The document discusses how Markov chains are used as the methodology behind PageRank to rank web pages on the internet. It provides an overview of key concepts, including defining Markov chains and stochastic processes. It explains the idea behind PageRank, treating each web page as a journal and measuring importance based on the number of citations/links to other pages. The PageRank algorithm models web surfing as a Markov chain and the steady-state probabilities of the chain indicate the importance of each page.

1. Markov Chains as methodology used by PageRank to
rank the Web Pages on Internet.
Sergio S. Guirreri - www.guirreri.host22.com
Google Technology User Group (GTUG) of Palermo.
5th March 2010
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 1 / 14

2. Overview
1 Concepts on Markov-Chains.
2 The idea of the PageRank algorithm.
3 The PageRank algorithm.
4 Solving the PageRank algorithm.
5 Conclusions.
6 Bibliography.
7 Internet web sites.
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3. Concepts on Markov-Chains.
Stochastic Process and Markov-Chains.
Let assume the following stochastic process
{Xn; n = 0, 1, 2, . . . }
with values in a set E, called the state space, while its elements are called
state of the process.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14

4. Concepts on Markov-Chains.
Stochastic Process and Markov-Chains.
Let assume the following stochastic process
{Xn; n = 0, 1, 2, . . . }
with values in a set E, called the state space, while its elements are called
state of the process.
Let assume the set E is finite or countable.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14

5. Concepts on Markov-Chains.
Stochastic Process and Markov-Chains.
Let assume the following stochastic process
{Xn; n = 0, 1, 2, . . . }
with values in a set E, called the state space, while its elements are called
state of the process.
Let assume the set E is finite or countable.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14

6. Concepts on Markov-Chains.
Stochastic Process and Markov-Chains.
Let assume the following stochastic process
{Xn; n = 0, 1, 2, . . . }
with values in a set E, called the state space, while its elements are called
state of the process.
Let assume the set E is finite or countable.
Definition
A Markov Chain is a stochastic process Xn that hold the following feature:
Prob{Xn+1 = j|Xn = i, Xn−1 = in−1, . . . , X0 = i0} =
= Prob{Xn+1 = j|Xn = i} = pij(n)
where E is the state space set and j, i, in−1, . . . , i0 ∈ E, n ∈ N.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14

7. Concepts on Markov-Chains.
Stochastic Process and Markov-Chains.
Let assume the following stochastic process
{Xn; n = 0, 1, 2, . . . }
with values in a set E, called the state space, while its elements are called
state of the process.
Let assume the set E is finite or countable.
Definition
A Markov Chain is a stochastic process Xn that hold the following feature:
Prob{Xn+1 = j|Xn = i, Xn−1 = in−1, . . . , X0 = i0} =
= Prob{Xn+1 = j|Xn = i} = pij(n)
where E is the state space set and j, i, in−1, . . . , i0 ∈ E, n ∈ N.
The transition probability matrix P of the process Xn is composed of pij,
∀i, j ∈ E.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14

8. The idea of the PageRank algorithm.
PageRank’s idea.
The idea behind the PageRank algorithm is similar to the idea of the impact
factor index used to rank the Journals [Page et al.(1999)]
[Brin and Page(1998)] [Langville et al.(2008)].
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 4 / 14

9. The idea of the PageRank algorithm.
PageRank’s idea.
The idea behind the PageRank algorithm is similar to the idea of the impact
factor index used to rank the Journals [Page et al.(1999)]
[Brin and Page(1998)] [Langville et al.(2008)].
PageRank the impact factor of Internet.
The impact factor of a journal is defined as the average number of citations
per recently published papers in that journal.
By regarding each web page as a journal, this idea was then extended to
measure the importance of the web page in the PageRank Algorithm.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 4 / 14

10. The idea of the PageRank algorithm.
Elements of the PageRank.
To illustrate the PageRank algorithm I define the following variables
[Ching and Ng(2006)]:
let be N the total number of web pages in the web.
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11. The idea of the PageRank algorithm.
Elements of the PageRank.
To illustrate the PageRank algorithm I define the following variables
[Ching and Ng(2006)]:
let be N the total number of web pages in the web.
let be k the outgoing links of web page j.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14

12. The idea of the PageRank algorithm.
Elements of the PageRank.
To illustrate the PageRank algorithm I define the following variables
[Ching and Ng(2006)]:
let be N the total number of web pages in the web.
let be k the outgoing links of web page j.
let be Q the so called hyperlink matrix with elements:
Qij =



1
k if web page i is an outgoing link of web page j;
0 otherwise;
Qi,i > 0 ∀i.
(1)
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14

13. The idea of the PageRank algorithm.
Elements of the PageRank.
To illustrate the PageRank algorithm I define the following variables
[Ching and Ng(2006)]:
let be N the total number of web pages in the web.
let be k the outgoing links of web page j.
let be Q the so called hyperlink matrix with elements:
Qij =



1
k if web page i is an outgoing link of web page j;
0 otherwise;
Qi,i > 0 ∀i.
(1)
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14

14. The idea of the PageRank algorithm.
Elements of the PageRank.
To illustrate the PageRank algorithm I define the following variables
[Ching and Ng(2006)]:
let be N the total number of web pages in the web.
let be k the outgoing links of web page j.
let be Q the so called hyperlink matrix with elements:
Qij =



1
k if web page i is an outgoing link of web page j;
0 otherwise;
Qi,i > 0 ∀i.
(1)
The hyperlink matrix Q can be regarded as a transition probability matrix of
a Markov chain.
One may regard a surfer on the net as a random walker and the web pages as
the states of the Markov chain.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14

15. The PageRank algorithm.
The PageRank with irreducible Markov Chain.
Assuming that the Markov chain is irreduciblea
and aperiodicb
then the
steady-state probability distribution (p1, p2, . . . , pN )T
of the states (web
pages) exists.
aA Markov chain is irreducible if all states communicate with each other.
bA chain is periodic if there exists k > 1 such that the interval between two visits to some
state s is always a multiple of k. Therefore a chain is aperiodic if k=1.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14

16. The PageRank algorithm.
The PageRank with irreducible Markov Chain.
Assuming that the Markov chain is irreduciblea
and aperiodicb
then the
steady-state probability distribution (p1, p2, . . . , pN )T
of the states (web
pages) exists.
aA Markov chain is irreducible if all states communicate with each other.
bA chain is periodic if there exists k > 1 such that the interval between two visits to some
state s is always a multiple of k. Therefore a chain is aperiodic if k=1.
The PageRank
Each pi is the proportion of time that the surfer visiting the web page i.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14

17. The PageRank algorithm.
The PageRank with irreducible Markov Chain.
Assuming that the Markov chain is irreduciblea
and aperiodicb
then the
steady-state probability distribution (p1, p2, . . . , pN )T
of the states (web
pages) exists.
aA Markov chain is irreducible if all states communicate with each other.
bA chain is periodic if there exists k > 1 such that the interval between two visits to some
state s is always a multiple of k. Therefore a chain is aperiodic if k=1.
The PageRank
Each pi is the proportion of time that the surfer visiting the web page i.
The higher the value of pi is, the more important web page i will be.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14

18. The PageRank algorithm.
The PageRank with irreducible Markov Chain.
Assuming that the Markov chain is irreduciblea
and aperiodicb
then the
steady-state probability distribution (p1, p2, . . . , pN )T
of the states (web
pages) exists.
aA Markov chain is irreducible if all states communicate with each other.
bA chain is periodic if there exists k > 1 such that the interval between two visits to some
state s is always a multiple of k. Therefore a chain is aperiodic if k=1.
The PageRank
Each pi is the proportion of time that the surfer visiting the web page i.
The higher the value of pi is, the more important web page i will be.
The PageRank of web page i is then defined as pi.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14

19. The PageRank algorithm.
The PageRank with reducible Markov Chain
Since the matrix Q can be reducible to ensure that the steady-state
probability exists and is unique the following matrix P must be considered:
P = α




Q11 Q12 . . . Q1N
Q21 Q22 . . . Q2N
. . . . . . . . . . . .
QN1 QN2 . . . QNN



 +
(1 − α)
N




1 1 . . . 1
1 1 . . . 1
. . . . . . . . . . . .
1 1 . . . 1



 (2)
Where 0 < α < 1 and the most popular values of α are 0.85 and (1 − 1/N).
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 7 / 14

20. The PageRank algorithm.
The PageRank with reducible Markov Chain
Since the matrix Q can be reducible to ensure that the steady-state
probability exists and is unique the following matrix P must be considered:
P = α




Q11 Q12 . . . Q1N
Q21 Q22 . . . Q2N
. . . . . . . . . . . .
QN1 QN2 . . . QNN



 +
(1 − α)
N




1 1 . . . 1
1 1 . . . 1
. . . . . . . . . . . .
1 1 . . . 1



 (2)
Where 0 < α < 1 and the most popular values of α are 0.85 and (1 − 1/N).
Interpretation of PageRank
The idea of the PageRank (2) is that, for a network of N web pages, each web
page has an inherent importance of (1 − α)/N.
If a page Pi has an importance of pi, then it will contribute an importance of
α pi which is shared among the web pages that it points to.
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21. The PageRank algorithm.
The PageRank with reducible Markov Chain
Solving the following linear system of equations subject to the normalization
constraint one can obtain the importance of web page Pi :





p1
p2
...
pN





= α




Q11 Q12 . . . Q1N
Q21 Q22 . . . Q2N
. . . . . . . . . . . .
QN1 QN2 . . . QNN









p1
p2
...
pN





+
(1 − α)
N





1
1
...
1





(3)
Since
N
i=1
pi = 1
the (3) can be rewritten as
(p1, p2, . . . , pN )T
= P(p1, p2, . . . , pN )T
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22. Solving the PageRank algorithm.
The power method.
The power method is an iterative method for solving the dominant eigenvalue
and its corresponding eigenvectors of a matrix.
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23. Solving the PageRank algorithm.
The power method.
The power method is an iterative method for solving the dominant eigenvalue
and its corresponding eigenvectors of a matrix.
Given an n × n matrix A, the hypothesis of power method are:
there is a single dominant eigenvalue. The eigenvalues can be sorted:
|λ1| > |λ2| ≥ |λ3| ≥ . . . |λn|
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14

24. Solving the PageRank algorithm.
The power method.
The power method is an iterative method for solving the dominant eigenvalue
and its corresponding eigenvectors of a matrix.
Given an n × n matrix A, the hypothesis of power method are:
there is a single dominant eigenvalue. The eigenvalues can be sorted:
|λ1| > |λ2| ≥ |λ3| ≥ . . . |λn|
there is a linearly independent set of n eigenvectors:
{u(1)
, u(2)
, . . . , u(n)
}
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14

25. Solving the PageRank algorithm.
The power method.
The power method is an iterative method for solving the dominant eigenvalue
and its corresponding eigenvectors of a matrix.
Given an n × n matrix A, the hypothesis of power method are:
there is a single dominant eigenvalue. The eigenvalues can be sorted:
|λ1| > |λ2| ≥ |λ3| ≥ . . . |λn|
there is a linearly independent set of n eigenvectors:
{u(1)
, u(2)
, . . . , u(n)
}
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14

26. Solving the PageRank algorithm.
The power method.
The power method is an iterative method for solving the dominant eigenvalue
and its corresponding eigenvectors of a matrix.
Given an n × n matrix A, the hypothesis of power method are:
there is a single dominant eigenvalue. The eigenvalues can be sorted:
|λ1| > |λ2| ≥ |λ3| ≥ . . . |λn|
there is a linearly independent set of n eigenvectors:
{u(1)
, u(2)
, . . . , u(n)
}
so that
Au(i)
= λiu(i)
, i = 1, . . . , n.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14

27. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

28. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
iterating the initial vector with the A matrix:
Ak
x(0)
= a1Ak
u(1)
+ a2Ak
u(2)
+ · · · + anAk
u(n)
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

29. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
iterating the initial vector with the A matrix:
Ak
x(0)
= a1Ak
u(1)
+ a2Ak
u(2)
+ · · · + anAk
u(n)
= a1λk
1u(1)
+ a2λk
2u(2)
+ · · · + anλk
nu(n)
.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

30. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
iterating the initial vector with the A matrix:
Ak
x(0)
= a1Ak
u(1)
+ a2Ak
u(2)
+ · · · + anAk
u(n)
= a1λk
1u(1)
+ a2λk
2u(2)
+ · · · + anλk
nu(n)
.
dividing by λk
1
Ak
x(0)
λk
1
= a1u(1)
+ a2
λ2
λ1
k
u(2)
+ · · · + an
λn
λ1
k
u(n)
,
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

31. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
iterating the initial vector with the A matrix:
Ak
x(0)
= a1Ak
u(1)
+ a2Ak
u(2)
+ · · · + anAk
u(n)
= a1λk
1u(1)
+ a2λk
2u(2)
+ · · · + anλk
nu(n)
.
dividing by λk
1
Ak
x(0)
λk
1
= a1u(1)
+ a2
λ2
λ1
k
u(2)
+ · · · + an
λn
λ1
k
u(n)
,
Since
|λi|
|λ1|
< 1 →
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

32. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
iterating the initial vector with the A matrix:
Ak
x(0)
= a1Ak
u(1)
+ a2Ak
u(2)
+ · · · + anAk
u(n)
= a1λk
1u(1)
+ a2λk
2u(2)
+ · · · + anλk
nu(n)
.
dividing by λk
1
Ak
x(0)
λk
1
= a1u(1)
+ a2
λ2
λ1
k
u(2)
+ · · · + an
λn
λ1
k
u(n)
,
Since
|λi|
|λ1|
< 1 → lim
k→∞
|λi|k
|λ1|k
= 0 →
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

33. Solving the PageRank algorithm.
The power method.
The initial vector x0 can be wrote:
x(0)
= a1u(1)
+ a2u(2)
+ · · · + anu(n)
iterating the initial vector with the A matrix:
Ak
x(0)
= a1Ak
u(1)
+ a2Ak
u(2)
+ · · · + anAk
u(n)
= a1λk
1u(1)
+ a2λk
2u(2)
+ · · · + anλk
nu(n)
.
dividing by λk
1
Ak
x(0)
λk
1
= a1u(1)
+ a2
λ2
λ1
k
u(2)
+ · · · + an
λn
λ1
k
u(n)
,
Since
|λi|
|λ1|
< 1 → lim
k→∞
|λi|k
|λ1|k
= 0 → Ak
≈ a1λk
1u(1)
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14

34. Conclusions.
The power method and PageRank.
Results.
The matrix P of the PageRank algorithm is a stochastic matrix therefore
the largest eigenvalue is 1.
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35. Conclusions.
The power method and PageRank.
Results.
The matrix P of the PageRank algorithm is a stochastic matrix therefore
the largest eigenvalue is 1.
The convergence rate of the power method depends on the ratio of λ2
λ1
.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14

36. Conclusions.
The power method and PageRank.
Results.
The matrix P of the PageRank algorithm is a stochastic matrix therefore
the largest eigenvalue is 1.
The convergence rate of the power method depends on the ratio of λ2
λ1
.
It has been showed by [Haveliwala and Kamvar(2003)] that for the second
largest eigenvalue of P, we have
|λ2| ≤ α 0 ≤ α ≤ 1.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14

37. Conclusions.
The power method and PageRank.
Results.
The matrix P of the PageRank algorithm is a stochastic matrix therefore
the largest eigenvalue is 1.
The convergence rate of the power method depends on the ratio of λ2
λ1
.
It has been showed by [Haveliwala and Kamvar(2003)] that for the second
largest eigenvalue of P, we have
|λ2| ≤ α 0 ≤ α ≤ 1.
Since λ1 = 1 the converge rate depends on α.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14

38. Conclusions.
The power method and PageRank.
Results.
The matrix P of the PageRank algorithm is a stochastic matrix therefore
the largest eigenvalue is 1.
The convergence rate of the power method depends on the ratio of λ2
λ1
.
It has been showed by [Haveliwala and Kamvar(2003)] that for the second
largest eigenvalue of P, we have
|λ2| ≤ α 0 ≤ α ≤ 1.
Since λ1 = 1 the converge rate depends on α.
The most popular value for α is 0.85. With this value it has been proved
that the power method on web data set of over 80 million pages converges
in about 50 iterations.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14

39. Conclusions.
Really thanks to GTUG Palermo
and
see you to the next meeting!
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 12 / 14

40. Bibliography.
Bibliography.
Brin, S. and Page, L. (1998).
The anatomy of a large-scale hypertextual Web search engine.
Computer networks and ISDN systems, 30(1-7), 107–117.
Ching, W. and Ng, M. (2006).
Markov Chains: Models, Algoritms and Applications.
Springer Science + Business Media, Inc.
Haveliwala, T. and Kamvar, M. (2003).
The second eigenvalue of the google matrix.
Technical report, Stanford University.
Langville, A., Meyer, C., and Fern´Andez, P. (2008).
Google’s PageRank and beyond: the science of search engine rankings.
The Mathematical Intelligencer, 30(1), 68–69.
Page, L., Brin, S., Motwani, R., and Winograd, T. (1999).
The PageRank Citation Ranking: Bringing Order to the Web.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 13 / 14

41. Internet web sites.
Internet web sites.
Jon Atle Gulla (2007) - From Google Search to Semantic Exploration. -
Norwegian University of Science Technology -
www.slideshare.net/sveino/semantics-and-search?type=presentation
Steven Levy (2010) - Exclusive: How Google’s Algorithm Rules the Web - Wired
Magazine - www.wired.com/magazine/2010/02/ff_google_algorithm/
Ann Smarty (2009) - Let’s Try to Find All 200 Parameters in Google Algorithm -
Search Engine Journal -
www.searchenginejournal.com/200-parameters-in-google-algorithm/15457/.
Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 14 / 14