Mathematical Properties & Analysis of Google s PageRank
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1 Mathematical Properties & Analysis of Google s PageRank Ilse Ipsen North Carolina State University, USA Joint work with Rebecca M. Wills Cedya p.1
2 PageRank An objective measure of the citation importance of a web page [Brin & Page 1998] Assigns a rank to every web page Influences the order in which Google displays search results Based on link structure of the web graph Topic independent Cedya p.2
3 Overview Simple Web Model Google Matrix Stability of PageRank Eigenvector Problem: Power Method Linear System: Jacobi Method Cedya p.3
4 Simple Web Model Construct matrix S Page i has d 1 outgoing links: If page i has link to page j then s ij = 1/d else s ij = 0 Page i has 0 outgoing links: s ij = 1/n (dangling node) s ij : probability that surfer moves from page i to page j Cedya p.4
5 Matrix S S is stochastic: 0 s ij 1 S1 = 1 Left eigenvector: ω T S = ω T Ranking: ω i is probability that surfer visits page i But: S does not model surfing behaviour properly Rank sinks, and pages with zero rank Several eigenvalues with magnitude 1 ω not unique Remedy: Change the matrix Cedya p.5
6 Google Matrix Convex combination G = αs + (1 α)1v T Stochastic matrix S Damping factor 0 < α < 1, e.g. α =.85 Personalization vector v 0 v 1 = 1 TrustRank (to combat web spam) v i = 0 if page i is spam page [Gyöngyi, Garcia-Molina, Pedersen 2004] Cedya p.6
7 Properties of G G = αs + (1 α)1v T Stochastic, reducible Eigenvalues: 1 > αλ 2 (S) αλ 3 (S)... [Elden 2003] Unique left eigenvector: π T G = π T π 0 π 1 = 1 ith entry of π: PageRank of page i PageRank. = largest left eigenvector of G Cedya p.7
8 Expression for π Jordan decomposition: X = (1 x 2... x n ) G = αs + (1 α)1v T S = XJX 1 [X 1 ] T = (y 1 y 2... y n ) PageRank: π T = y T 1 + n j=2 β j y T j β j = (1 α) vt x j + α J j,j 1 β j 1 1 α λ j [Serra-Capizzano 2004] Cedya p.8
9 Expression for π as α 1 G = αs + (1 α)1v T Eigenvalue 1 of S has multiplicity m As α 1 : π T y T 1 + (vt x 2 ) y (v T x m ) y m Limit depends on personalization vector v [Serra-Capizzano 2004] Cedya p.9
10 Properties of PageRank π G = αs + (1 α)1v T π T G = π T π 0 π 1 = 1 π is unique ith entry of π: PageRank of page i As α 1 : limit of π depends on v Cedya p.10
11 Stability of PageRank How sensitive is PageRank π to Round off errors Changes in damping factor α Changes in personalization vector v Addition/deletion of links Cedya p.11
12 Perturbation Theory For Markov chains Schweizer 1968, Meyer 1980 Haviv & van Heyden 1984 Funderlic & Meyer 1986 Golub & Meyer 1986 Seneta 1988, 1991 Ipsen & Meyer 1994 Kirkland, Neumann & Shader 1998 Cho & Meyer 2000, 2001 Kirkland 2003, 2004 Cedya p.12
13 Perturbation Theory For Google matrix Chien, Dwork, Kumar & Sivakumar 2001 Ng, Zheng & Jordan 2001 Bianchini, Gori & Scarselli 2003 Boldi, Santini & Vigna 2004 Langville & Meyer 2004 Golub & Greif 2004 Kirkland 2005 Chien, Dwork, Kumar, Simon & Sivakumar 2005 Cedya p.13
14 Changes in the Matrix S Exact: π T G = π T G = αs + (1 α)1v T Perturbed: π T G = π T G = α(s + E) + (1 α)1v T Error: π T π T = α π T E(I αs) 1 π π 1 α 1 α E Cedya p.14
15 Changes in Damping Factor α Exact: π T G = π T G = αs + (1 α)1v T Perturbed: π T G = π T G = (α+µ)s+(1 (α + µ)) 1v T Error: π π α µ [Langville & Meyer 2004] Cedya p.15
16 Changes in Vector v Exact: π T G = π T G = αs + (1 α)1v T Perturbed: π T G = π T G = αs + (1 α)1(v + f) T Error: π π 1 f 1 Cedya p.16
17 Sensitivity of PageRank π π T G = π T G = αs + (1 α)1v T Changes in S: condition number α/(1 α) α: condition number 2/(1 α) v: condition number 1 α =.85: condition numbers 14 α =.99: condition numbers 200 PageRank insensitive to perturbations Cedya p.17
18 Adding an In-Link i π i > π i Adding an in-link increases PageRank (monotonicity) Removing an in-link decreases PageRank [Chien, Dwork, Kumar & Sivakumar 2001] [Chien, Dwork, Kumar, Simon & Sivakumar 2005] Cedya p.18
19 Adding an Out-Link π 3 = 1 + α + α2 3(1 + α + α 2 /2) < π 3 = 1 + α + α2 3(1 + α) Adding an out-link may decrease PageRank Cedya p.19
20 Justification for TrustRank Adjust personalization vector to change PageRank Increase v for page i: v i := v i + φ Decrease v for page j: v j := v j φ PageRank of page i increases: π i > π i PageRank of page j decreases: π j < π j Total change in PageRank π π 1 2φ Cedya p.20
21 Power Method Eigenvector problem: π T G = π T Power method: Pick x (0) > 0 x (0) 1 = 1 Repeat [x (k+1) ] T = [x (k) ] T G until x (k+1) x (k) τ [x (k+1) ] T [x (k) ] T = [x (k) ] T G [x (k) ] T residual Cedya p.21
22 Why is an Iteration Cheap? Google matrix G = αs + (1 α)1v T S = H + dw T }{{} dangling nodes w 0 w 1 = 1 often: w = 1 n 1 Matrix H: models webgraph substochastic dimension: several billion very sparse Cedya p.22
23 Matrix Vector Multiplication Vector x > 0 x 1 = 1 x T G = x T [ α(h + dw T ) + (1 α)1v T] = α x T H + α x T d }{{} scalar Cost: # non-zeros in H w T + (1 α)v T Cedya p.23
24 Error in Power Method π T G = π T G = αs + (1 α)1v T Forward error in iteration k e k x (k) π Norms: 1, e k α k e 0 2 α k [Bianchini, Gori & Scarselli 2003] Cedya p.24
25 Error in Power Method π T G = π T G = αs + (1 α)1v T Backward error (residual) in iteration k r T k = [x(k) ] T G [x (k) ] T r k α k r 0 2 α k Norms: 1, Cedya p.25
26 Termination Stop when r k 10 8 (α =.85) n k Residual norm r k 2α k 2α k 10 8 when k 119 Fewer iterations than predicted by bound Cedya p.26
27 Iteration Counts for Different α G = αs + (1 α)1v T α n = n = bound bound: k such that 2 α k 10 8 Fewer iterations than predicted by bound Cedya p.27
28 Properties of Power Method Converges to unique vector Convergence rate α Vectorizes Storage for only a single vector Matrix vector multiplication with very sparse matrix Accurate (no subtractions) Simple (few decisions) But: can be slow Cedya p.28
29 PageRank Computation Power method Page, Brin, Motwani & Winograd 1999 Acceleration of power method Kamvar, Haveliwala, Manning & Golub 2003 Haveliwala, Kamvar, Klein, Manning & Golub 2003 Brezinski & Redivo-Zaglia 2004 Brezinski, Redivo-Zaglia & Serra-Capizzano 2005 Aggregation/Disaggregation Langville & Meyer 2002, 2003, 2004 Ipsen & Kirkland 2004 Cedya p.29
30 PageRank Computation Methods that adapt to web graph Broder, Lempel, Maghoul & Pedersen 2004 Kamvar, Haveliwala & Golub 2004 Haveliwala, Kamvar, Manning & Golub 2003 Lee, Golub & Zenios 2003 Lu, Zhang, Xi, Chen, Liu, Lyu & Ma 2004 Krylov methods Golub & Greif 2004 Cedya p.30
31 PageRank from Linear System Eigenvector problem: π T (αs + (1 α)1v T ) }{{} G = π T π 0 π 1 = 1 Linear system: π T (I αs) = (1 α)v T I αs nonsingular M-matrix [Arasu, Novak, Tomkins & Tomlin 2002] [Bianchini, Gori & Scarselli 2003] Cedya p.31
32 Stationary Iterative Methods π T (I αs) = (1 α)v T Can be faster than power method Can be faster than Krylov space methods Predictable, monotonic convergence Can converge even for α 1 Less storage than Krylov space methods Accurate (no subtractions) [Gleich, Zhulov & Berkhin 2005] Cedya p.32
33 Example [Gleich, Zhukov & Berkhin 2005] Web graph: 1.4 billion nodes 6.6 billion edges Beowulf cluster with 140 processors Stopping criterion: residual norm 10 7 BiCGSTAB: 28.2 minutes (preconditioner?) Jacobi method: 35.5 minutes Cedya p.33
34 Jacobi Method Assume no page has a link to itself π T (I αs) = (1 α)v T I αs = D O [x (k+1) ] T = [x (k) ] T OD 1 + (1 α)v T D 1 I αs is M-matrix Jacobi converges No dangling nodes: D = I O = αs Jacobi method = power method Cedya p.34
35 Iteration Counts for Jacobi n Power Jacobi r k 10 8, α =.85 Jacobi: fewer iterations than power method Cedya p.35
36 Summary Google Matrix G = α S + (1 α) 1v T PageRank = left eigenvector of G PageRank insensitive to perturbations in G Adding in-links increases PageRank Adding out-links may decrease PageRank Justification for TrustRank Cedya p.36
37 Summary, ctd Power method Computes PageRank as eigenvector Forward, backward errors 2 α k Jacobi Method Computes PageRank from linear system No dangling nodes: Jacobi = power method Cedya p.37
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