The PageRank Citation Ranking: Bringing Order to the Web Page L' , Brin S' , Motwani R' , Winograd T

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The PageRank Citation Ranking Bringing Order to
the WebPage L. , Brin S. , Motwani R. ,
Winograd T. Stanford Digital Library
Technologies Projecthttp//dbpubs.stanford.edu/pu
b/1999-66

• Presented by Zheng Zhao
• Originally designed by Soumya Sanyal
• http//ranger.uta.edu/gdas/Courses/Spring2005/DBI
  R/slides/The20PageRank20Citation20Ranking20-2
  0Redone.ppt

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Outline

• Paper Citations and the Web Motivation
• PageRank Why it should be considered?
• More PageRank Nuts and bolts
• PageRank Unleashed Looking under the hood
• Convergence and Random Walks Why does it work?
• Implementation Getting your hands dirty
• Personalized PageRank The invisible source
• Applications What wasnt apparent already
• Conclusions

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Paper Citations and the Web Motivation

• Academic Citations link to other well known
  papers
• But they are peer reviewed and have quality
  control
• Web of academic documents are homogeneous in
  their quality, usage, citation length

• Most web pages link to web pages as well
• Quality measure of a web page is subjective to
  the user though
• Importance of a page is a quantity that isnt
  intuitively possible to capture

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Contd.

• An user wants to see what is most applicable to
  her needs first.
• The job of the retrieval system is to present the
  more relevant documents up front.
• The notion of quality or relative importance of a
  web page magnifies
• The average quality experienced by an user is
  higher than the average quality of the average
  web page.

• Notations Used
• Backlinks (inedges) Links that point to a
  certain page
• Forward Links (outedges) Links that emanate
  from that page

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PageRank Why it should be considered?

• Think of a color palette
• Colors are formed by the mixture of one or more
  colors
• The amount and intensity of each color you mix
  ultimately governs the color of the final mixture
  not the number of colors !!!
• Now think of a Web Page
• A number of back links (inedges) point to this
  webpage
• Say a certain back link came from Yahoo! and
  another came from an obscure home page.
• Think of the importance of the Yahoo! Page as
  opposed to the importance of the home page.
• Now say the importance of the Yahoo! Page was
  mapped to the amount (intensity) of one color and
  the home page to another color
• Importance of back links rather than their
  number.

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More PageRank Nuts and bolts

• Say for any Web Page u the number of forward
  links is given by Fu and the number of back links
  be Bu and Nu Fu
• R() Rank of page u c Normalization
  Constant
• Note c lt 1 to cover for pages with no outgoing
  links

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Contd..

• So what does the overall picture look like?
• A is designated to be a matrix, u and v
  correspond to the columns of this matrix

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Contd.. (Matrices Revisited)

• Eigenvectors and eigenvalues
• Given that A is a matrix, and R be a vector over
  all the Web pages, the dominant eigenvector is
  the one associated with the maximal eigenvalue.
• It can be found out by recursing the previous
  equation till the recurrence converges.
• A set of eigenvalues form what is called the
  eigenspace.

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Contd.. (A Walk Through Example)

• Lets take an example

AT
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Contd..

• Matrix Notation
• R c A R M R
• c eigenvalue
• R eigenvector of A
• A x ? x
• A - ?I x 0

A
R
Normalized
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Contd.. (Markov Chains)

• Random surfer model
• Description of a random walk through the Web
  graph
• Interpreted as a transition matrix with
  asymptotic probability that a surfer is currently
  browsing that page
• The above notion is fundamental to any Markovian
  System. For a discrete notion of the above, the
  following is assumed.
• Rt M Rt-1 M transition matrix for a
  first-order Markov chain (stochastic)
• The question is does it converge to some sensible
  solution (as t??) regardless of the initial ranks
  ?

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Contd..(Issues..)

• The above equation would converge were it not for
  a little problem
• This problem is called the Rank Sink Problem.
• The sink accumulates rank, but never distributes
  it!

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Contd..()

• In general many Web pages dont have either
  backlinks or forward links.
• Results in dangling edges of the graph
• no parent ? rank 0
• MT converges to a matrix whose last column is all
  zero
• no children ? no solution
• MT converges to zero matrix

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Contd..(More Random Surfer)

• How do we escape from this ?
• A We actually escape from it.
• Say a surfer is randomly clicking and hopping
  from one page to the other.
• If this surfer keeps going back to the same set
  of pages, she will get bored (in reality too) and
  try and escape from this set of pages.
• Hence, we associate an escape factor E to
  account for this boredom.
• How do we model this escape probability
• We term this E to be a vector over all the web
  pages that accounts for each pages escape
  probability.

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Contd..

• Given this Escape vector, how do we associate
  this with the original model
• In matrix notation
  where
• It can be rewritten as
• Hence

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PageRank Unleashed Looking under the hood
The main algorithm

• What can we say about d and ? ?
• d1 is called the eigengap and it controls the
  rate of convergence
• ? is the convergence threshold

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Convergence and Random Walks Why does it work?

• Irreducible Aperiodic Markov Chains with a
  Primitive transition probability matrix
• What is the issue all about?
• We need a transition matrix model that is
  guaranteed convergence and does indeed converge
  to a unique stationary distribution vector.

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Contd..

• Addition of the escape vector E, allows us to
  make the original matrix A be both primitive and
  stochastic
• This guarantees convergence
• What about the addition of new links
• Whether the link analysis algorithms based on
  eigenvectors are stable in the sense that results
  dont change significantly?
• The connectivity of a portion of the graph is
  changed arbitrary
• How will it affect the results of algorithms?
• Ng et al. (2001) IJCAI and Bianchini et al.
  (2002) WWW02
• It is possible to perturb a symmetric matrix by a
  quantity that grows as d1 that produces a
  constant perturbation of the dominant eigenvector

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Contd..

• Convergence Experiment(s)
• Expander graphs and d1 (every subset S has a
  neighborhood bounded by some factor ? times S)
• Rapidly mixing random walk Convergence is
  guaranteed in logarithmic time in the order of
  the size of the graph

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Implementation Getting your hands dirty

• In 1998
• 24 million web pages
• Crawler builds an index of links
• To do this in 5 days, 50 Web pages/second need to
  be crawled
• 11 is the average outdegree, 550 links/second
• 75 million unique URLs to be compared against
• URLs are hashed to unique integer ID
• No dangling links are kept initially
• Vector E will help in convergence issues also
• Weights were kept for 75 million URLs _at_ 4
  bytes/weight (300MB)
• Access to link Database is linear since it is
  sorted
• 99 800 million pages 00 - 2 billion 01 4
  billion

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Personalized PageRank The invisible source

• E10.15
• Web Pages are valued because they exist!
• Web Pages with many related links receive an
  overly high ranking
• The other extreme E for just one web page
• Netscape Home Page and John McCarthys home page

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Applications What wasnt apparent already

• Estimating Web Traffic
• How PageRank corresponds to actual usage
• Internet proxy cache from NLANR compared to
  PageRank
• 2.6 million pages intersect with PageRanks
  indexed 75 mil.
• Web based email access is one plausible reason
  for this disparity
• People look at certain pages but never link them
• Backlink Predictor
• PageRank is a better predictor for future
  citation counts than citation counts themselves.
• Experiment starts out with one URL and no other
  information
• Goal is to crawl the Web in the order of their
  importance
• Importance being an Evaluation function on the
  number of citation counts (number of backlinks)
• PageRank escapes local minima, citation count get
  stuck in these.

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Conclusions

• In essence, the importance of one page being
  dependent on the importance of its predecessors
  is like a peer review.
• NASDAQ 17th February, 2005 - 197.41 Need I
  say More?