PAGERANK PARAMETERS. Amy N. Langville. American Institute of Mathematics Workshop on Ranking Palo Alto, CA August 17th, 2010

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1 PAGERANK PARAMETERS 100David F. Gleich 120 Amy N. Langville American Institute of Mathematics Workshop on Ranking Palo Alto, CA August 17th, 2010 Gleich & Langville AIM 1 / 21

2 The most important page on the web Gleich & Langville Recap AIM 2 / 21

3 The most important page on the web Gleich & Langville Recap AIM 2 / 21

4 PageRank details /6 1/ / / /6 1/2 0 1/3 0 0 P j 0 1/6 0 1/ e T P=e T 1/6 0 1/2 1/ /6 6 } 0 0 {{ } P Markov chain Linear system Ignored jump v = [ 1 n... 1 n ] T 0 αp + (1 α)ve T x = x unique x j 0, e T x = 1. ( αp)x = (1 α)v dangling nodes patched back to v algorithms later e T v=1 Gleich & Langville Recap AIM 3 / 21

5 NM_ Contig32125_RC NM_ AB U82987 Contig55377_RC NM_ NM_ Contig48328_RC NM_ Contig46223_RC NM_ NM_ NM_0181 AF NM_ AF Contig63102_RC Contig34634_RC NM_ NM_ AB NM_ AL NM_ Contig57595 AF NM_0012 AJ Contig49670_RC U45975 Contig25055_RC Contig753_RC Contig53646_RC Contig42421_RC Contig51749_RC NM_ AL NM_ Contig41887_RC NM_ NM_ NM_ AB Contig43747_RC NM_ AB NM_ AL NM_0044 Contig37063_RC NM_ NM_ Contig502_RC AB Contig53742_RC NM_ Contig51963 NM_ Contig53268_RC NM_ Contig55813_RC NM_ Contig27312_RC Contig464_RC NM_ NM_ NM_ AL Contig475_RC Contig55829_RC NM_ Contig45347_RC Contig37598 NM_ NM_ AL0110 Contig17359_RC AL NM_ NM_ Contig55313_RC AF NM_ Contig50106_RC NM_ NM_ NM_ Contig64688 U533 Contig3902_RC NM_ Contig41413_RC NM_ NM_0178 NM_ NM_ Contig45816_RC L275 NM_ AL NM_ Contig51519_RC NM_ Contig1778_RC NM_ NM_ NM_ NM_ NM_0043 NM_0096 AB Contig462_RC D25328 NM_0104 X94232 Contig55188_RC Contig8581_RC Contig53226_RC Contig50410 NM_ NM_ Contig134_RC NM_ Contig128_RC AL NM_ Contig2504_RC NM_ NM_ AL1333 R70506_RC NM_ NM_ NM_ Contig21812_RC NM_ NM_0052 Contig864_RC Contig4595 NM_ NM_ U96131 Contig44799_RC NM_ NM_ NM_ NM_ Contig25343_RC NM_ Contig57864_RC NM_ NM_ Contig58368_RC NM_0028 Contig46653_RC NM_ NM_ M21551 NM_ NM_ NM_0198 NM_ AF NM_ NM_ NM_ AF Contig44289_RC NM_ Contig33814_RC NM_ NM_0030 NM_ NM_ NM_ NM_ NM_ NM_ NM_ NM_ Contig2399_RC NM_ Contig20217_RC NM_0019 NM_ NM_ NM_ NM_ AF NM_ Contig56457_RC NM_ Contig24252_RC NM_ Contig55725_RC NM_ NM_ NM_ Contig51464_RC AL0079 NM_ NM_ NM_ X05610 Contig831_RC NM_ AK Contig32185_RC NM_ AF NM_0037 AF NM_ Contig25991 NM_ Contig35251_RC NM_ NM_ NM_ NM_ NM_ NM_ Contig28552_RC AL Contig38288_RC AA555029_RC Contig46218_RC NM_ Contig63649_RC AL Other uses for PageRank What else people use PageRank to do GeneRank EventRank Morrison et al. GeneRank, 2005 Note Use ( αgd 1 )x = w to find nearby important genes. New paper LabRank with a random scientist? ProteinRank ObjectRank IsoRank Clustering Sports ranking Food webs Centrality Reverse PageRank FutureRank SocialPageRank BookRank ArticleRank ItemRank SimRank DiffusionRank TrustRank TweetRank Gleich & Langville Recap AIM 4 / 21

6 Ulam Networks Chirikov map Ulam network y t+1 = ηy t +k sin( t +θ t ) 1. divide phase space into100 uniform cells 120 t+1 = t + y t+1 2. form P based on trajectories. Note log(e [x(a)]) A Bet (2, 16) log(std [x(a)]))/ log(e [x(a)]) White is larger, black is smaller Google matrix, dynamical attractors, and Ulam networks, Shepelyansky and Zhirov, arxiv Gleich & Langville Recap AIM 5 / 21

7 Choosing alpha Choosing alpha Slide 6 of 21 Choosing personalization Related methods Open issues

8 What is alpha? There s no single answer. Ask yourself, why am I computing PageRank? Then use the best value for your application tune α for the best feature web-search vector node centrality find important nodes in a web-graph understand what random jumps mean in your graph use the random surfer interpretation Author α Brin and Page (1998) Najork et al. (2007) Litvak et al. (2006) 0.5 Pan el al. (2004) 0.15 Algorithms (...) 0.85 Experiment??? Gleich & Langville Choosing alpha AIM 7 / 21

9 The PageRank limit value Singular? ( αp)x = (1 α)v P = X X 0 J αx X 1 x = (1 α)v X α α 0 J J J 1 X 1 x = (1 α)v y = (1 α)z (1 α)y 1 = (1 α)z 1 ( αj 2 )y 2 = (1 α)z 2 Boldi et al. 2003: PageRank as a function of the damping parameter Gleich & Langville Choosing alpha AIM 8 / 21

10 TotalRank t = x(α) dα 0 Proposed by Boldi et al. (2005) as a parameter free PageRank. Gleich & Langville Choosing alpha AIM 9 / 21

11 Generalized PageRank PageRank ( αp)x = (1 α)v x = =0 (1 α)(α )P v Generalized PageRank y = =0 ƒ ( )P v ƒ ( ) < TotalRank ƒ ( ) = LinearRank... HyperRank... Baeza-Yates et al Gleich & Langville Choosing alpha AIM 10 / 21

12 Pick a distribution Multiple surfers should have an impact! Each person picks α from distribution 100 A 120 x(e [A]) x(e [A]) = E [x(a)] TotalRank : E [x(a)] : A U[0, 1] Constantine & Gleich, Internet Mathematics, in press.... E [x(a)] Gleich & Langville Choosing alpha AIM 11 / 21

13 From users density Raw α Sample mean μ = Gleich et al., WWW2010 Note 257,664 users from Microsoft toolbar data Gleich & Langville Choosing alpha AIM 12 / 21

14 Choosing alpha Choosing personalization Slide 13 of 21 Choosing personalization Related methods Open issues

15 Personalization choices Application specific GeneRank : v = normalized microarray weights TopicRank: v = pages on the same topic TrustRank: v = only pages known to be good BadRank: v = only pages known to be bad (an reverse the graph) Super-personalized Set v to have only a single non-zero : v = e. Gleich & Langville Choosing personalization AIM 14 / 21

16 Personalized PageRank B = (1 α)( αp) 1 B j = personalized score of page when jumping to page Gleich & Langville Choosing personalization AIM 15 / 21

17 Choosing alpha Related methods Slide 16 of 21 Choosing personalization Related methods Open issues

18 PageRank history See Vigna 2010: Spectral Ranking and Franceschet 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)ve T )x = x Seeley 1949 Wei 1952 Katz 1953 Hubbell 1965 ( αa)x = e Px = x A T x = x A T x = x + v Gleich & Langville Related methods AIM 17 / 21

19 Graph centrality For a graph G, a score assigned to each vertex V is a centrality score if larger scores are more central vertices and the score is independent of the labeling on the vertices. Gleich & Langville Related methods AIM 18 / 21

20 Choosing alpha Open issues Slide 19 of 21 Choosing personalization Related methods Open issues

21 Vigna, A history of spectral ranking, MMDS2010 Fro

22 Other issues Gleich & Langville Open issues AIM 21 / 21

23 QUESTIONS?