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Internet: Search Engine: Google: Algorithm: Search: Pagerank: Link: Can you show a simple example?

Mar 31st, 2009 10:45
engatoo engatoo, Roj Merry, Knud van Eeden,


----------------------------------------------------------------------
--- Knud van Eeden --- 15 July 2006 - 06:11 pm -----------------------
Internet: Search Engine: Google: Algorithm: Search: Link: Pagerank: 
Can you show a simple example?
---
PageRank is just one factor (amongst more than 100) used to calculate
the rank of a page in results of searches.
Other factors of influence are e.g. the text content of your page.
===
This pagerank calculation shows only the 'Simplified PageRank
algorithm', and the demping factor.
===
Given a universe of totally N pages.
This pages can give each other pagerank.
The total pagerank all pages can give away together is (by definition)
maximally 1 pagerank.
Each page is able to give away maximally 1/N pagerank.
In this example N is chosen to be 4, thus each page can give away
maximally 1/4 pagerank
Each page has initially 0 pagerank for its own, all its
own pagerank is built up by the gifts of the other pages
===
Thus initially
  +----------+   +----------+   +----------+  +----------+
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  +----------+   +----------+   +----------+  +----------+
       A               B              C             D
A has received 0 PR
B has received 0 PR
C has received 0 PR
D has received 0 PR
===
Now say that page B, C and D decide to link to page A
      +----------------------<----------------------+
      |+---------------<--------------+             |
      ||+-------<------+              |             |
      |||              |              |             |
  +---+++----+   +-----+----+   +-----+----+  +-----+----+
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  +----------+   +----------+   +----------+  +----------+
       A               B              C             D
Thus
A gives away   0 PR
B gives away 1/4 PR to A,
C gives away 1/4 PR to A,
D gives away 1/4 PR to A.
After this the score is:
Received total
A has received 1/4 + 1/4 + 1/4 PR = 3/4 PR = 0.75 PR
B has received 0 PR
C has received 0 PR
D has received 0 PR
===
Now say that after this page B decides to link C, and D decides to 
link to the 3 other pages
      +----------------------<----------------------+
      |+---------------<--------------+             |
      ||+-------<------+              |             |
      |||              |              |             |
  +---+++----+   +-----+----+   +-----+----+  +-----+----+
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  |          |   |          |   |          |  |          |
  +----------+   +--++------+   +---+-----++  +---+---+--+
       A            || B            | C   |       | D |
                    |+------->------+     +---<---+   |
                    +-----------------<---------------+
Thus
A gives away 0 PR
B gives away (1/2) . (1/4 PR) to A, and (1/2) . (1/4 PR) to C
C gives away its (1/1) . (1/4 PR) to A,
D gives away its (1/3) . (1/4 PR) to A, (1/3) . (1/4 PR) to B, (1/3) . 
(1/4 PR) to C
After this the score is:
===
http://www.webs4soft.com/Links4.htm
http://hotelsinindia.webs4soft.com/hotels-in-delhi.htm
http://indiatravel.webs4soft.com/Resources.htm
http://indianmovies.webs4soft.com/Kuch-Kuch-Hota-Hai.htm
http://real-estate.webs4soft.com/property-tips.htm
http://foodhealthcaretips.webs4soft.com/Resources.htm
http://www.websitecompanyindia.com/seo-Links.htm
http://www.websitecompanyindia.com/seo-Links.htm
http://indianjewelry.websitecompanyindia.com/dimond-ring.htm
http://www.bestindiaeducation.com/Link-Exchange.htm
http://bestjobconsultant.bestindiaeducation.com/PARTNERS.htm
http://eloctronicandmobilestore.bestindiaeducation.com/mobile-links.htm
http://easyloanservice.bestindiaeducation.com/Home-Insurance-Links.htm
http://www.rajhealthcenter.com/Cosmetic-Surgery.htm
http://creativebusinessgroup.rajhealthcenter.com/Business-Links.htm
http://onlinefreeinternetgames.rajhealthcenter.com/Games-Links.htm
http://top-beauty-tips.rajhealthcenter.com/Beauty-Links.htm
http://www.indiatourpoint.com/Travel-Links.htm
http://four-wheeler-buy-tips.indiatourpoint.com/Auto-Links.htm
http://watch-online-free-cricket-match.indiatourpoint.com/Sports-
Links.htm
http://automobile.easyfindindia.com/
http://bollywood.easyfindindia.com/
http://cloth-shopping.onlineindiadiary.com/
http://easyfind.j4ulive.com/
http://www.onlineindiadiary.com/
http://finance-guide.easyfindindia.com/
http://freelance-jobs.onlineindiadiary.com/
http://gadgets.easyfindindia.com/
http://www.easyfindindia.com/
http://gift-ideas.easyfindindia.com/
http://health-fitness.easyfindindia.com/
http://home-improvement.j4ulive.com/
http://homedecor.easyfindindia.com/
http://mortgage.onlineindiadiary.com/
http://online-business.j4ulive.com/
http://online-education.j4ulive.com/
http://onlinein.j4ulive.com/
http://pets-online.easyfindindia.com/
http://realestate.easyfindindia.com/
http://sports-in-india.easyfindindia.com/
http://travel-guide.easyfindindia.com/
http://wedding-shopping.j4ulive.com/
http://www.j4ulive.com/
http://wildlifeholidayindia.blogspot.com/
http://enjoywildlife.blogspot.com/
http://enjoytourismindia.blogspot.com/
http://directory.indiatourpoint.com/
http://www.bestlifeindia.com/Resources1.htm
http://directory.bestlifeindia.com/
http://freeorkutscrapandsms.bestlifeindia.com/SMS-Links.htm
http://onlinegiftshop.bestlifeindia.com/Gift-Links.htm
http://www.freemusicpoint.com/Music-Links.htm
http://love-dating.freemusicpoint.com
http://online-art-presentation.freemusicpoint.com/
http://onlinefurnitureshop.freemusicpoint.com/
Received total
---
A has received from respectively B, C and D =
(1/2) . (1/4 PR) +
(1/1) . (1/4 PR) +
(1/3) . (1/4 PR)
= ( 1/8 + 1/4 + 1/12 ) PR
= ( 3/24 + 6/24 + 2/24 ) PR
= 11/24 PR
= 0.4583333333... PR
or about 46 percent of the total PR
---
B has received from respectively A, C and D =
(0  ) . (1/4 PR) +
(0  ) . (1/4 PR) +
(1/3) . (1/4 PR)
= ( 0/4 + 0/4 + 1/12 ) PR
= ( 0/24 + 0/24 + 2/24 ) PR
= 2/24 PR
= 0.0833333333... PR
or about 8 percent of the total PR
---
C has received from respectively A, B and D =
(0  ) . (1/4 PR) +
(1/2) . (1/4 PR) +
(1/3) . (1/4 PR)
= ( 0/4 + 1/8 + 1/12 ) PR
= ( 0/24 + 3/24 + 2/24 ) PR
= 5/24 PR
= 0.2083333333... PR
or about 21 percent of the total PR
---
D has received from respectively A, B and C =
(0  ) . (1/4 PR) +
(0  ) . (1/4 PR) +
(0  ) . (1/4 PR)
= ( 0/4 + 0/4 + 0/4 ) PR
= ( 0/24 + 0/24 + 0/24 ) PR
= 0/24 PR
or 0 percent of the total PR
===
Note:
Checking this result:
As only 3 pages (that is B, C and D) have been given away their 
pagerank
(because A does not give away anything)
the total pagerank to be divided between the pages is 3/4 PR.
And indeed, if you add together the pageranks of A, B, C and D 
respectively
you get 11/24 + 2/24 + 5/24 + 0/24 = 18/24 = 3/4 PR.
===
In general you can say
(given that the the total of outgoing links of a page is one or more
(otherwise you would divide by zero in this formula))
you have
(pagerank A ) =
= (total of received pagerank of the other pages)
=
(total pagerank of page B) / (total of outgoing links of page B) +
(total pagerank of page C) / (total of outgoing links of page C) +
(total pagerank of page D) / (total of outgoing links of page D)
---
(pagerank B ) =
= (total of received pagerank of the other pages)
=
(total pagerank of page A) / (total of outgoing links of page A) +
(total pagerank of page C) / (total of outgoing links of page C) +
(total pagerank of page D) / (total of outgoing links of page D)
---
(pagerank C ) =
= (total of received pagerank of the other pages)
=
(total pagerank of page A) / (total of outgoing links of page A) +
(total pagerank of page B) / (total of outgoing links of page B) +
(total pagerank of page D) / (total of outgoing links of page D)
---
(pagerank D ) =
= (total of received pagerank of the other pages)
=
(total pagerank of page A) / (total of outgoing links of page A) +
(total pagerank of page B) / (total of outgoing links of page B) +
(total pagerank of page C) / (total of outgoing links of page C)
===
The modified pagerank formula
This basically adds some constant values to the final formula for the
pagerank of each page.
The next step in calculating the pagerank value is to add a so called
demping factor.
This is a number believed to be about 0.85.
===
(pagerank of any given page ) =
(1 - (demping factor)) / (total pages) +
(demping factor) . (total of received pagerank of the other pages)
===
Or thus in this example:
(pagerank A ) =
(1 - (demping factor)) / (total pages) +
(demping factor) .
(
(total pagerank of page B) / (total of outgoing links of page B) +
(total pagerank of page C) / (total of outgoing links of page C) +
(total pagerank of page D) / (total of outgoing links of page D) )
)
or thus filling in using the earlier example:
= ((1 - 0.85) / 4) + 0.85 . (11/24) = 0.4270833333 PR
===
(pagerank B ) =
(1 - (demping factor)) / (total pages) +
(demping factor) .
(
(total pagerank of page A) / (total of outgoing links of page A) +
(total pagerank of page C) / (total of outgoing links of page C) +
(total pagerank of page D) / (total of outgoing links of page D)
)
or thus filling in using the earlier example:
= ((1 - 0.85) / 4) + 0.85 . (2/24) = 0.1083333333 PR
===
(pagerank C ) =
(1 - (demping factor)) / (total pages) +
(demping factor) .
(
(total pagerank of page A) / (total of outgoing links of page A) +
(total pagerank of page B) / (total of outgoing links of page B) +
(total pagerank of page D) / (total of outgoing links of page D)
)
or thus filling in using the earlier example:
= ((1 - 0.85) / 4) + 0.85 . (5/24) = 0.2145833333 PR
===
(pagerank D ) =
(1 - (demping factor)) / (total pages) +
(demping factor) .
(
(total pagerank of page A) / (total of outgoing links of page A) +
(total pagerank of page B) / (total of outgoing links of page B) +
(total pagerank of page C) / (total of outgoing links of page C)
)
or thus filling in using the earlier example:
= ((1 - 0.85) / 4) + 0.85 . (0/24) = 0.03750 PR
===
Thus if you wanted to simulate this system yourself, you would have to
 1. Create N pages containing links to each other
    1. -E.g. page A
  <HTML>
  </HTML>
    2. -E.g. page B
  <HTML>
   <A href="http://localhost/pageA.html"> outgoing link to page A </A>
   <A href="http://localhost/pageC.html"> outgoing link to page C </A>
  </HTML>
    3. -E.g. page C
  <HTML>
   <A href="http://localhost/pageA.html"> outgoing link to page A </A>
  </HTML>
    4. -E.g. page D
  <HTML>
   <A href="http://localhost/pageA.html"> outgoing link to page A </A>
   <A href="http://localhost/pageB.html"> outgoing link to page B </A>
   <A href="http://localhost/pageC.html"> outgoing link to page C </A>
  </HTML>
 2. Extract the links in each page
    1. -You extract using a parser (which you could write yourself,
        e.g. using PHP, Perl, C++, Delphi, Java, BBCBASIC, ...) all
        occurrences of the <A http="">...</A>
    2. -You extract the links http://... and store them in
        a table counting the total links on that page
        1. -Page A
             ------------------------------------------
             | LINKS FOUND IN PAGE A
             ------------------------------------------
             ------------------------------------------
             thus outgoing linksum = 0
        2. -Page B
             ------------------------------------------
             | LINKS FOUND IN PAGE B
             ------------------------------------------
               http://localhost/pageA.html
               http://localhost/pageC.html
             ------------------------------------------
             thus outgoing linksum = 2
        3. -Page C
             ------------------------------------------
             | LINKS FOUND IN PAGE C
             ------------------------------------------
               http://localhost/pageA.html
             ------------------------------------------
             thus outgoing linksum = 1
        4. -Page D
             ------------------------------------------
             | LINKS FOUND IN PAGE D
             ------------------------------------------
               http://localhost/pageA.html
               http://localhost/pageB.html
               http://localhost/pageC.html
             ------------------------------------------
             thus outgoing linksum = 3
    3. -You add each found new link to an all links table
             ------------------------------------------
             | ALL LINKS                   | PAGERANK |
             ------------------------------------------
               http://localhost/pageA.html
               http://localhost/pageB.html
               http://localhost/pageC.html
               http://localhost/pageD.html
               ...
             ------------------------------------------
    4. -Each time you finished parsing that web page, you know the
        total of outgoing links, so to each of the corresponding links
        in the pagerank column in the table you can add the pagerank 
value
             ------------------------------------------
             | ALL LINKS                   | PAGERANK |
             ------------------------------------------
               http://localhost/pageA.html   11/24
               http://localhost/pageB.html    2/24
               http://localhost/pageC.html    5/24
               http://localhost/pageD.html    0/24
             ------------------------------------------
               ...
 3. Finally you can go through the links in that table to add
    the demping factor
             ---------------------------------------------
             | ALL LINKS                   | PAGERANK    |
             ---------------------------------------------
               http://localhost/pageA.html   0.4270833333
               http://localhost/pageB.html   0.1083333333
               http://localhost/pageC.html   0.2145833333
               http://localhost/pageD.html   0.03750
               ...
             ---------------------------------------------
 4. -Thus if asked for an overview of the page ranks, you
     sort this table on the pagerank value (thus the
     page with the highest pagerank comes on top)
             ---------------------------------------------
             | ALL LINKS                   | PAGERANK    |
             ---------------------------------------------
               http://localhost/pageA.html   0.4270833333
               http://localhost/pageC.html   0.2145833333
               http://localhost/pageB.html   0.1083333333
               http://localhost/pageD.html   0.03750
               ...
             ---------------------------------------------
 5. -You could create this simulation system using e.g. PHP, Java or
     Perl (running the parser to extract the links (<A HREF=http://...,
     www..., ...) in each web page and running the database commands
     like CREATE TABLE ..., INSERT INTO ..., UPDATE ... WHERE ...,
     ORDER BY) and a database (e.g. MySql or Microsoft SQL server,
     Oracle, ...), and possibly a web server (e.g. IIS or Apache).
===
Internet: see also:
---
PageRank
http://en.wikipedia.org/wiki/PageRank
---
BBCBASIC: Windows: Algorithm: Search: Google: Link: Pagerank: Can you 
show a simple program?
http://www.faqts.com/knowledge_base/view.phtml/aid/41709/fid/768
---
TSE: Search: Regular expression: HTML: URL: Which possible regular 
expression to extract anchor <A>?
http://www.faqts.com/knowledge_base/view.phtml/aid/31432/fid/865
---
PHP: Install: How to install PHP on Microsoft Internet Information 
Server (IIS)?
http://www.faqts.com/knowledge_base/view.phtml/aid/28265/fid/51
---
Database: Relational: MySql: Link: Overview: Can you give an overview 
of links?
http://www.faqts.com/knowledge_base/view.phtml/aid/35331/fid/52
----------------------------------------------------------------------
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