Page Ranking
       presented
           by


 Arvind,Chintan,Raveendra
Motivation

●   World Wid e Web was released in 1991 an d with in
    few years, it ju stified its n am e.

●   In Jan u a...
Goals of p age ran kin g
• A p age m u st h ave a h igh PageRan k if th ere are m an y
   p ages th at p oin t to it.
• If...
Exam p le
• A Prob lem sim ilar to p age ran kin g arises in
   ratin g sp ort team s
• Con sid er th e ran kin g of crick...
Exam p le(...con td.)
         • Consider the graph, where an edge 
           is drawn from loser to winner. 
         • ...
Exam p le(...con td.)
          • Weigh ts get refin ed in
            su ccessive iteration s as
            sh own in di...
Eigen Vectors
• Sp eakin g in term s of m atrices, we are u sin g a m atrix
  n orm alized alon g th e colu m n s
        ...
Eigen Vectors (...con td.)

• Th is is n oth in g b u t th e eigen -valu e p roblem with
  eigen -valu e 1.
• Th at is we ...
Exten din g to web p ages
• We can u se th e sam e con cep t to fin d weigh ts for web
  p ages an d ran k th em .

      ...
Add- on s
• Page Ran k com p u tation can be con sidered as a
  station ary d istribu tion of Markov ch ain s.
• Th e eige...
Referen ce
• Lawren ce Page, Sergey Brin , Rajeev Motwan i, an d
  Terry Win ograd . “Th e PageRan k Citation Ran kin g:
 ...
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Pageranking

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how page ranking in search engines work

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Pageranking

  1. 1. Page Ranking presented by Arvind,Chintan,Raveendra
  2. 2. Motivation ● World Wid e Web was released in 1991 an d with in few years, it ju stified its n am e. ● In Jan u ary 2001, th e n u m ber of h osts stood at 110 m illion an d th e n u m ber of web-sites had reach ed 30 m illion . ● A search en gin e m u st search th rou gh th ese m illion s of sites to give m ost 'relevan t' resu lts to th e u ser. ● Th ere th e con cep t of ran kin g of p ages com es u sefu l.
  3. 3. Goals of p age ran kin g • A p age m u st h ave a h igh PageRan k if th ere are m an y p ages th at p oin t to it. • If th ere are som e p ages th at p oin t to it an d h ave a h igh PageRan k th en also it m u st h ave h igh ran k. • Pages th at are well cited from m an y p laces (like h ttp :/ / www.iisc.ern et.in / ) arou n d th e web are worth lookin g at. • Pages th at h ave p erh ap s on ly on e citation from som eth in g like th e Yah oo! h om ep age are also gen erally worth lookin g at.
  4. 4. Exam p le • A Prob lem sim ilar to p age ran kin g arises in ratin g sp ort team s • Con sid er th e ran kin g of cricket team s. Th eir p erform an ce in a tou rn am en t is sh own below. W ) /Lost(L) on(W India Australia Pakistan Kenya Total Wins India ( A ) - W W L 2 Australia ( B ) L - W W 2 Pakistan ( C ) L L - W 1 Kenya ( D ) W L L - 1 • Can win cou n t alon e su ffice for ratin g team s? • Here we like to rate A h igh er th an B sin ce A won again st B.
  5. 5. Exam p le(...con td.) • Consider the graph, where an edge  is drawn from loser to winner.  • First assign equal weights (w) to  every one and then assign them  new weights(w') as follows: w' a w i k i where i lost against a i k(i) is total losses of team i This is because, we want a team to  go higher up the ranking for  winning against a team which is  already higher up the ranking than  for winning against a team which  is not highly rated as shown in 2nd  figure.
  6. 6. Exam p le(...con td.) • Weigh ts get refin ed in su ccessive iteration s as sh own in diagram s beside. • Con tin u in g like th is we con verge to an equ ilibriu m state as sh own in figu re below:
  7. 7. Eigen Vectors • Sp eakin g in term s of m atrices, we are u sin g a m atrix n orm alized alon g th e colu m n s 0 1 ½ 0 0 0 ½ ½ M = 0 0 0 ½ 1 0 0 0 • Th en we are m u ltip lyin g M b y th e in itial weigh t vector W to get a n ew weigh t vector W' wh ich is again m u ltip led b y M to get W'' an d so on u n til we get a vector Wi' su ch th at W ' = M *W ' = W ' i+1 i i
  8. 8. Eigen Vectors (...con td.) • Th is is n oth in g b u t th e eigen -valu e p roblem with eigen -valu e 1. • Th at is we wan t to solve th e equ ation M *W = W i.e. we wan t to fin d an eigen vector W with eigen - valu e 1. ● We cou ld d irectly h ave u sed th is con cep t to fin d th e requ ired weigh ts for th e team s.
  9. 9. Exten din g to web p ages • We can u se th e sam e con cep t to fin d weigh ts for web p ages an d ran k th em . Here, M = csa_showcase.com 1/ 3 bogus.com Solvin g th e equ ation M*W=W 1/ 3 1/ 3 1/ 3 1/ 2 1/ 2 gives u s th e followin g weigh ts: 1 waste.com 1/ 3 csa_showcase.com 0.4 linux.org 0.4 1 yahooindia.com 0.2 yahooindia.com lin ux.org bogus.com 0 1/ 3 waste.com 0
  10. 10. Add- on s • Page Ran k com p u tation can be con sidered as a station ary d istribu tion of Markov ch ain s. • Th e eigen -valu e com p u tation W=M.W can be con sid ered as fin din g th e fixed p oin t. Th is is sim ilar to th e equ ation X=F(X) ( wh ose con vergen t valu e can b e iteratively fou n d, say by Newton Rap h son m eth od)
  11. 11. Referen ce • Lawren ce Page, Sergey Brin , Rajeev Motwan i, an d Terry Win ograd . “Th e PageRan k Citation Ran kin g: Brin gin g Order to th e Web” Tech n ical Rep ort, Stan ford Un iversity, 1998
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