791 Contig40831_RC AK000745 NM_015984 NM_016577 Contig32185_RC AF052162 AF073519 NM_003607 NM_006101 NM_003875 Contig25991 Contig35251_RC NM_004994 NM_000436 NM_002073 NM_002019 NM_000127 NM_020188 S AL137718 Contig28552_RC Contig38288_RC AA555029_RC NM_016359 Contig46218_RC Contig63649_RC AL080059 10 20 30 40 50 60 70 Use ( GD 1 )x = w to find “nearby” important genes. 11/40Note Conjectured new papers: TweetRank (Done, WS UTRC Seminar David Gleich, Purdue
    • Richardson is a robust, simplealgorithm to compute PageRankGiven α, P, v (I ↵P)x = (1 ↵)v Richardson ) (k+1) (k) x = ↵Px + (1 ↵)v (k) k error = kx xk1  2↵ 12/40 UTRC Seminar David Gleich, Purdue
    • Sensitivity 13/40 UTRC Seminar David Gleich, Purdue
    • Which sensitivity? PageRank circa 2006 ( P)x = (1 )v Sensitivity to the links : examined and understood Sensitivity to the jump : examined, understood, and useful Sensitivity to : less well understood 14/40 For information about how to compute the PageRank derivative, see: Gleich, Glynn, Golub, Greif. Three results on the PageRank vector, 2007. UTRC Seminar David Gleich, Purdue
    • Wikipedia test case PageRank on Wikipedia = 0.50 = 0.85 = 0.99 United States United States C:Contents C:Living people C:Main topic classif. C:Main topic classif. France C:Contents C:Fundamental Germany C:Living people United States England C:Ctgs. by country C:Wikipedia admin. United Kingdom United Kingdom P:List of portals Canada C:Fundamental P:Contents/Portals Japan C:Ctgs. by topic C:Portals Poland C:Wikipedia admin. C:Society Australia France C:Ctgs. by topic Note Top 10 articles on Wikipedia with highest PageRank 15/40 David F. Gleich (Sandia) Sensitivity Purdue 11 / 36 UTRC Seminar David Gleich, Purdue
    • What is alpha?What is alpha? The teleportation parameter! Author Brin and Page (1998) 0.85 Najork et al. (2007) 0.85 Litvak et al. (2006) 0.5 Experiment (slide 19) 0.63 Algorithms (...) 0.85 For you,αis clear. or you, is clearoogle Google wants PageRank for everyone wants PageRank for everyone 16/40 UTRC Seminar David Gleich, Purdue
    • What about me?Multiple surfers should have an impact! Each person picks from distribution A ... # # x(E [A]) E [x(A)] & . 17/40 x(E [A]) 6= E [x(A)] David F. Gleich (Sandia) Random sensitivity Purdue 15 / 36 UTRC Seminar David Gleich, Purdue
    • alpha PageRank PageRa RandomPageRankdom alpha alpha Random alpha PageRank RAPr or PageRank meets UQs the random variables as the random variables Model PageRankageRank as the random variables x(A) x(A) x(A) and look atk E [x(A)] and Std [x(A)] . at E [x(A)] and Std [x(A)] . E [x(A)] and Std [x(A)] . 18/40 Explored in Constantine and Gleich, WAW2007; and " Constantine and Gleich, J. Internet Mathematics 2011. UTRC Seminar David Gleich, Purdue
    • Alpha, measured from users! What is alpha based on users? 3.0 InfBeta( 3.2 , 2.0 , 1.9e−05 , 0.0019 ) mean 0.63 2.5 mode 0.69 2.0 density 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Raw α 19/40 see Gleich et al. WWW2010 for more Constantine, Flaxman, Gleich, Gunawardana, Tracking the Random Surfer, WWW2010. UTRC Seminar David Gleich, Purdue
    • What is A? A simple model for alpha       20/40 Bet ( , b, , r) UTRC Seminar David Gleich, Purdue
    • An Examplerandom variables The PageRank x 1 3 x 2 2 5 x 3 4 x4 1 6 x 5 x 6 21/40 0 0.5 UTRC Seminar David Gleich, Purdue
    • A theoretical concernJust one a problem isn’t really second ... Z 1 Z 1 1 E [x( )] = x( ) ( ) d = (1 )( P) v ( )d 0 0 = 1 ( P) 1 ! P stochastic singular? Yes, but ... 1 lim (1 )( P) v=x is unique !1 22/40 (Think about P = 1, use Jordan Form of P to generalize) UTRC Seminar David Gleich, Purdue
    • Many PageRank properties areWhat changes? unchanged by a random alphaReally, what stays the same! x(A) A ⇠ Bet ( , b, , r) with 0  < r  1 1. E [ (A)] 0 and kE [x(A)]k = 1; thus E [x(A)] is a probability distribution. P î ó 2. E [x(A)] = =0 E A A +1 P v; thus we can interpret E [x(A)] in length- paths. 3. for page with no in-links, (A) = (1 A) ; thus E [ (A)] = (E [A]) and Std [ (A)] = Std [A] 23/40 But is this one useful? UTRC Seminar David Gleich, Purdue
    • Wikipedia test case (take 2) RAPr on WikipediaRAPr on Wikipedia EE [x(A)] [x(A)] Std [x(A)] Std [x(A)] United States United States United States United States C:Living people C:Living people C:Living people C:Living people France France C:Main topic classif. C:Main topic classif. United Kingdom United Kingdom C:Contents C:Contents Germany Germany C:Ctgs. by country C:Ctgs. by country England England United Kingdom United Kingdom Canada Canada France France Japan Japan C:Fundamental C:Fundamental Poland Poland England England 24/40 Australia Australia C:Ctgs. by topic C:Ctgs. by topicNote A A ⇠ Bet(0.5, 1.5, [0, 1]) ⇡ ⇡ empirical distribution on WikipediaGleich, Purdue Note ⇠ Bet (0.5, 1.5, [0, 1]) empirical distribution Seminar David UTRC on Wikipedia
    • Ulam NetworksUlam NetworksUlam Networks PageRank on a dynamical system Networks yt+1 Chirikov map Chirikov map Ulam networ yt+1 = yt +k sin( t + t ) 1. divide phas Ulam Ulam network Ulam t+1 = t + network 2. form P basehirikov mapChirikov map = Chirikov +k sin( t Ulam phase Ulam Networksyt+1 = ytyt illustrates map1.1. divide networkspace into uniform c nicely +k sin(t + +t ) t ) divide phase space into uniform cel Ulam network+1 = = t Ulam Networks based ontrajectories. the uncertainty. NetworksP based onUlam network +Ulam + yt+1 2.2. formmap+1 yt+1 = yt +k sin( t + t ) 1. divide phase space into uniform cells t+1 y yt+1 t +t+1 = t t+1 ChirikovP P form form 2. based on trajectories. trajectories. Chirikov map Chirikov map yt+1 = yt +k sin( t + t ) 1. divide phase space Ulam network Ulam network 1. = t + yt+1 t t ) divide phase space into form P based yt+1t+1 = t +k+k sin(+ t +)t+1 1. divide phase space into uniform cells on tr y = y yt sin( t 2. uniform cells t+1 = = +t yt+1 t+1 t + yt+1 form P P based trajectories. 2. 2. form based onon trajectories. log(E [x(A)]) log( log(E [x(A)]) log(Std [x(A)]))/ log(E Bet (2, 1 A ⇠ [x(A)]) Note Bet (2, 16) A ⇠ White is larger, black is smaller Note White is larger, black is smaller Google matrix, dynamical attractors, and Google matrix, dynamical attractors, and Ulam networks, Shepelyansky and Zhirov, arXiv David F. Gleich (Sandia) Random sensitivity log(E [x(A)]) log(E [x(A)]) log(E [x(A)]) log(E [x(A)]) log(Std [x(A)]))/ log(E [x(A)]) [x(A) log(Std[x(A)]))/ log(E [x(A)]) [x( log(Std [x(A)]))/ log(E 23 [x( log(Std log(E 25/40 David F. Gleich (Sandia) log(E [x(A)]) [x(A)]))/ log(Std/ 36 Random sensitivity Purdue White is larger, black is smaller ⇠ Bet (2, 16) A A ⇠ Bet (2, 16) Note White is larger, black is is Note White is larger, black Bet (2, 16) A ⇠ Bet (2, 16) Model from Shepelyasky and Zhirov, Bet(2, 16) Asmaller " Asmaller ⇠⇠ Phy. Rev. E. 2011. Google matrix, dynamical attractors, andUTRCnetworks,smaller Gleich, PurduearXiv Ulam Seminar David GoogleNote dynamical attractors, andblack is Shepelyansky and and Zhirov, matrix, White is larger, Ulam networks, Shepelyansky Zhirov, arXiv
    • Convergence 0 10 Algorithms & " Convergence −5 10 Monte Carlo −10 10 1. Monte Carlo E [x(A)] −15 1 PN 10 ⇡ N =1 x( ⇠A 0 1 2 3 4 5 ) 0 10 10 10 10 10 10 10 2. Path Damping E [x(A)] 10 −5 PN î ó ⇡ =0 E A A +1 P v Path Damping −10 10 3. Quadrature E [x(A)] 10 −15 (No Std) Rr 10 0 1 10 2 10 10 3 ⇡ x( ) d ( ) 0 10 PN C ⇡ =1 x( ) −5 s 10 (hConvergence toto semi-exact Convergence semi-exactsolutions on a 335-nodestrong solution on a 335-node graph −10 10 Quadrature component.(harvard500 strong component). 26/40 Blue = Beta(2,16) 16) Blue Bet (2, −15 10 Green = Beta(1,1,0.1,0.9) 0.9) Green Bet (1, 1, 0.1, 0 10 20 30 40 50 60 70 80 90 100 Salmon = uniform (0.6, 0.9) Salmon Uniform(0.6,0.9) David F. Gleich (Sandia) Random sensitivity Red = Beta(-0.5, -0.5, 0.2, 0.7) Red Bet ( 0.5, 0.5, 0.2, 0.7) UTRC Seminar David Gleich, Purdue
    • f(α) ￿ ⋅ ￿￿￿￿￿￿￿￿ ￿￿￿￿￿￿￿￿￿￿ g(α)￿￿ ￿.￿ ⋅ ￿￿￿￿￿￿￿￿ ￿￿￿￿￿￿￿￿ ￿￿ f (α) = 1724683103168320512000α 102 − 351689859974563275916800α 101 + 1046657678560756011923040α 100 ￿(α) = 21252680112847680000α 102 +332821515558986503317268308α 99 + 202994690094545539249274953458α 98 + 701216550622104187641429941160α 97 −3542775096896042918400α 101 − 377301357230918051819160α 100 + 62030166204003769204027938α 99 + 301903572553392042618587937α 98 +38942435173273232195508862504752α 96 − 5204876256969489587508598423780757α 95 − 53419116345848724180375395029139614α 94 −27515144995670593102754792187α 97 − 1391342388530090922919905979557α 96 − 11397010225845179645798293856049α 95 +1621997105501543781796265745838677670α + 17992097277595516775992937444966323725α 92 93 +487046819801240647260974920877667α 94 + 8641748415645906110710596472701695α 93 − 14615573868254463557271968794871527α 92 −228388738389199148614341585444680228464α 91 − 2572935401339464873388154472765864295466α 90 −1455304405730842808585234463006780870α 91 − 16140532952116322684344866986683755014α 90 −18662047188535851000868073690251020472621α 89 − 155192964832717622674637679380949267008397α 88 −107685923577790689207116358432796101348α 89 + 3574857500140390342079726927167132783327α 88 +13633798075806927018912795365187923947976816α 87 + 153692481592717017931843564092779914769739855α 86 +76245995916566900197088870723441134067760α 87 − 320477613697118756563592647774688786780579α 86 −2424702525231324896856434133527720085459106818α 85 − 34112664906875644324640001664890877920583430935α 84 −14315018719450474212530996756919665488506623α 85 − 12271042346558183829899943919127664848771235α 84 +222921632950502905446093540571509314548545319158α 83 + 4458381340774458139955262362762709170337141183042α 82 +1538719934896052457300693234469902122130588440α 83 + 7259823837632938466306787148779956756499503259α 82 −9722398912749159172830586061232227612575398195577α 81 − 402863595222192101330043246404750577170418624210463α 80 −91383277962053778179963631846131934198363974003α 81 − 912158632690159715631486922494993985581191177254α 80 −241296146875962767748365749082981265577900593669099α 79 + 26884891161116233003550134767867058390000240645389885α 78 +1124589169570249225316595386438810701468062018941α 79 − 55599491760340084897708205765116975153096053881206α 78 +75002935639704657680175868562515328344632861061620026α 77 − 1355245718493528694128677343628002432897202221776993666α 76 +254197028878341726795811304127085084201803714274594α 77 − 1155102780712932745491921904562487673324953687625090α 76 −6666337432948865424681896342751813538288258918631143898α 75 + 50876562123828411130342908134923596879946044492587906688α 74 −19623309116424352882311523132748440745863270150867432α 75 − 72367264828688457023192884699324797029606326773402260α 74 +385972738637461890892793659070699381929652086327544953064α 73 − 1324370012053495348856190918458325441254102678707139546912α 72 +510591330662979105902331311824358111451756310585317896α 73 + 6560635654785580651459993551515346226540950556472012168α 72 −16416792980158036153780188009203628703318521649963318398744α 71 + 17510197624369310054645143199845105805941154913191274775360α 70 +11841946546859350197679256661965428675545845230913012752α 71 − 222422692257166102165445803087102201095333519552710152624α 70 +533320137070985354296793454864336229974212018883255863520736α 69 + 275502212308122569075672900514808641788656066608417565862128α 68 −1447290325427425453794609658098719385231428839474861685840α 69 + 2125011726240928873652963898522501443619028980101705108896α 68 −13429082722840051523544458153489421210623008268881676515202688α 67 +56163879158282775333105949842095267377034088228166264755488α 67 + 133653341840138472687713523321901358136789047544268798190144α 66 −23110058843365910555627839838104471746030299594537756688223008α 66 −1165851790876533575106055126719543401792990924852555883239232α 65 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−35756856984770583727093678769849105127720172150476292008503798661120000000000000000α −5203808713264169193283107063136995887025759130647063545708229427200000000000000000 +6649311133615327302528414580675050300088470000271247863960515379200000000000000000 Figure 2.5 – A PageRank function. x 1 (α) = (−23￿6030) f (α)￿￿(α), see section ￿.￿ x1(α) = -23/6030 f(α)/g(α) Figure 2.5 (continued). UTRC Seminar David Gleich, Purdue
    • Random alpha PageRankhas a rigorous convergence Convergencetheory. theory Method Conv. Work Required What is N? 1 number of Monte Carlo p N PageRank systems N samples from A Path Damping r N+2 N + 1 matrix vector terms of (without N1+ products Neumann series Std [x(A)]) number of Gaussian r 2N N PageRank systems quadrature Quadrature points and r are parameters from Bet ( , b, , r) 28/40 David F. Gleich (Sandia) Random sensitivity UTRC Seminar David Gleich, Purdue 27 / 36 Purdue
    • Convergence of quadrature in the r=1 regime is matrix dependent. Singularities10 0.03 8 6 0.02 4 0.01 2 0 0 1.00129 2 3−2 4 5 6 7 8 −0.01−4 9 10−6 −0.02−8−10 −0.03 −10 −5 0 5 10 0.97 0.98 0.99 1 1.01 1.02 1.03 29/40 log10(9+|1/λ|)eiarg(1/λ) 1/λNote 500-node harvard500 graph from Cleve Moler, left plot is Gleich, Purdue UTRC Seminar David
    • Establishing this theoreticalconvergence provedindependently useful. Constantine, Gleich, and Iaccarino. Spectral Methods for Parameterized Matrix Equations, SIMAX, 2010. A(s)x(s) = b(s) , A(J 1 )x(J 1 ) = b(J 1 ) ) A(J N )x(J N ) = b(J N ) or ) AN (J 1 )xN (J 1 ) = bN (J 1 ) Constantine, Gleich, and Iaccarino. A factorization of the spectral Galerkin system for parameterized matrix equations: derivation and applications, SISC 2011. 30/40 How to compute the Galerkin solution in a weakly intrusive manner.! UTRC Seminar David Gleich, Purdue
    • A real test-case Webspam application Hosts of uk-2006 are labeled as spam, not-spam, other P R f FP FN Baseline 0.694 0.558 0.618 0.034 0.442 Beta(0.5,1.5) 0.695 0.561 0.621 0.034 0.439 Beta(1,1) 0.698 0.562 0.622 0.033 0.438 Beta(2,16) 0.699 0.562 0.623 0.033 0.438 31/40 Note Bagged (10) J48 decision tree classifier in Weka, mean of 50 repetitions from 10-fold cross-validation of 4948 non-spam and 674 spam hosts (5622 total). Becchetti et al. Link analysis for Web spam detection, 2008. David F. Gleich (Sandia) Random sensitivity UTRC Seminar David Gleich, Purdue Purdue 28 / 36
    • New directions 32/40 UTRC Seminar David Gleich, Purdue
    • Data driven surrogate functionsBeyond spectral methods for UQ 33/40 UTRC Seminar David Gleich, Purdue
    • j r Square s) t t A L B Network alignment 34/40 m ximize w T x + 1 xT Sx UTRC Seminar David Gleich, Purdue
    •       Nuclear-norm & " matrix completion" based ranking Gleich and Lim, KDD2011avid F. Gleich (Purdue) KDD 2011 16/20 Overlapping clusters" for distributed computation Andersen, Gleich, and Mirrokni, WSDM2012 35/40 UTRC Seminar David Gleich, Purdue
    • Local methods for massive FOR KATZ TOP-K ALGORITHMnetwork analysis Approximate                                                 where       is sparse Keep       sparse too Ideally, don’t “touch” all of       This is possible for " personalized PageRank! 36/40 David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 34 of 47 UTRC Seminar David Gleich, Purdue
    • Graph spectraGraph spectra 37/40 UTRC Seminar David Gleich, Purdue
    • What about time?Real networks evolve in time.What to do?Look towards dynamical systems! 38/40 UTRC Seminar David Gleich, Purdue
    • What about time?Real networks evolve in time.What to do?Look towards dynamical systems! Now I must be preaching to the choir! 39/40 UTRC Seminar David Gleich, Purdue
    • Questions?www.cs.purdue.edu/homes/dgleich Google “David Gleich” 40 UTRC Seminar David Gleich, Purdue