Random web surfer pagerank algorithm

International Journal of Computer Applications (0975 – 8887)
Volume 35– No.11, December 2011
36
Random Web Surfer PageRank Algorithm
Navadiya Hareshkumar, Dr. Deepak Garg
Computer Science and Engineering Department
Thapar University, Punjab
ABSTRACT
In this paper analyzes how the Google web search engine
implements the PageRank algorithm to define prominent status to
web pages in a network. It describes the PageRank algorithm as a
Markov process, web page as state of Markov chain, Link
structure of web as Transitions probability matrix of Markov
chains, the solution to an eigenvector equation and Vector
iteration power method.
It mainly focus on how to relate the eigenvalues and eigenvector
of Google matrix to PageRank values to guarantee that there is a
single stationary distribution vector to which the PageRank
algorithm converges and efficiently compute the PageRank for
large sets of web Pages. Finally, it will demonstrate example of
the PageRank algorithm.
Keywords
PageRank, Markov chains, Power method, Google matrix,
stationary distribution vector, Eigen Vector and Values
1. INTRODUCTION
Paper begins with a review of the most basic PageRank model for
determining the importance of a webpage.
Paper provides comprehensive survey of all issues associated with
PageRank, available and recommended solution methods,
changing α value and convergence rate, storage issues and
compares PageRank to an idealized random Web surfer and
shows how to efficiently compute PageRank for large numbers of
pages.
2. PAGERANK
A search query with Google‘s search engine usually returns a very
large number of pages. E.g., a search on ‗University‘ returns 225
million pages. Although the search returns several million pages,
the most relevant pages are usually found within the top ten or
twenty pages in the list of results. How does the search engine
know which pages are the most important?
Google assigns a number to each individual webpage based on the
link structure of the web, expressing its importance. This number
is known as the PageRank and is computed via the PageRank
Algorithm. PageRank has applications in search, browsing, and
traffic estimation.
3. BASIC PAGERANK ALGORITHM
MODEL
An intuitive definition of PageRank algorithm shows how one
might measure the importance of a web page on the Internet.
A webpage U‘s PageRank is calculated base on how many other
WebPages Backlink (inedges) into U. The PageRank of U is the
sum of the PageRanks of each webpage Vi that backlinks to U
divided by the number of WebPages to which Vi links [1]. This
intuitive definition says that: if webpage U is linked to only by
WebPages with low PageRanks, then U may not get more
importance. Further, if U is linked to by a page Vj with a high
PageRank, but Vj links to many other pages, U should not receive
the full weight (benefits) of Vj‘s PageRank.
For example, if a webpage has a link to the Facebook Home
page, it may be just one link but it is a very important one. This
page should be ranked higher than many pages with more links
but from lower important pages. PageRank is an attempt to see
how good an approximation to ―importance‖ can be obtained from
the link structure.
3.1 FORMAL DEFINITION
Now that we have an idea of how PageRank works, we will
formally define the algorithm. Drawing on [2], we define
PageRank as follows:
 Let U be a web page.
 Let FU be the set of Forward links (outedges) from U.
 Let BU be the set of Backlinks into U.
 Let NU = |FU| be the number of forward links from U.
 Let c be a normalization factor so that ―the total rank of
all web pages is constant‖ [2].
Back Links
Forward Links
Fig 1: A Webpage’s Link Structure
We begin by defining a simple ranking, R which is a slightly
simplified version of PageRank:
R U = c
R(V)
NV
VϵBU
(1)
Note that here c < 1 because there are number of pages with no
forward links and their weight is lost from the system (see Section
4.2). The equation is recursive but it may be computed by starting
with any set of ranks and iterating the computation until it
converges. Fig 2 demonstrates the propagation of rank from one
pair of pages to another.
Webpage U

37
50
3
50
3
3
Fig 2: Simplified PageRank Calculation
There is a small problem with this simplified ranking algorithm.
Consider two web pages that point to each other but to no other
page. And suppose there is some web page which points to one of
them. Then, during iteration, this loop will accumulate rank but
never distribute any rank (since there are no outedges). The loop
forms a sort of trap which is called a Rank Sink.
Fig 3: Loop which acts as a Rank Sink
To overcome this problem of rank sinks, we introduce a rank
source:
 Let E(U) be ―some vector over the Web pages that
corresponds to a source of rank‖ [2].
Then, the PageRank of page U is given by
R′
U = c
R′
V
NV
VϵBU
+ c E U (2)
This is iterated until the values have converged sufficiently [3].
0.2
0.2 0.2
0.4
Fig4: Simplified PageRank Calculation
Notice that PageRank can be described using equation 2, the
summation method is neither the most interesting nor the most
illustrative of the algorithm‘s properties. So, it is preferred to
rewrite the sum as Matrix Multiplication.
4. RANDOM WEB SURFER MODEL
PageRank also has a second definition based upon the model of a
random web surfer navigating the Internet. In short, the model
states that PageRank models the behavior of someone who ―keeps
clicking on successive links at random‖. However, occasionally
the surfer ―gets bored and jumps to a random page chosen based
on the distribution in E.‖ [2]. so, surfer will not continue in the
loop forever.
Consider the illustration in Fig 5 of a simple Link Structure of
web pages; we can represent this structure using an N × N
adjacency matrix A, where Aij = 1 if there is a link (Edge) from
webpage (Vertex) i to webpage j, and 0 otherwise [4].
Fig 5: A Simple Link Structure
Thus, the 6×6 adjacency matrix for the link structure in Fig 5 is
A =
0 1 0 0 1 0
0 0 1 1 0 0
0 0 0 1 1 1
1 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 0
Notice that the diagonal entries are all zero. We assume that links
from a page to itself are ignored when constructing an adjacency
matrix.
To illustrate the PageRank algorithm based on second definition,
the following variables are defined:
 Let N be the total number of web pages in the web.
 Let πT
be the 1×N PageRank row vector (Stationary
vector).
 Let H the N × N row-normalized adjacency matrix.
(Transition Probability Matrix)
100 53
9 50
∞ ∞ ∞
A
0.4
B
0.2
C
0.4
Webpage A
Webpage E Webpage F
Webpage D
Webpage B
Webpage C

38
Then we can describe the PageRank vector at the kth
iteration as
π(k)T
= π(k−1)T
H (3)
So that successive iterations π(k)T
converge to the PageRank
vector πT
[3].
4.1 ROW-NORMALITY
Let to build a Transition Probabilities matrix,
Hi =
Ai
Aik
N
k=1
(4)
So, that each row 𝐴i of A is divided by its row sum. Apply
equation 4 on Adjacency matrix A. so we have, sparse matrix
H =
0 1/2 0 0 1/2 0
0 0 1/2 1/2 0 0
0 0 0 1/3 1/3 1/3
1 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 0
Matrix H whose element Hij is the probability of moving from
state i (page i) to state j (page j) in one time step (In Fig 5, assume
that, starting from any webpage, it is Half the probability to arrive
from page A to page B in one time step).
4.2 DANGLING LINKS
One issue with this model is dangling links. Dangling links are
simply links that point to any page with no outgoing links (In
Fig 5, Link to Webpage F) and Pages with no outlinks are called
dangling nodes (Webpage F). They affect the model because it is
not clear where their weight should be distributed. A page of data,
a page with a postscript graph, a page with jpeg pictures, a pdf
document, a page that has been fetched by a crawler but not yet
explored - these are all examples of possible dangling nodes. In
fact, for some subsets of the web, dangling nodes make up 80% of
the collection‘s pages [5].
4.3 STOCHASTICITY
Using the matrix H is insufficient for the PageRank algorithm,
however, because the iteration using H alone might not converge
properly — ―it may be dependent on the startingvector‖[3]. Here
matrix H is not yet stochastic [3]. A matrix is stochastic when it
is a ―matrix of transition probabilities for a Markov chain,‖ [A
Markov chain is ―collection of random variables‖ whose future
values are independent of their past values [6].] with the property
that ―all elements are non-negative and all its row sums are unity‖
(one) [7]. Thus, to ensure that H is stochastic, we must ensure that
every row sums to one. But here, the sum of the last row of matrix
H is Zero because of Dangling Node (see Section 4.2).
To overcome this problem, pages with no Forward links
(Webpage F) are assigned artificial links or ―teleporters‖ to all
other pages.
Artificial Links
Fig 6: A StochasticLink Structure
Therefore, we define the stochastic S as,
S = H +
a ∗ eT
N
(5)
Where,
a = N×1 Column Vector such that
ai = 1 if Hik = 0N
k=1 (i.e. Page i is Dangling Page)
= 0 otherwise
e = N×1 Column Vector of one‘s [3].
Apply equation 5 on matrix H. so we have stochastic matrix S as,
S =
0 1/2 0 0 1/2 0
0 0 1/2 1/2 0 0
0 0 0 1/3 1/3 1/3
1 0 0 0 0 0
1 0 0 0 0 0
1/6 1/6 1/6 1/6 1/6 1/6
It makes sure that surfer‘s random walk process does not get stuck
and the web pages are the states of the Markov chain.
4.4 THE GOOGLE MATRIX
There is no guarantee that S has a unique stationary distribution
vector (i.e., there might not be a single ―correct‖ PageRank
vector) [3]. For us to guarantee that there is a single stationary
distribution vector πT
to which we can converge, we must ensure
that S is irreducible as well as stochastic [3]. A matrix is
irreducible if and only if its graph is strongly connected [8].
Webpage A
Webpage E Webpage F
Webpage D
Webpage B
Webpage C

39
So we define the irreducible row-stochastic matrix G as
G = αS + 1 − α E ; 0 ≤ α ≤ 1 6
E =
e ∗ eT
N
G is the Google matrix, and we can use it to define
π(k)T
= π(k−1)T
G (7)
as the new iterative method for PageRank to replace equation 3.
Google uses the value of 0.85 for α [3] (see Section 6), and we
will use this same value. Thus, apply equation6 on matrix S, our
G is, G =
1/40 9/20 1/40 1/40 9/20 1/40
1/40 1/40 9/20 9/20 1/40 1/40
1/40 1/40 1/40 77/250 77/250 77/250
7/8 1/40 1/40 1/40 1/40 1/40
7/8 1/40 1/40 1/40 1/40 1/40
83/500 83/500 83/500 83/500 83/500 83/500
Since the matrix G is completely dense, and the graph of G now
strongly connected and G is irreducible.
5. THE POWER METHOD
The Google matrix G is currently of size more than eight billion
web pages [9] and therefore the Eigen value computation is not
trivial. To find an approximation of the principal Eigenvector the
Power Methodis used.
We iterate using the Google matrix G by writing equation 7. Let
us take 1×N initial row vectorπ(0)T
=
eT
N
Based on it result found
is, 𝛑(𝟏𝟗)𝐓 =
0.320 0.170 0.107 0.137 0.200 0.064
The PageRank of page i is given by the ith
element of πT
.
Table 1.PageRank of Tiny 6 web pages Link Structure
Starting anywhere within Fig 5 and randomly surfing the web
pages in given network, the likelihood of ending up at any of the
Pages are, respectively, about 32%, 17%, 11%, 14%, 20 % and
6%.
5.1 IMPROVEMENT OF POWER
METHOD
When dealing with large data sets, it is difficult to form a matrix
G and find its dominant eigenvector. It is more efficient to
compute the PageRank vector using the power method, where we
iterate using the sparse matrix H by rewriting equation 7.
π k T
= π k−1 T
G
= π k−1 T
αS + 1 − α E
= π k−1 T
αS + 1 − α
e ∗ eT
N
=απ k−1 T
S + 1 − α π k−1 T e∗eT
N
=απ k−1 T
S + 1 − α
eT
N
=απ k−1 T
( H +
a∗eT
N
) + 1 − α
eT
N
=απ k−1 T
H + απ k−1 T
a + 1 − α
eT
N
(8)
Since π(k−1)T is a probability vector, and thus, π(k−1)T e = 1
Fortunately, since H is sparse, each vector-matrix multiplication
required by the power method can be computed in nnz(H) flops,
where nnz(H) is the number of nonzeros in H and since the
average number of nonzeros per row in H is 3-10, O(nnz(H)) ≈
O(n) [5]. Furthermore, at each iterationthe power method only
requires the storage of one vector, the current iterate. The
founders of Google, Lawrence Page and Sergey Brin, use α = 0.85
and report success using only 50 to 100 power iterations [2].
6. CHANGING 𝛂 VALUE
ANDCONVERGENCE RATE
One of the most obvious places to begin fiddling with the basic
PageRank model is α. Brin and Page, have reported using α =
0.85. One wonders why this choice for α? Might a different
choice produce a very different ranking of retrieved WebPages?
As mentioned in sections The Google Matrix and The Power
Method, there are good reasons for using α = 0.85, one being the
speedy convergence of the power method. With this value for α,
rough estimate of the number of iterations needed to converge to a
tolerance level τ (measured by the residual,π(k)T
− π
(k−1)T
) is
log10τ / log10α. For τ = 10-6
and α = 0.85, one can expect roughly
-6 / log100.85 ≈ 85 iterations until convergence to the PageRank
vector. For τ = 10-8
, about 114 iterations and for τ = 10-10
, about
142 iterations. Brin and Page report success using only 50 to 100
power iterations, implying that τ could range from 10-3
to 10-7
.
Obviously, this choice of α brings faster convergence than higher
values of α.
Compare with α = 0.99, whereby roughly 1833 iterations are
required to achieve a residual less than 10-8
. When working with a
sparse 8 billion by 8 billion matrix, each iteration counts; over a
few hundred power iterations is more than Google is willing to
compute. However, in addition to the computational reasons for
choosing α = 0.85, this choice for α also carries some intuitive
weight: α = 0.85 implies that roughly 5/6 of the time a Web surfer
randomly clicks on hyperlinks (i.e., following the structure of the
Web, as captured by the 𝛂𝐒 part of the equation 6), while 1/6 of
the time this Web surfer will go to the URL line and type the
address of a new page to ―Artificial Link‖ to (as captured by the
Web Page PageRank Relative importance
A 0.320 1
B 0.170 3
C 0.107 5
D 0.137 4
E 0.200 2
F 0.064 6
Total = 1

40
𝟏 − 𝛂 𝐄 part of the equation 6). Perhaps this was the original
motivation behind Brin and Page‘s choice of α = 0.85; it produces
an accurate model for Web surfing behavior. Whereas α = 0.99,
not only slows convergence of the power method, but also places
much greater emphasis on the link structure of the Web and much
less on the teleportation tendencies of surfers. The PageRank
vector derived from α = 0.99 can be vastly different from that
obtained using α = 0.85. Perhaps it gives a ―truer‖ PageRanking.
An Experiment with various α‘s show significant variation in
rankings produced by different values of α[10].
7. ALTERNATIVE OF POWER METHOD
The iterative aggregation/disaggregation (IAD) method: is an
improvement of the PageRank algorithm used bythe search engine
Google to compute stationary probabilities of very large Markov
chains.
It is shown that the power method applied to the Google matrix
always converges, and that the convergence rate of theIAD
method is at least as good as that of the power method.
Furthermore, by exploiting the hyperlink structure of the web it
can be shown that the convergence rate of the IAD method
applied to the Google matrix can be made strictly faster than that
of the power method [11].
The adaptive randomized method: for finding
eigenvectorcorresponding to Eigen Value 1 for stochastic matrices
hasbeen proposed. The upper bound of its rate of convergenceis of
non-asymptotic type and has the explicit factor. Moreover, the
bound is valid for the whole class of stochastic matrices and does
not depend on propertiesof the individual matrix. The method can
be applied for PageRank computation with small parameter α.
Further workon acceleration of the method is possible [12].
DYNA-RANK: focuses upon efficiently calculating and updating
Google‘s PageRank vector using ―peer to peer‖ system. The
changes inthe web structure will be handled incrementally
amongstthe peers. DYNA-RANK produces the relative PageRank
oneach peer. DYNA-RANK is proven to take less computation
time and less number of iterations compared to centralized
approach [13].
8. STORAGE ISSUES OF GOOGLE
MATRIX
The size of the Markov matrix makes storage issues nontrivial.
For subsets of the web, the transition probability matrix H may or
may not fit in main memory. When a large matrix exceeds a
machine‘s memory, researchers usually try one of two things:
1. Compress the data needed so that the compressed
representation fits in main memory and then creatively
implement a modified version of PageRank on this
compressed representation.
2. Keep the data in its uncompressed form and develop
I/O-efficient implementations of the computations that
must take place on the large, uncompressed data.
For modern web structure for which the transition probability
matrix H can be stored in main memory, compression of the data
is not essential.
Rather than storing the full matrix or a compressed version of the
matrix, Web-sized implementations of the PageRank model store
the H or A matrix in an adjacency list of the columns of the
matrix [14]. In order to compute the PageRank vector, the
PageRank power method requires vector-matrix multiplications of
𝛑 𝐤−𝟏 𝐓
𝐇 at each iteration k. Therefore, quick access to the
columns of the matrix H (or A) is essential to algorithm speed.
Column i contains the inlink (Backlink) information for page i,
which, for the PageRank system of ranking WebPages, is more
important than outlink (Forwardlink) information contained in the
rows of H or A. For the tiny 6-node web from Fig 5, an
adjacency list representation of the columns of matrix A is:
Table 2.Adjacency list of matrix A
Cleve Moler gives one possible implementation of the power
method applied to an adjacency list, along with sample Matlab
code [15]. When the adjacency list does not fit in main memory
[14], suggest methods for compressing the data. Reference [16]
take the other approach and suggest I/O-efficient implementations
of PageRank. Since the PageRank vector itself is large and
completely dense, containing over 8 billion pages, and must be
consulted in order to process each user query, Reference [16] has
also suggested a technique to compress the PageRank vector. This
encoding of the PageRank vector hopes to keep the ranking
information cached in main memory, thus speeding query
processing.
9. CONCLUSION
In this paper, we have taken on the audacious task of condensing
every page on the World Wide Web into a single number, its
PageRank. PageRank is a global ranking of all web pages,
regardless of their content, based solely on their location in the
Web's link structure. Using PageRank, we are able to order search
results so that more important and central WebPages are given
preference.
The intuition behind Page Rank is that it uses information which
is external to the Web pages themselves – their backlinks, which
provide a kind of peer review. Furthermore, backlinks from
―important‖ pages are more significant than backlinks from
average pages. This is encompassed in the recursive definition of
PageRank (equation 2).
In the proposed algorithm a value of α is usedthat play a very
important role on the analysis of thealgorithm. After the analysis
it is concluded that α must not be selected closer to zero. At α = 1,
the system enters into the ideal state and theranking provided is
insignificant. As per evaluation α must be selected greater than or
equals to 0.5. However, if we consider convergence speed as only
factor toevaluate the performance than the best factor α will be
.85. The proposed algorithm is query independent algorithmand
does not consider query during ranking.As the Web continues its
amazing growth, the need for smarter storage schemes and
evenfaster numerical methods will become more evident. Both are
exciting areas for computer scientists and numerical analysts
interested in information retrieval.
Web Page BackLink From
A D, E
B A
C B
D B, C
E A, C
F C

41
As future work, it is desirable to conduct more experiments using
larger and various kinds of datasets in order to further validate our
conclusions in this paper.
10. REFERENCES
[1] Desmond J. Higham and Alan Taylor, The Sleekest Link
Algorithm (2003).
[2] Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry
Winograd, The PageRank Citation Ranking: Bringing Order
to the Web (1998).
[3] Amy N. Langville and Carl D. Meyer, The Use of the Linear
Algebra by Web Search Engines (2004).
[4] Eric W. Weisstein, Adjacency Matrix, From MathWorld- A
Wolfram Web Resource.
[5] Amy N. Langville, Carl D. Meyer, Deeper
InsidePageRank(2004).
[6] Eric W. Weisstein, Markov Chain, From MathWorld– A
Wolfram Web Resource.
[7] David Nelson, editor, The Penguin Dictionary of
Mathematics (Penguin Books Ltd, London, 2003).
[8] Sergio S. Guirreri, Markov Chains as methodology used
byPageRank to rank the Web Pages on Internet (2010).
[9] Bill Coughran, Google’s index nearly doubles, Google
Inc.(2004)
[10] Kristen Thorson. Modeling the Web and the computation of
PageRank (Hollins University, 2004).
[11] Ilse C.F. Ipsen, Steve Kirkland, Convergence Analysis Of An
Improved PageRank Algorithm (2003)
[12] Alexander Nazin, Boris Polyak, Adaptive Randomized
Algorithm for Finding Eigenvector of Stochastic Matrix with
Application to PageRank (48th IEEE Conference- December
16-18, 2009)
[13] MandarKale, Mrs.P.SanthiThilagam, DYNA-RANK: Efficient
calculation and updation of PageRank(International
Conference on Computer Science and Information
Technology 2008)
[14] Sriram Raghavan, Hector Garcia-Molina. Compressing the
graph structure of the Web. In Proceedings of the IEEE
Conference on Data Compression, pages 213–222, (March
2001).
[15] Cleve B. Moler. Numerical Computing with MATLAB.
(SIAM, 2004).
[16] Taher H. Haveliwala. Efficient encodings for
documentranking vectors. (Technical report, CS
Department,Stanford University, November 2002).

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