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Implementing PageRank
Algorithm Using Hadoop
MapReduce
FARZAN HAJIAN
FARZAN.HAJIAN@GMAIL.COM
Introduction
โ€ข An algorithm for ranking web pages based on their importance
โ€ข Developed by Lawrence Page and Sergey Brin (founders of Google)
โ€ข Being used In Google to sort search results
โ€ข Describes how probable web pages are to be visited by a random
web surfer
โ€ข It is an iterative graph processing algorithm
Ranking Web Pages
โ€ข Web pages are not equally โ€œImportantโ€
โ€ข www.amazon.com
โ€ข www.my-personal-weblog.com
โ€ข It is more likely that amazon.com is visited than the other web page
โ€ข So it is more important (it has more weight)
โ€ข WHY?
Ranking Web Pages
โ€ข Inbound links count
โ€ข The more inbound link a page has, the more important (probable to
be visited) it become
โ€ข Imagine two web pages
โ€ข Page โ€œAโ€ (2 inbound links)
โ€ข Page โ€œBโ€ (10 inbound links)
โ€ข Which page is more important?
โ€ข Page โ€œBโ€
Ranking Web Pages
โ€ข Now suppose this condition
โ€ข Page โ€œAโ€ (2 inbound links)
โ€ข amazon.com
โ€ข facebook.com
โ€ข Page โ€œBโ€ (10 inbound linked)
โ€ข my-personal-weblog1.com
โ€ข โ€ฆ
โ€ข my-personal-weblog10.com
โ€ข Now which page is more weighted?
Ranking Web Pages
โ€ข Inbound links count
โ€ข But not all inbound links are equal
โ€ข So โ€œimportanceโ€ (PageRank) of page โ€œPโ€ depends on
โ€ข โ€œimportanceโ€ (PageRank) of the pages that link to page โ€œPโ€ (not barely on the
count of the pages that link to page โ€œPโ€)
Simple Recursive Formula
โ€ข Each linkโ€™s weight is proportional to the importance of its source
page
โ€ข If page โ€œPโ€ with importance โ€œxโ€ has โ€œnโ€ outbound links, each link
gets โ€œx/nโ€ weight
โ€ข Page โ€œPโ€โ€™s own importance is the sum of the weight on its inbound
links
The Random Surfer Model
โ€ข Consider PageRank as a model of user behavior
โ€ข Where a surfer clicks on links at random with no regard towards
content
โ€ข The random surfer visits a web page with a certain probability which
derives from the page's PageRank
โ€ข The probability that the random surfer clicks on one link is solely
given by the number of links on that page
โ€ข This is why one page's PageRank is not completely passed on to a
page it links to, but is divided by the number of links on the page
The Random Surfer Model
โ€ข So, the probability for the random surfer reaching one page is the
sum of probabilities for the random surfer following links to this
page
โ€ข The surfer does not click on an infinite number of links, but gets
bored sometimes and jumps to another page at random
โ€ข The probability for the random surfer not stopping to click on links is
given by the โ€œdamping factorโ€ (set between 0 and 1)
โ€ข The โ€œdamping factorโ€ is usually set to 0.85
The Final Formula
โ€ข PR(A) =
1โˆ’๐‘‘
๐‘
+ d (
๐‘ƒ๐‘…(๐‘‡๐‘–)
๐ถ(๐‘‡๐‘–)
)
โ€ข PR(A) is the PageRank of page A
โ€ข PR(Ti) is the PageRank of page Ti which link to page A
โ€ข C(Ti) is the number of outbound links on page Ti
โ€ข N is the number of web pages
โ€ข d is a damping factor which can be set between 0 and 1
Example
โ€ข PR(A) โ‰ˆ PR(C)
โ€ข PR(B) โ‰ˆ 0.5* PR(A)
โ€ข PR(C) โ‰ˆ 0.5*PR(A) , PR(B)
Example
โ€ข To keep the calculation simple we set the damping factor
to 0.5 and the number of nodes is ignored
โ€ข PR(A) = (1-0.5) + 0.5 (
๐‘ƒ๐‘…(๐‘‡๐‘–)
๐ถ(๐‘‡๐‘–)
)
โ€ข PR(A) = 0.5 + 0.5 PR(C) = 1.07692308
PR(B) = 0.5 + 0.5 (PR(A) / 2) = 0.76923077
PR(C) = 0.5 + 0.5 (PR(A) / 2 + PR(B)) = 1.15384615
The Iterative Computation of PageRank
โ€ข In practice, the web consists of billions of pages and it is not possible
to find a solution by using equation systems
โ€ข Google search engine uses an approximative, iterative computation
of PageRank
โ€ข Each page is assigned an initial starting value (usually
1
# ๐‘œ๐‘“ ๐‘›๐‘œ๐‘‘๐‘’๐‘ 
) and
the PageRanks of all pages are then calculated in several
computation circles based on the equations determined by the
PageRank algorithm values
The Iterative Computation of PageRank
Iteration PR(A) PR(B) PR(C)
0 1 1 1
1 1 0.75 1.125
2 1.0625 0.765625 1.1484375
3 1.07421875 0.76855469 1.15283203
4 1.07641602 0.76910400 1.15365601
5 1.07682800 0.76920700 1.15381050
6 1.07690525 0.76922631 1.15383947
7 1.07691973 0.76922993 1.15384490
8 1.07692245 0.76923061 1.15384592
9 1.07692296 0.76923074 1.15384611
10 1.07692305 0.76923076 1.15384615
11 1.07692307 0.76923077 1.15384615
12 1.07692308 0.76923077 1.15384615
Implementing PageRank Using MapReduce
โ€ข Multiple stages of mappers and reducers are needed
โ€ข Output of reducers are feed into the next stage mappers
โ€ข The initial input data for the previous example will be
organized as
A B C
B C
C A
โ€ข In each row
โ€ข The first column contains our nodes
โ€ข Other columns are the nodes that the main node has an outbound link to
Implementing PageRank Using MapReduce
โ€ข The initial PageRank values are calculated (
1
# ๐‘œ๐‘“ ๐‘›๐‘œ๐‘‘๐‘’๐‘ 
) and added to
the file
A 1/3 B C
B 1/3 C
C 1/3 A
โ€ข In each row
โ€ข The first column contains our nodes
โ€ข Other columns are the nodes that the main node has an outbound link to
Implementing PageRank Using MapReduce
โ€ข Mappers receive values as follows
โ€ข (y, PR(y) x1 x2 โ€ฆ xn)
โ€ข And emit the following values for each row
โ€ข (y, PR(y) x1 x2 โ€ฆ xn)
โ€ข for i = 1 โ€ฆ n
(xi,
๐‘ƒ๐‘…(๐‘ฆ)
๐ถ(๐‘ฆ)
)
Implementing PageRank Using MapReduce
โ€ข Reducers receive values from mappers and use the PageRank
formula to aggregate values and calculate new PageRank values
โ€ข New Input file for the next phase is created
โ€ข The differences between New PageRanks and old PagesRanks are
compared to the convergence factor
Implementing PageRank Using MapReduce
โ€ข Mappers in our example
โ€ข A 1/3 B C => (A, 1/3 B C)
(B, 1/6)
(C, 1/6)
โ€ข B 1/3 C => (B, 1/3 C)
(C, 1/3)
โ€ข C 1/3 A => (C, 1/3 A)
(A, 1/3)
Implementing PageRank Using MapReduce
โ€ข Reducers in our example
โ€ข (A, 1/3 B C) => (A, 1/3 B C)
(A, 1/3)
โ€ข (B, 1/3 C) => (B, 1/6 C)
(B, 1/6)
โ€ข (C, 1/3 A) => (C, 1/6+1/3 A)
(C, 1/6)
(C, 1/3)
Implementing PageRank Using MapReduce
โ€ข The new input file for mappers in the next phase will be
โ€ข A 0.3333 B C
B 0.1917 C
C 0.4750 A
Thank You

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Implementing page rank algorithm using hadoop map reduce

  • 2. Introduction โ€ข An algorithm for ranking web pages based on their importance โ€ข Developed by Lawrence Page and Sergey Brin (founders of Google) โ€ข Being used In Google to sort search results โ€ข Describes how probable web pages are to be visited by a random web surfer โ€ข It is an iterative graph processing algorithm
  • 3. Ranking Web Pages โ€ข Web pages are not equally โ€œImportantโ€ โ€ข www.amazon.com โ€ข www.my-personal-weblog.com โ€ข It is more likely that amazon.com is visited than the other web page โ€ข So it is more important (it has more weight) โ€ข WHY?
  • 4. Ranking Web Pages โ€ข Inbound links count โ€ข The more inbound link a page has, the more important (probable to be visited) it become โ€ข Imagine two web pages โ€ข Page โ€œAโ€ (2 inbound links) โ€ข Page โ€œBโ€ (10 inbound links) โ€ข Which page is more important? โ€ข Page โ€œBโ€
  • 5. Ranking Web Pages โ€ข Now suppose this condition โ€ข Page โ€œAโ€ (2 inbound links) โ€ข amazon.com โ€ข facebook.com โ€ข Page โ€œBโ€ (10 inbound linked) โ€ข my-personal-weblog1.com โ€ข โ€ฆ โ€ข my-personal-weblog10.com โ€ข Now which page is more weighted?
  • 6. Ranking Web Pages โ€ข Inbound links count โ€ข But not all inbound links are equal โ€ข So โ€œimportanceโ€ (PageRank) of page โ€œPโ€ depends on โ€ข โ€œimportanceโ€ (PageRank) of the pages that link to page โ€œPโ€ (not barely on the count of the pages that link to page โ€œPโ€)
  • 7. Simple Recursive Formula โ€ข Each linkโ€™s weight is proportional to the importance of its source page โ€ข If page โ€œPโ€ with importance โ€œxโ€ has โ€œnโ€ outbound links, each link gets โ€œx/nโ€ weight โ€ข Page โ€œPโ€โ€™s own importance is the sum of the weight on its inbound links
  • 8. The Random Surfer Model โ€ข Consider PageRank as a model of user behavior โ€ข Where a surfer clicks on links at random with no regard towards content โ€ข The random surfer visits a web page with a certain probability which derives from the page's PageRank โ€ข The probability that the random surfer clicks on one link is solely given by the number of links on that page โ€ข This is why one page's PageRank is not completely passed on to a page it links to, but is divided by the number of links on the page
  • 9. The Random Surfer Model โ€ข So, the probability for the random surfer reaching one page is the sum of probabilities for the random surfer following links to this page โ€ข The surfer does not click on an infinite number of links, but gets bored sometimes and jumps to another page at random โ€ข The probability for the random surfer not stopping to click on links is given by the โ€œdamping factorโ€ (set between 0 and 1) โ€ข The โ€œdamping factorโ€ is usually set to 0.85
  • 10. The Final Formula โ€ข PR(A) = 1โˆ’๐‘‘ ๐‘ + d ( ๐‘ƒ๐‘…(๐‘‡๐‘–) ๐ถ(๐‘‡๐‘–) ) โ€ข PR(A) is the PageRank of page A โ€ข PR(Ti) is the PageRank of page Ti which link to page A โ€ข C(Ti) is the number of outbound links on page Ti โ€ข N is the number of web pages โ€ข d is a damping factor which can be set between 0 and 1
  • 11. Example โ€ข PR(A) โ‰ˆ PR(C) โ€ข PR(B) โ‰ˆ 0.5* PR(A) โ€ข PR(C) โ‰ˆ 0.5*PR(A) , PR(B)
  • 12. Example โ€ข To keep the calculation simple we set the damping factor to 0.5 and the number of nodes is ignored โ€ข PR(A) = (1-0.5) + 0.5 ( ๐‘ƒ๐‘…(๐‘‡๐‘–) ๐ถ(๐‘‡๐‘–) ) โ€ข PR(A) = 0.5 + 0.5 PR(C) = 1.07692308 PR(B) = 0.5 + 0.5 (PR(A) / 2) = 0.76923077 PR(C) = 0.5 + 0.5 (PR(A) / 2 + PR(B)) = 1.15384615
  • 13. The Iterative Computation of PageRank โ€ข In practice, the web consists of billions of pages and it is not possible to find a solution by using equation systems โ€ข Google search engine uses an approximative, iterative computation of PageRank โ€ข Each page is assigned an initial starting value (usually 1 # ๐‘œ๐‘“ ๐‘›๐‘œ๐‘‘๐‘’๐‘  ) and the PageRanks of all pages are then calculated in several computation circles based on the equations determined by the PageRank algorithm values
  • 14. The Iterative Computation of PageRank Iteration PR(A) PR(B) PR(C) 0 1 1 1 1 1 0.75 1.125 2 1.0625 0.765625 1.1484375 3 1.07421875 0.76855469 1.15283203 4 1.07641602 0.76910400 1.15365601 5 1.07682800 0.76920700 1.15381050 6 1.07690525 0.76922631 1.15383947 7 1.07691973 0.76922993 1.15384490 8 1.07692245 0.76923061 1.15384592 9 1.07692296 0.76923074 1.15384611 10 1.07692305 0.76923076 1.15384615 11 1.07692307 0.76923077 1.15384615 12 1.07692308 0.76923077 1.15384615
  • 15. Implementing PageRank Using MapReduce โ€ข Multiple stages of mappers and reducers are needed โ€ข Output of reducers are feed into the next stage mappers โ€ข The initial input data for the previous example will be organized as A B C B C C A โ€ข In each row โ€ข The first column contains our nodes โ€ข Other columns are the nodes that the main node has an outbound link to
  • 16. Implementing PageRank Using MapReduce โ€ข The initial PageRank values are calculated ( 1 # ๐‘œ๐‘“ ๐‘›๐‘œ๐‘‘๐‘’๐‘  ) and added to the file A 1/3 B C B 1/3 C C 1/3 A โ€ข In each row โ€ข The first column contains our nodes โ€ข Other columns are the nodes that the main node has an outbound link to
  • 17. Implementing PageRank Using MapReduce โ€ข Mappers receive values as follows โ€ข (y, PR(y) x1 x2 โ€ฆ xn) โ€ข And emit the following values for each row โ€ข (y, PR(y) x1 x2 โ€ฆ xn) โ€ข for i = 1 โ€ฆ n (xi, ๐‘ƒ๐‘…(๐‘ฆ) ๐ถ(๐‘ฆ) )
  • 18. Implementing PageRank Using MapReduce โ€ข Reducers receive values from mappers and use the PageRank formula to aggregate values and calculate new PageRank values โ€ข New Input file for the next phase is created โ€ข The differences between New PageRanks and old PagesRanks are compared to the convergence factor
  • 19. Implementing PageRank Using MapReduce โ€ข Mappers in our example โ€ข A 1/3 B C => (A, 1/3 B C) (B, 1/6) (C, 1/6) โ€ข B 1/3 C => (B, 1/3 C) (C, 1/3) โ€ข C 1/3 A => (C, 1/3 A) (A, 1/3)
  • 20. Implementing PageRank Using MapReduce โ€ข Reducers in our example โ€ข (A, 1/3 B C) => (A, 1/3 B C) (A, 1/3) โ€ข (B, 1/3 C) => (B, 1/6 C) (B, 1/6) โ€ข (C, 1/3 A) => (C, 1/6+1/3 A) (C, 1/6) (C, 1/3)
  • 21. Implementing PageRank Using MapReduce โ€ข The new input file for mappers in the next phase will be โ€ข A 0.3333 B C B 0.1917 C C 0.4750 A