1. PageRank Algorithm
Prepared By: Mai Mustafa

2. Contents:
• Background
• Introduction to PageRank
• PageRank Algorithm
• Power iteration method
• Examples using PageRank and iteration
• Exercises
• Pseudo code of PageRank algorithm
• Searching with PageRank
• Application using PageRank
• Advantages and disadvantages of PageRank algorithm
• References

3. Background
PageRank was presented and published by Sergey Brin and
Larry Page at the Seventh International World Wide Web
Conference (WWW7) in April 1998.
The aim of this algorithm is track some difficulties with the
content-based ranking algorithms of early search engines
which used text documents for webpages to retrieve the
information with no explicit relationship of link between
them.

4. Introduction to PageRank
• PageRank is an algorithm uses to measure the
importance of website pages using hyperlinks between
pages.
• Some hyperlinks point to pages to the same site (in link)
and others point to pages in other Web sites(out link).
• PageRank is a “vote”, by all the other pages on the Web,
about how important a page is.
• A link to a page counts as a vote of support

5. PageRank Algorithm
The main concepts:
• In-links of page i : These are the hyperlinks that point to page i from other
pages. Usually, hyperlinks from the same site are not considered.
• Out-links of page i : These are the hyperlinks that point out to other pages
from page i .
The following ideas based on rank prestige are used to derive the
PageRank algorithm:
• A hyperlink from a page pointing to another page is an implicit conveyance
of authority to the target page. Thus, the more in-links that a page i
receives, the more prestige the page i has.
• Pages that point to page i also have their own prestige scores. A page with
a higher prestige score pointing to i is more important than a page with a
lower prestige score pointing to i .

7. Cont. PageRank Algorithm
To formulate the above ideas, we treat the Web as
a directed graph G = (V, E), where V is the set of vertices
or nodes, i.e., the set of all pages, and E is the set of
directed edges in the graph, i.e., hyperlinks. Let the
total number of pages on the Web be n (i.e., n = |V|).
The PageRank score of the page i (denoted by P(i)) is defined by:
1-
Oj is the number of out-links of page j.

8. Cont. PageRank Algorithm
Mathematically, we have a system of n linear equations (1) with n
unknowns. We can use a matrix to represent all the equations. Let P be
a n-dimensional column vector of PageRank values
Let A be the adjacency matrix of our graph with
2-
We can write the system of n equations with
3-

9. Cont. PageRank Algorithm
the above three conditions come from Markov chains Model, in it; each
Web page in the Web graph is regarded as a state. A hyperlink is a
transition, which leads from one state to another state with a
probability. Thus, this framework models Web surfing as a stochastic
process. It models a Web surfer randomly surfing the Web as a state
transition in the Markov chain .so on, this three conditions are not
satisfied. Because First of all, A is not a stochastic matrix. A stochastic
matrix is the transition matrix for a finite Markov chain whose entries
in each row are nonnegative real numbers and sum to 1. This requires
that every Web page must have at least one out-link. This is not true on
the Web because many pages have no out-links, which are reflected in
transition matrix A by some rows of complete 0’s. Such pages are called
dangling pages (nodes).

10. Cont. PageRank Algorithm
We can see that A is not a stochastic
matrix because the fifth row is all 0’s,
that is, page 5 is a dangling page.
We can fix this problem by adding a
complete set of outgoing links from
each such page i to all the pages on
the Web. Thus, the transition
probability of going from i to every
page is 1/n, assuming a uniform
probability distribution. That is, we
replace each row containing all 0’s
with e/n, where e is n-dimensional
vector of all 1’s.

11. Cont. PageRank Algorithm
Another problems:
A is not irreducible, which means that the Web graph G is not strongly
connected. And to be strongly connected it must have a path from u
to v. (if there is a non-zero probability of transitioning from any state to
any other state).
A is not aperiodic. A state i in a Markov chain being periodic means
that there exists a directed cycle that the chain has to traverse. To be
aperiodic all paths leading from state i back to state i have a length that
is a multiple of k.
It is easy to deal with the above two problems with a single strategy.
We add a link from each page to every page and give each link a small
transition probability controlled by a parameter d, it is used to model
the probability that at each page the surfer will become unhappy with
the links and request another random page.
The parameter d, called the damping factor, can be set to a value
between 0 and 1.
Always d = 0.85.

12. Cont. PageRank Algorithm
• The PageRank model:
• (t: Transpose)
• The PageRank formula for each page i :
T
A

13. Power iteration method:
The PageRank algorithm must be able to deal with billions of pages, meaning
incredibly immense matrices; thus, we need to find an efficient way to calculate the
eigenvector of a square matrix with a dimension in the billions. Thus, the best
option for calculating the eigenvector is through the power method. The power
method is a simple and easy to implement algorithm. Additionally, it is effective in
that it is not necessary to compute a matrix decomposition, which is near-
impossible for matrices containing very few values, such as the link matrix we
receive. The power method does have downsides, however, in that it is only able to
find the eigenvector of the largest absolute-value eigenvalue of a matrix. Also, the
power method must be repeated many times until it converges, which can occur
slowly.
Fortunately, as we are working with a stochastic matrix, the largest eigenvalue is
guaranteed to be 1. Since this is the eigenvector we are searching for, the power
method will return the importance vector we are looking for. Additionally, it has
been proven that the speed of convergence for the Google PageRank matrix is
slower the closer α gets to 0. Since we have set d to be equal to 0.15, we can expect
the speed of convergence to be approximately 50 - 100 iterations, which is the
number of iterations reported by the creators of PageRank to be necessary for
returning sufficiently close values.

15. Simpleexampleusing PageRankwithiteration
2 pages A,B:
• P(A)=(1-d)+d(pagerank(B)/1)
 P(A)=0.15+0.85*1=1
• P(B)=(1-d)+d(pagerank(A)/1)
 P(B)=0.15+0.85*1=1
When we calculate the PageRank of A and B is 1. now, we plug in 0 as
the guess and calculate again:
 P(A)=0.15+0.85*0=0.15
 P(B)=0.15+0.85*0.15=0.2775
Continue the second iteration:
 P(A)=0.15+0.85*0.2775=0.3859
 P(B)=0.15+0.85*0.3859=0.4780
If we repeat the calculations, eventually the PageRank for both the pages
converge to 1.

16. Anotherexampleusing PageRankwithiteration
Three pages A,B And C
• P(A)=(1-d)+d(pagerank(B)+pagerank(C)/1)
• P(B)=(1-d)+d(pagerank(A)/2)
• P(C)=(1-d)+d(pagerank(A)/2)
Begin with the initial value as 0:
1st iteration:
P(A)=0.15+0.85*0=0.15
P(B)=0.15+0.85*(0.15/2)=0.21
P(c)=0.15+0.85*(0.15/2)=0.21
2nd iteration:
P(A)=0.15+0.85*(0.21*2)=0.51
P(B)=0.15+0.85*(0.51/2)=0.37
P(C)=0.15+0.85*(0.51/2)=0.37

17. Cont.example
3rd iteration:
P(A)=0.15+0.85*(0.37*2)=0.78
P(B)=0.15+0.85*(0.87/2)=0.48
P(C)=0.15+0.85*(0.87/2)=0.48
And so on.. After 20 iterations
P(A)=1.46
P(B)=0.77
P(C)=0.77
The total PageRank =3, but we can see A has much larger proportion of
the PageRank than B and C, because they are passing to A not to any
other pages.

18. Exercise:
Given A below, obtain P by solving Equation PageRank model directly.
first: we will represent the matrix
as graph:

19. pdAedp T
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Find first then find e:T
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20. 
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002125.0000
425.002125.0000
425.00000283.0
0425.02125.00425.0283.0
0425.02125.085.00283.0
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And we know that d=0.85

21. 
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025.0025.02375.0025.0025.0025.0
45.0025.02375.0025.0025.0025.0
45.0025.0025.0025.0025.0308.0
025.045.02375.0025.045.0308.0
025.045.02375.0875.0025.0308.0
025.0025.0025.0025.045.0025.0
p

22. Exercise2:
Given A as in problem 1 in the last exercise, use the power iteration method
to show the first 5 iterations of P.
• First iteration:
• Second iteration:
0k
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0.363
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025.045.02375.0025.045.0308.0
025.045.02375.0875.0025.0308.0
025.0025.0025.0025.045.0025.0
0k
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0.333
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0.467
1.647
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0.966
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0.788
0.858
1.496
1.921
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025.0025.02375.0025.0025.0025.0
45.0025.02375.0025.0025.0025.0
45.0025.0025.0025.0025.0308.0
025.045.02375.0025.045.0308.0
025.045.02375.0875.0025.0308.0
025.0025.0025.0025.045.0025.0
* 01 kPk

23. • third iteration:
• Fourth iteration:
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0.250
0.391
0.565
1.623
2.130
1.043
0.333
0.487
0.467
1.647
2.102
0.966
025.0025.02375.0025.0025.0025.0
45.0025.02375.0025.0025.0025.0
45.0025.0025.0025.0025.0308.0
025.045.02375.0025.045.0308.0
025.045.02375.0875.0025.0308.0
025.0025.0025.0025.045.0025.0
* 12 kPk
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0.270
0.377
0.551
1.637
2.111
1.055
0.250
0.391
0.565
1.623
2.130
1.043
025.0025.02375.0025.0025.0025.0
45.0025.02375.0025.0025.0025.0
45.0025.0025.0025.0025.0308.0
025.045.02375.0025.045.0308.0
025.045.02375.0875.0025.0308.0
025.0025.0025.0025.045.0025.0
* 23 kPk

24. • Fifth iteration:
We would then continue this iterating until the values are approximately
stable, and we would be able to determine the importance ranking using the
resulting vector. With this, we can see that even with a small count of 5
iterations, our vector was already converging towards the eigenvector. Since
this is the importance vector of our network, we can see that the PageRank
importance ranking of our pages would thus be
2 > 3 > 1 > 4 > 5 > 6
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0.267
0.382
0.563
1.623
2.118
1.047
0.270
0.377
0.551
1.637
2.111
1.055
025.0025.02375.0025.0025.0025.0
45.0025.02375.0025.0025.0025.0
45.0025.0025.0025.0025.0308.0
025.045.02375.0025.045.0308.0
025.045.02375.0875.0025.0308.0
025.0025.0025.0025.045.0025.0
* 34 kPk

25. Pseudo code of PageRank
algorithm:

26. Searching with PageRank
Two search engines:
• Title-based search engine
• Full text search engine
• Title-based search engine
 Searches only the “Titles”
 Finds all the web pages whose titles contain all the query words
 Sorts the results by PageRank
 Very simple and cheap to implement
 Title match ensures high precision, and PageRank ensures high
quality
• Full text search engine
• Called Google
• Examines all the words in every stored document and also
performs PageRank (Rank Merging)
• More precise but more complicated

27. Cont. searching with PageRank

28. Application using PageRank
• the first and most obvious application of the PageRank algorithm is
for search engines. As it was developed specifically by Google for
use in their search engine, PageRank is able to rank websites in
order to provide more relevant search results faster.
• applied PageRank algorithm is towards searching networks outside
of the internet. this can be applied towards academic papers; by
using citations as a substitute for links, PageRank can determine the
most effective and referenced papers in an academic area.
• real-world application of the PageRank algorithm; for example,
determining key species in an ecology. By mapping the relationships
between species in an ecosystem, applying the PageRank algorithm
allows the user to identify the most important species. Thus, being
able to assign importance towards key animal and plant species in
an ecosystem allows for easier forecasting of consequences such as
extinction or removal of a species from the ecosystem.

29. AdvantageanddisadvantagesofPageRank
algorithm:
Advantages of PageRank:
1. The algorithm is robust against Spam since its not easy for a
webpage owner to add in links to his/her page from other
important pages.
2. PageRank is a global measure and is query independent.
Disadvantages of PageRank:
1. it favors the older pages, because a new page, even a very good
one will not have many links unless it is a part of an existing site.
2. It is very efficient to raise your own PageRank, is ’buying’ a link on
a page with high PageRank.

30. References:
• Comparative Analysis Of Pagerank And HITS Algorithms,
by: Ritika Wason. Published in IJERT, October - 2012.
• The top ten algorithms in data mining, by: Xindong wu
and vipin kumar.
• Building an Intelligent Web: Theory and Practice, By
Pawan Lingras, Saint Mary.
• Hyperlink based search algorithms-PageRank and
HITS, by: Shatakirti.