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)].
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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.
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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)
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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.
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
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).
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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
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