Personalized PageRank Seminar: Link mining

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PersonalizedPageRankSeminar Link mining
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Contents

• PageRank Overview
• Motivation for PPVs
• PPVs
• Main Theorems
• Algorithms
• Experimental Result
• Other Related Work
• Conclusion
• References

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PageRank Overview

• Method to rank web pages giving to it a numeric
  value that represent their importance
• Based on the link structure of the web
• A page X has a high rank if
• It has many In-links or few but highly ranked
• Has few Out-links
• Dumping factor Probability that the random
  surfer picks a web page and keeps clicking on
  links inside of this web
• Dangling links web pages that doesnt have any
  Out-link

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Motivation for PPVs

• PR reflects a democratic importance
• Random user may have a set of preferred pages
• Create "personalized views, i.e. consider user
  preferences
• Idea
• PPV - Personalized PageRank vector
• Have a vector with length n pages in web
  graph
• ppvp i-th component of vector ppv

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PPVs

• Problem
• PPVs too large to compute and store offline
• Computing PPVs during query time takes too long
• Solution Method from Jeh and Widom
• Computes only limited amount of PageRank vectors
  offline
• Theorems and a couple of algorithms for the
  computation
• How
• Generalize preference set P to a preference
  vector u
• u 1 and u(p) preference for page p 1/p
  for p ? P
• Let A be the link matrix corresponding to the web
  graph
• Calculate PPV with v cu (1-c)Av

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• Example
• P calvinandhobbes.com, tomtu.de, yahoo.com
• u 0,1/3,1/2,1/60
• Link matrix A calculated from crawling the web
• Apply the formula
• v cu (1-c)Av
• v 0,9,8,10 0
• Tomtu is more preferred but global PR of yahoo is
  bigger

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Main Theorems

• Linearity Theorem
• The Hubs Theorem
• The Decomposition Theorem

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• Linearity Theorem
• if u is a linear combination of u1 and u2, then
  (v1, v2) are also a linear combination with the
  same constants (a1, a2)
• (a1 v1 a2 v2) (1-c)A (a1 v1 a2v2) c (a1
  u1 a2u2)
• Represent PPVs as linear combination of basis
  vectors
• Basis vector ri represent importance of entry j
  in i view
• Still to hard to compute

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• The Hubs Theorem
• Choose set P from set H
• Basis vector ? hub vector rp for each p ? H
• Calculate the hub vectors of H ahead of time and
  store them
• However each hub vector requiring multiple scans
  of the web
• Computing and storing all hub vectors is
  impractical

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• The Decomposition Theorem
• Pages important in p are important also in p
  Out-neighbours
• Identifies relationships among basis vectors
• To reduce redundancy hub vectors can be encoded
  as
• partial vectors encodes the part of rp unique
  to p (capture distances from hub to arbitrary
  node without traveling through another hub)
• and hubs skeleton encodes the
  interrelationships among hub vectors (captures
  distances from hub to hub)

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Algorithms

• Computing Basis vectors
• Selective expansion algorithm
• Repeat squaring algorithm
• Dynamic programming algorithm
• Partial Quantities
• Partial vectors
• Hub skeleton
• Web skeleton

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• Computing Basis vectors
• Selective expansion algorithm
• Expand hub pages to out-neighbours if pass
  through H
• p deliver q sooner (High rank page) as a random
  page t
• Repeat squaring algorithm
• Takes H as set and computes reducing error
• Dynamic programming algorithm
• Compute a basis vector for each p using vectors
  from p's neighbours
• Compute all the possible hub vectors into a table
  for fast lookup

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• Partial Quantities
• Computing Partial vectors
• Specialization of the selective expansion
  algorithm
• Uses high PR pages as hub pages to improve
  performance
• Hub skeleton
• Specialization of the repeated squaring algorithm
• Interested only in the computation of a subset of
  the web skeleton
• Web skeleton
• Specialization of the Basic dynamic algorithm
• Yield an approximation of PPVs distances for
  arbitrary nodes
• Restrict the computation eliminating all p ? P

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• Construction of PPVs
• Now, the ppv using the Hubs equation where
• u a1 p1 ... az pz
• h hub pages from H
• 1
• And ai weight of page i
• ppv Si ai(partial vector) i 1/c Sh (hub
  skeleton)h

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Experimental Results

• Stanfords web data (120 millions pages)
• Taking out dangling links 80 millions pages
• Using a 1.4 GHz CPU with 3.5 GB memory, with
  H100,000
• Computation of Full vectors aprox. 8 hrs
• Computation of Partial vectors
• Using the specialization of the selective
  expansion algorithm
• Take about 50 min
• Computation of Hubs Skeleton
• Using the specialization for the repeated
  squaring algorithm
• Take about 10 hrs
• Result Hubs vector containing 14 million entries
  can be constructed in 6 sec.

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Other Related Work

• Personalized PageRank scores were use to enable
• Topic-Sensitive PageRank
• The Weighting of Links Based on Content Analyses
• Intelligent Surfer
• Personalized Web Search engines

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Conclusion

• Personalized PageRanks (for all pages in the web
  graph) can be calculated from an initial set P of
  personally chosen pages and out link information
  of the web graph
• Advantages
• Efficient calculations for computing PPV
• Experimentation on this has shown its utility and
  promise
• Disadvantages
• needs outside info like the calculation of link
  matrix A
• performance depends on choice of personal pages

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References

• Jeh, Widom. Scaling Personalized Web Search, 2003
• Chirita, Olmedilla, Nejdl. PROS A Personalized
  Ranking Platform for Web Search
• Aktas, Nacar, Menczer. An Application of
  Personalized PageRank Vectors Personalized
  Search Engine
• L.Pager S. Brin,The PageRank citation ranking
  Bringing order to the web , Stanford Digital
  Library Technique, Working paper 1999-0120, 1998
• Haveliwala, Topic-sensitive PageRank, 2002
• http//pr.efactory.de/
• http//www-db.stanford.edu/backrub/google.html

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Summary of Terms