How does Google rank webpages?
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1 Linear Algebra Spring 016 How does Google rank webpages? Dept. of Internet and Multimedia Eng. Konkuk University 1
2 Background on search engines Outline HITS algorithm (Jon Kleinberg) PageRank algorithm (Brin and Page) Open problems [LM06] Google s PageRank and Beyond: The Science of Search Engine Rankings by Langville and Meyer, 006. [Chiang1] Networked Life: 0 Questions and Answers, by M. Chiang, 01.
3 Traditional Search Methods
4 Boolean Search Engines Operators AND, OR, NOT Example car AND maintenance returns all documents containing both car and maintenance Advantages Easy to program Disadvantages Can fail to return documents containing the words with the same meaning (e.g., automobile maintenance) Can return irrelevant documents (e.g., bank has several meanings) 4
5 Vector Space Model Search Engines Key idea Mapping of textual data to numeric vectors or matrices Relevance scoring ordered list of documents Example Terms Documents F11 F1 F1n F1 F Fn Fij: # times term i appears in document j Drawback: poor scalability Fm1 Fm Fmn 5
6 Web Information Retrieval 6
7 Web Information Retrieval World Wide Web (WWW) Huge: ~60 trillion pages (01) - Dynamic - Content changes: 40% weekly, % daily - Size changes: billions of pages added each year Self-organized - No standards - Errors, spammers, link rot Hyperlinked 7
8 Elements of a Web Search Engine 8 Figure from [LM06]
9 Crawler Why crawler? Self-organized web and no central controller Web lives in a cyber space and can grow very quickly Need to collect and categorize web s documents Functions Spiders scour the Web gathering new information and webpages, and returning to store them in a central repository MATLAB example if Internet access available surfer.m at 9
10 Requesting Google to Add Your Site No guarantee about when it will be crawled 10
11 Repository and Indexing Module Page repository Temporarily store new webpages that stay in the repository until being sent to the indexing module Indexing module Extract vital descriptors and create a compressed description of each new page Processed page is tossed out or returned to the repository 11
12 Functions Indexes Hold valuable compressed information for each webpage Three types of indexes Content index - Contents such as keyword, title, and anchor text are stored using an inverted file structure Structure index - Hyperlink structure of pages is stored in compressed form Special-purpose index - Indexes such as image index and pdf index that are useful for particular query tasks Google: over 100 million gigabytes 1
13 Search Engine Size War Search Size War I ( 97-99) Search Size War II ( 99-00) Search Size War I ( 0) 1
14 Query and Ranking Modules Query module Convert user s language query into search system language Consult indexes to find pages relevant to query Pass relevant pages to ranking module Ranking module (Google: based on over 00 factors) Rank relevant pages according to some criterion Two scores: content score + popularity score Content score - Example: (page title has query word) > (page body has query word) Popularity (or importance) score - Determined via analysis of hyperlink structure 14
15 Ranking Webpages by Popularity 15
16 Pre-1998 A Bit of History Yahoo: hierarchies of sites organized by humans Text-based search Search engines are largely based on term-matching - The number of relevant pages are far too large for a human user to digest How can a search engine provide a small set of the most authoritative or definitive pages? 16
17 Post-1998: link analysis A Bit of History PageRank Sergey Brin and Lawrence Page, The anatomy of a large-scale hypertextual Web search engine, Computer Networks and ISDN systems, 1998 HITS Jon Kleinberg, Authoritative sources in a hyperlinked environment, Journal of the ACM, 1999 Modified figure from Langville slides 07 17
18 Key Idea of Link Analysis Use hyperlink structure to focus the relevant page set Enable a mathematical definition of quality of search results Key idea Seminal (landmark) paper Survey paper Authority Hub Good hubs point to good authorities Good authorities are pointed by good hubs 18
19 Hypertext Induced Topic Search (HITS) 19
20 Hypertext Induced Topic Search (HITS) How to measure importance Two popularity scores: hubs + authorities Hub: page with many outlinks Authority: page with many inlinks Hub Authority Good hubs point to good authorities Good authorities are pointed to by good hubs 0
21 Authority and Hub Scores Web graph (V, E) 1 V: set of webpages E: set of directed hyperlinks 4 V={1,,,4} E={(1,), (1,), (,1), (,), (,), (,4), (4,)} ai: authority score of page i hi: hub score of page i a i = X h j, 8i V h i = X a j, 8i V j:(j,i)e j:(i,j)e 1
22 Example Example web graph and equations 1 4 a 1 = h a = h 1 + h + h 4 a = h 1 + h a 4 = h h 1 = a + a h = a 1 + a h = a + a 4 h 4 = a
23 Matrix Representation: Authority Matrix representation a 1 = h a = h 1 + h + h 4 a = a = h 1 + h a 4 = h a 1 6a 7 4a 5 = a h 1 6h 7 4h 5 = LT h h P1 P P P4 P P P P =L Adjacency matrix
24 Matrix Representation: Hub Matrix representation h 1 = a + a h = a 1 + a h = a + a 4 h 4 = a h = h 1 6h 7 4h 5 = h a 1 6a 7 4a 5 = La a P1 P P P4 P P P P =L 4
25 HITS Algorithm Authority and hub scores equations a = L T h h = La L: adjacency matrix HITS iteratively solves the system a (1) = L T h (0) h (1) = La (1) a () = L T h (1)... 5
26 HITS Algorithm Original HITS algorithm Initialize: h (0) = e,k = 1; Do: a (k) = L T (k 1) h h (k) = La (k) Iterative equations Normalize a (k), h (k) k = k +1 a (k) = L T h (k 1) = L T (k 1) La h (k) = La (k) = LL T (k 1) h 6
27 Analysis of HITS Algorithm Power method with normalization a (k) = L T h (k 1) = L T (k 1) La h (k) = La (k) = LL T (k 1) h a (k) a (k) a (k) h(k) h (k) h (k) a (k) = (LT L) k 1 L T h (0) (L T L) k 1 L T h (0) = Ak 1 a (0) A k 1 a (0)!? h (k) = (LLT ) k h (0) (LL T ) k h (0) = Hk h (0) H k h (0)!? Need to understand the behavior of the power of a matrix 7
28 Review of Linear Algebra Will consider only real matrices A matrix A is invertible (or nonsingular) if there exists a matrix B such that AB = I & BA = I B is the inverse of A and denoted as A - 1 A is singular if it is not invertible The nullspace N(A) of A is the set of vectors x such that Ax=0 Fact N(A) ={x Ax = 0} N(A) 6= {0}, A singular, A =0 8
29 Review of Linear Algebra (contd.) Eigenvalue and eigenvector of A R n n Ax = x : eigenvalue x 6= 0 x : eigenvector (, x) : eigenpair Finding eigenvalues Ax = x (A I)x = 0 (A I) singular A I =0 9
30 Review of Linear Algebra (contd.) Facts There are n eigenvalues 1,,..., n There are at most n distinct eigenvalues (A) ={ 1,,..., m}, m apple n (A) : set of distinct eigenvalues (λ,x) eigenpair (λ,cx) eigenpair for any nonzero c A single eigenvalue can have multiple linearly independent eigenvectors as eigenpair - Example: A = =1, 1, x = 405, 4 15,
31 Review of Linear Algebra (contd.) A is symmetric if A T =A A is positive semidefinite (PSD) if Facts about symmetric and PSD A A has only nonnegative real eigenvalues, i.e., A has n orthonormal eigenvectors A can be diagonalized as A = Q Q T Q = x 1 x x n Q T Q = I ( i, x i ) : eigenpair 1 x T Ax where 0, 8x R n i R +,i=1,...,n x 1, x,...,x n x T i x j = = n ( 1, if i = j 7 5 0, if i 6= j
32 Review of Linear Algebra (contd.) Perron s theorem (specialized to nonnegative symmetric PSD matrices) If A( 0) is nonnegative, symmetric and positive semidefinite, then - The largest eigenvalue, say λ1, is positive - There exists an eigenvector x1 0 such that Ax1= λ1x1 Will use a nonnegative dominant eigenvector in the diagonalization of A
33 Back to HITS Two matrices A and H in HITS are symmetric and PSD A k a (0) A k a (0)!? H k h (0) H k h (0)!? Ordered eigenvalues of A 1 n 0 For simplicity, assume λ1>λ (a similar result holds true w/o this assumption) Diagonalization of A implies A = L T L H = LL T A = Q Q T A = Q Q T Q Q T = Q Q T A k = Q k Q T
34 Applying Diagonalization Two matrices A and H in HITS are symmetric and PSD A k a (0) A k a (0) = Q k Q T a (0) 1 Q k Q T a (0) = k Q k Q T a (0) 1 1 k Q k Q T a (0) 1 The numerator is 1 Q k Q T a (0) k 1 = 4x 1 x x n ( / 1 ) k... ( n / 1 ) k T 7 4x 1 x x n 5 5 a (0) 4
35 Applying Diagonalization (contd.) As k tends to infinity 4x 1 x x n T x 1 x x n a (0) = 4x x T 6 4. x T a(0) = x 1 x T 1 a (0) x T n This is an eigenvector of A corresponding to the largest eigenvalue, i.e., a dominant eigenvector (as long as a (0) is not orthogonal to x1) 5
36 Convergence Convergence a (k) = Ak 1 a (0) A k 1 a (0)! x 1x T 1 a (0) x 1 x T 1 a(0) h (k) = Hk h (0) H k h (0)! y 1y T 1 h (0) y 1 y T 1 h(0) Authority and hub scores vectors converge to dominant eigenvectors of authority and hub matrices, respectively k! 0 The iteration converges at the rate at which 1 6
37 Discussion Does it depend on the initial vector? x1x1 T is the projection matrix onto the subspace spanned by x1 The limit point is the projection of the initial vector onto the x1-line subspace, and it is normalized. Thus, the limit point is unique irrespective of the initial vector, and it is in fact the dominant eigenvector x1 If the dominant eigenvalue is not simple, then the limit point can possibly change depending on the initial vector. The limit point will have a form of, for example, (x 1 x T 1 + x x T )a (0) (x 1 x T 1 + x x T )a(0) which is a projection onto the subspace spanned by x1 and x 7
38 Discussion HITS problem is eigenvalue/eigenvector problem a = L T h a = L T La h = La h = LL T h Why not directly solve eigenvalue/eigenvector problem? What is important is raking not absolute scores Convergence is not necessary 8
39 Implementation Query Text-based search, e.g., AltaVista Relevant pages Add adjacent pages - Authority pages - Hub pages Apply HITS Expanded web graph 9
40 Example Expanded web graph (A) = (H) ={.7,, 1, 0.8, 0, 0} x 1 = T y 1 = T L =
41 Score Vectors Initial vector: h (0) =[ ]/6 Authority Score Page 1 Page Page Page 4 Page 5 Page 6 Hub Score ,5,6 Page 1 Page Page Page 4 Page 5 Page Iteration Iteration 41
42 Initial vector: h (0) =rand(6,1) Score Vectors Authority Score Page 1 Page Page Page 4 Page 5 Page 6 Hub Score ,5,6 Page 1 Page Page Page 4 Page 5 Page Iteration Iteration 4
43 Discussion Pros Two opinions for users Small web graph Cons Query-dependence - At query time, a neighborhood graph must be built Vulnerable to spamming - Can easily manipulate scores by creating a hub Two eigenvectors must be computed Dependence on the initial vector - Can be addressed by modifying the two matrices to be irreducible 4
44 Google s PageRank 44
45 Key Idea of PageRank Random walk on web graph 1 4 Web surfers do random walk on a web graph along hyperlinks between web pages Important web pages are likely to be visited more frequently 45
46 Important web pages Key Idea of PageRank Pointed to by many important web pages The score of a page Sum of scores of pages pointing to the page Equally distributed to outlinks
47 PageRank Formula Formula r(p i )= X j:(j,i)e r(p j ) P j r(p i ) : rank of page i P i : number of outlinks from page i Matrix form = H = r(p 1 ) r(p ) r(p n ) 1 H = L diag P 1, 1 P,..., 1 P n H ij =1/ P i if P i! P j 47
48 Example 6-page example 1 H = P1 P P P4 P5 P6 P1 P P P4 P5 P6 0 1/ 1/ / 1/ 0 0 1/ / 1/ / 0 1/ r(p 1 )= r(p ) r(p )= r(p 1). + r(p ) 48
49 Solving π =πh Solving π =πh gives a pagerank vector π is a left eigenvector of H corresponding to eigenvalue 1 Issue Too huge to apply known methods (e.g., elimination, inversion) Brin and Page took an iterative approach (k + 1) = (k)h ) (k) = (0)H k 49
50 Observations H is substochastic X H ij 0, 8i, j H ij apple 1, 8i H = j 0 1/ 1/ / 1/ 0 0 1/ / 1/ / 0 1/ H is sparse H has about 10n nonzero entries Each step in the iteration requires O(n) effort (k + 1) = (k)h 50
51 Problems Will this iterative process converge? If yes, under what conditions? Will it converge to something that makes sense? Will it converge to a unique vector irrespective of π(0)? How fast does it converge? These questions can be answered by studying the asymptotic behavior of H k 51
52 Dangling Cluster Dangling (or absorbing) cluster A set of pages such that there is no inlink from outside to the set Example Node sets {} and {4,5,6} are dangling clusters 1 Issue Random walk will get trapped in a dangling cluster Dangling cluster is a rank sink
53 Impact of Dangling Cluster Limit point of the iterative approach applied to the 6-page example Issues Only dangling cluster survives Others get zero score Easy to spam PageRank PageRank P4 P6 P5 P1 P P P4 P5 P Iteration Number
54 Stochasticity adjustment Dangling node has zero row Replace zero row by e/n, i.e., S = H + a(e T /n) a i = Adjustment I ( 1 if page i is a dangling node 0 otherwise e = T S is stochastic, i.e., each row sum is 1 54
55 Adjustment I (contd.) In the example Node is a dangling node 0 1/ 1/ /6 1/6 1/6 1/6 1/6 1/6 S = 1/ 1/ 0 0 1/ / 1/ / 0 1/ New PageRank update equation (k + 1) = (k)s 4 55
56 Example Dangling cluster still monopolizes PageRank Stochasticity is not enough to fix the problem PageRank P4 P6 P5 P1 P P P4 P5 P Iteration Number We need aperiodicity and irreducibility 4 56
57 Review of Markov Chain Elements of a discrete-time Markov chain Set of states: S = {1,,...,n} Transition probabilities: p ij = P(next state=j current state=i) Sequence of random variables X 0,X 1,..., each taking one of the states in S, that satisfy P(X t+1 = j X t = i, X t 1 = i t 1,...,X 0 = i 0 )=p ij, for all times t, for every pair of states i, j S and for all possible sequences i 0,...,i t 1 of earlier states 57
58 Review of Markov Chain (contd.) Transition probability matrix and graph P = p 11 p 1 p 1n p 1 p p n p n1 p n p nn P = 0 p 1 p p 0 5 p 1 p 0 1 p1 p1 p1 p p In the PageRank update equation, S is the transition probability matrix of the Markov chain modeling random walk on web graph S = 0 1/ 1/ /6 1/6 1/6 1/6 1/6 1/6 1/ 1/ 0 0 1/ / 1/ / 0 1/
59 Review of Markov Chain (contd.) Markov chain is irreducible if Each state can be reached from every other state Markov chain is reducible if It is not irreducible Markov chain is periodic if The set of states can be partitioned into d>1 subsets S1,...,Sd so that all transitions from one subset lead to the next subset Markov chain is aperiodic if It is not periodic (more precisely, it is defined for a single recurrent class) 59
60 Review of Markov Chain (contd.) Examples of Markov chain 1 p1 p1 p1 p 1 4 Reducible 5 Irreducible and periodic 1 Irreducible and aperiodic 60
61 Review of Markov Chain (contd.) Consider an irreducible and aperiodic Markov chain with transition probability matrix P Facts (first two: Perron-Frobenius Theorem) P has an eigenvalue 1, and it is the largest eigenvalue and simple There is a unique vector π, called the stationary distribution, such that = P, > 0, and 1 =1 The power series of P converges as follows: lim P k = k! = 1 n 1 n n 61 lim (0)P k = k!1 for any probability vector (0)
62 Example Probability matrix apple A = Irreducible and aperiodic Eigenvalues and eigenvectors x 1 = =1 x = 1 1 =0.5 6
63 Power series A = A k = lim k!1 apple x1 +0.x x 1 0.x A = apple k 1 Observation Example (contd.) 1 x k 1 x k 1 1 x 1 0. k 1! x apple x1 +0.x x 1 0.x A = Convergence not affected by the initial vector The second largest eigenvalue determines the convergence speed 6 apple 1x x 1x 1 0. x apple x1 x 1 x 1 = x 11 x 1 p 1 p A k = p 1 p apple x 11 x 1 x 11 x 1 = x 11 x 1 = x1
64 Adjustment II The example web graph is reducible 1 Irreducibility and aperiodicity adjustment G = S +(1 )ee T /n 0 apple < 1 Stochastic, irreducible and aperiodic 6 5 New update equation (k + 1) = (k)g 4 Absorbing pages 64
65 G in the 6-page example Example G =0.9 S +0.1 ee T /n 0 1/ 1/ /6 1/6 1/6 1/6 1/6 1/6 =0.9 1/ 1/ 0 0 1/ / 1/ / 0 1/ =
66 Google Matrix G Google matrix G>0 is stochastic, irreducible and aperiodic G has an eigenvalue 1, and it is strictly greater than any other eigenvalue For any probability vector π(0), π(k) converges to an eigenvector of dominant eigenvalue 1 Second largest eigenvalue is no greater than α - Small α leads to fast convergence, but artificial hyperlinks are more reflected - Large α leads to slow convergence, but the effect of artificial hyperlinks is reduced - This trade-off between relevance and convergence speed can be controlled by the choice of α. α =0.85 suggested in the original paper 66
67 Impact of Initial Vector PageRank P5 P4,P6 P1 P P P4 P5 P6 PageRank P5 P4,P6 P1 P P P4 P5 P Iteration Number Iteration Number (0) = e/6 (0) = rand(1, 6) Initial vector does not change the limit point 67
68 Impact of α PageRank P5 P4,P6 P1 P P P4 P5 P6 PageRank P5 P4,P6 P1 P P P4 P5 P Iteration Number α= Iteration Number α=0.95 α controls the tradeoff between relevance and convergence speed 68
69 Implementation Query Index of 60 trillion webpages 100 million gigabytes (constantly growing) Relevant pages Ranked web pages Ranking Ranking Uses over 00 factors - PageRank, geographic region,... About 500 improvements a year - Live traffic experiment, report from users, 69...
70 1996 Original Google Server 10 4-Gigabyte disk drives Now Millions of servers Text 70
71 Interesting Numbers Past 6 million pages ( 98) 00,000 searches a day ( 98) 1,907 employees ( 04) First public acquisition in 01 Kickoff street view effort ( 07) Revenue of $961.9 million ( 0) Google+ introduced in 11 Google Chrome launched in 08 YouTube: 0 million unique visitors a month ( 06) Present ( 1) 60 trillion pages 100 billion searches a month 44,777 employees More than 100 acquisitions so far Drove 6m miles (k cities, 5 countries) $50 billion ( 1) 500 million members 750 million users passing IE 1 billion unique visitors a month stream 6 billion hours of video 71 Google s First 15 Years, by the Numbers,
72 Issues Sensitivity of PageRank Storage issues Convergence criterion Accelerating the computation of PageRank Updating problems... 7
73 Conclusion We have discussed just a part of Google s PageRank There are a number of issues to be addressed for implementation of PageRank Refer to Google s PageRank and Beyond: The Science of Search Engine Ranking PageRank is a grand application of linear algebra, and the key to the success of Google 7
74 Thank you! Q&A The full version available at 74
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