This document discusses Google's approach to assigning importance scores to web pages and summarizing the key challenges faced by earlier unsuccessful approaches. It describes Google ranking pages based on the concept of importance scores derived from backlinks, and three early attempts - counting backlinks, summing the scores of backlinking pages, and distributing forward page scores to backlinks - that failed to adequately capture importance. The third approach ran into issues with dangling nodes having no outgoing links and disconnected webs lacking unique importance scores.
1. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Linear Algebra behind Google Search
Dr. V.N. Krishnachandran
Department of Computer Applications
Vidya Academy of Science and Technology
Thrissur - 680501, Kerala.
August 2011
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
2. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Outline
1 Web: An example
2 Importance score
3 First unsuccessful approach
4 Second unsuccessful approach
5 Third unsuccessful approach
6 Dangling nodes
7 Disconnected webs
8 Google approach
9 Computational scheme
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
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Web world
The web world consists of a number of pages and links from some
of the pages to some other pages.
In a diagrammatic representation of a web world, pages are denoted
by small squares or circles and links are indicated by arrows.
See a simplified web world in next slide.
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Linear Algebra behind Google Search
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Web world
Example 1: A web with four pages numbered 1,2,3,4.
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Linear Algebra behind Google Search
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Links
In the figure above, arrow denotes:
an incoming link (also called a backlink) to Page q.
an outgoing link from Page p.
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Linear Algebra behind Google Search
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Links
Outgoing links in Example 1
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Linear Algebra behind Google Search
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Links
Incoming links in Example 1
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Linear Algebra behind Google Search
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Importance score
In Google’s search algorithm, the most important concept is that
of the importance score of a page.
This we explain in the next few slides...
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
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Importance score
The importance score, or simply the score, of a page is a
number which is a measure of the relative importance of a
page.
The importance score is a nonnegative real number.
The importance score of a page is derived from the backlinks
for that page.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
10. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Importance score vector
We denote the importance score of Page k by xk.
Let there be n pages in the web. The column vector
x = [x1 x2 · · · xn]T
is called the importance score vector.
The importance score vector x is said to be normalised if
x1 + x2 + · · · xn = 1.
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Linear Algebra behind Google Search
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Unsuccessful attempts to define importance score
Before considering Google’s approach, we consider
three unsuccessful attempts to define the concept of the
importance score of a page.
A study of these unsuccessful attempts helps one appreciate the
significance of Google’s approach.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
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Importance score:
First unsuccessful approach
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Linear Algebra behind Google Search
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Importance score: First unsuccessful approach
Definition (First unsuccessful approach)
Importance score of Page k is the number of backlinks for Page k.
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Importance score: First unsuccessful approach
Importance scores in Example 1
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Linear Algebra behind Google Search
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Importance score
Importance score: A desirable property
“A link to Page k from an important page must increase Page k’s
score more than a link from an unimportant page.”
First unsuccessful approach does not have this property.
(see next slide)
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Linear Algebra behind Google Search
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Importance score: First unsuccessful approach
Importance score of Page 1 must be higher than that of Page 4.
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Linear Algebra behind Google Search
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Importance score:
Second unsuccessful approach
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Importance score: Second unsuccessful approach
Definition (Second unsuccessful approach)
The importance score of a page is the sum of the scores of all
pages linking to the page.
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Importance score: Second unsuccessful approach
Importance scores in Example 1
The importance scores in Example 1 (second approach) are
solutions of the following system of equations:
x1 = x3 + x4
x2 = x1
x3 = x1 + x2 + x4
x4 = x1 + x2
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Linear Algebra behind Google Search
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Importance score: Second unsuccessful approach
Importance scores in Example 1 : Matrix formulation
H =
0 0 1 1
1 0 0 0
1 1 0 1
1 1 0 0
x = [x1 x2 x3 x4]T
Hx = x
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Linear Algebra behind Google Search
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Importance score: Second unsuccessful approach
Importance scores in Example 1 : Matrix formulation
x is an eigenvector with eigenvalue 1 for the matrix H.
1 is not an eigenvalue of H.
There is no eigenvector with eigenvalue 1 for the matrix H.
The second approach does not produce importance scores to pages
in Example 1 .
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Linear Algebra behind Google Search
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Importance score: Second unsuccessful approach
Importance score: An undesirable property
“A page with many outgoing links has a bigger influence on the
scores of other pages than a page with less number of outgoing
links.”
This is undesirable.
The recommendation letter of a Professor who is choosy in giving
such letters carries higher value than that of a Professor who is
very liberal in issuing such letters.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
23. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Importance score:
Third unsuccessful approach
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Linear Algebra behind Google Search
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Importance score: Third unsuccessful approach
Notations
n = Number of pages in the web
Pages indexed by k = 1, 2, . . . , n.
nj = Number of outgoing links from page j
Lk = Set of indices of backlinks for page k
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Importance score: Third unsuccessful approach
Definition (Third unsuccessful approach)
Let the web contain n pages and let it be indexed by an integer k,
1 ≤ k ≤ n. Let Lk ⊆ {1, 2, . . . , n} be the set of backlinks for Page
k, and nj the number of outgoing links from Page j. Then
xk =
j∈Lk
xj
nj
, k = 1, 2, . . . , n.
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Linear Algebra behind Google Search
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Notations
n = 4, k = 1, 2, 3, 4.
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Notations
n1 = 3, n2 = 2, n3 = 1, n4 = 2
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Notations
L1 = {3, 4}, L2 = {1}, L3 = {1, 2, 4}, L4 = {1, 2}
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Equations
Expression to compute x1:
x1 =
j∈L1
xj
nj
=
j∈{3,4}
xj
nj
=
x3
n3
+
x4
n4
=
x3
1
+
x4
2
Similar expressions for x2, x3 and x4. (See next slide ...)
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Linear Algebra behind Google Search
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Equations
Linear system of equations to compute importance score:
x1 =
x3
1
+
x4
2
x2 =
x1
3
x3 =
x1
3
+
x2
2
+
x4
2
x4 =
x1
3
+
x2
2
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Linear Algebra behind Google Search
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Matrix formulation
The link matrix of web world in Example 1:
A =
0 0 1 1
2
1
3 0 0 0
1
3
1
2 0 1
2
1
3
1
2 0 0
x = [x1 x2 x3 x4]T
Ax = x
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
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Importance score: Third unsuccessful approach
Importance scores in Example 1 : Matrix formulation
x is an eigenvector with eigenvalue 1 for the link matrix A.
1 is indeed an eigenvalue of A.
All multiples of the vector [12 4 9 6] are eigenvectors of
A corresponding to the eigenvalue 1.
The normalised importance score vector for the web in
Example 1 is
x =
12
31
4
31
9
31
6
31
= [0.387 0.129 0.290 0.194] (approx.)
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Linear Algebra behind Google Search
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Limitations of
third unsuccessful approach
Third unsuccessful approach has two severe limitations:
Problem of dangling nodes: If there are dangling nodes in the
web, one cannot assign importance scores to any page.
Problem of disconnected web: If the web is disconnected, one
cannot assign unique importance scores to all the pages in the
web.
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Linear Algebra behind Google Search
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Dangling nodes
Definition
A dangling node is a page with no outgoing links.
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Dangling nodes
Example 2 : Web with dangling node
(Page 4 is a dangling node)
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Linear Algebra behind Google Search
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Dangling nodes
Importance scores in Example 2 : Equations
x1 = x3
x2 =
x1
3
x3 =
x1
3
+
x2
2
x4 =
x1
3
+
x2
2
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Linear Algebra behind Google Search
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Dangling nodes
Importance scores in Example 2 : Matrix formulation
Link matrix for the web in Example 2:
A =
0 0 1 0
1
3 0 0 0
1
3
1
2 0 0
1
3
1
2 0 0
x = [x1 x2 x3 x4]T
Ax = x
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Linear Algebra behind Google Search
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Dangling nodes
Importance scores in Example 2 : Values
x is an eigenvector with eigenvalue 1 for the matrix A.
1 is not an eigenvalue of A.
There is no eigenvector with eigenvalue 1 for the matrix A.
The definition (third approach) does not produce importance
scores to pages in Example 2 .
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Linear Algebra behind Google Search
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Dangling nodes
Mathematics
Definition
A square matrix is called a column-schochastic matrix if all its
entries are nonnegative and the entries in each column sum to 1.
Theorem
Every column-stochastic matrix has 1 as an eigenvalue.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
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Dangling nodes
Mathematics
Theorem
The link matrix for a web with no dangling nodes is
column-stochastic.
Theorem
The link matrix for a web with no dangling nodes has 1 as an
eigenvalue.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
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Disconnected webs
Definition
A web W is disconnected if W can be partitioned into two
nonempty subwebs W1 and W2 such that there is no outgoing link
from any page in W1 to any page in W2 and vice versa.
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Disconnected webs
Example 3 : A web with two disconnected subwebs
W1 (Pages 1, 2) and W2 (Pages 3, 4, 5)
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Disconnected webs
Importance scores in Example 3 : Equations
x1 = x2
x2 = x1
x3 = x4 +
x5
2
x4 = x3 +
x5
2
x5 = 0
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Disconnected webs
Importance scores in Example 3 : Matrix formulation
A =
0 1 0 0 0
1 0 0 0 0
0 0 0 1 1
2
0 0 1 0 1
2
0 0 0 0 0
x = [x1 x2 x3 x4]T
Ax = x
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Linear Algebra behind Google Search
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Disconnected webs
Importance scores in Example 3 : Values
Two linearly independent eigenvectors with eigenvalue 1:
x =
1
2
1
2
0 0 0
x = 0 0
1
2
1
2
0
These are linearly independent, normalised, importance score
vectors in Example 3 .
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Linear Algebra behind Google Search
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Disconnected webs
The third approach does not produce a unique importance score
for every page in a disconnected web.
In third approach:
Web is disconnected =⇒ Importance scores are not unique
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Linear Algebra behind Google Search
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Google’s approach
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Google matrix: Definition
Consider a web with n pages.
Let A be the link matrix of the web.
Let S be an n × n matrix with all entries equal to 1
n .
Let m be such that 0 ≤ m ≤ 1.
Definition
The Google matrix of the web is
M = (1 − m)A + mS.
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Google matrix: Damping factor
Definition
The constant 1 − m in the definition of the Google matrix is called
the damping factor of the Google matrix. (The creators of
Google’s search algorithm chose 0.85 as the damping factor.)
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Google’s approach: Importance score
Definition
Let M be the Google matrix of a web having n pages. Let xk be
the importance score of Page k in the web and let
x = [x1 x2 · · · xn]T . Then a solution of the matrix equation
Mx = x
is called the importance score vector of the web.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
51. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Google’s approach: Importance score
Definition (alternate)
Let M be the Google matrix of a web having n pages. Let xk be
the importance score of Page k in the web and let
x = [x1 x2 · · · xn]T . Then an eigenvector of the matrix M
having eigenvalue 1 is called the importance score vector of the
web.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
53. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Google’s approach: Example 1
The importance scores are solutions of the matrix equation
Mx = x,
which are the eigenvectors of M having the eigenvalue 1.
M is column stochastic.
M has 1 as an eigenvalue.
M has an eigenvector having eigenvalue 1.
The web in Example 1 has an importance score vector as per
Google’s approach.
Is the important score vector unique?
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Linear Algebra behind Google Search
54. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Google’s approach: Example 1
The eigenvector of M (in Example 1) having eigenvalue 1 is
x =
106613
58520
40
57
57
40
1 .
The normalised importance score vector is (approximately)
x = [0.368 0.142 0.288 0.202].
The importance scores of the web pages are
x1 = 0.368, x2 = 0.142, x3 = 0.288, x4 = 0.202.
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Linear Algebra behind Google Search
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Google’s approach: Example 2
Example 2
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Google’s approach: Example 3
M (in Example 3) is column stochastic.
M (in Example 3) has 1 as an eigenvalue.
The eigenvector of M (in Example 3) having eigenvalue 1 is
x = [0.200 0.200 0.285 0.285 0.030].
The importance scores of the web pages (in Example 3) are
x1 = 0.200, x2 = 0.200, x3 = 0.285, x4 = 0.285 x5 = 0.030.
The scores are all positive.
The scores are unique even though the web has disconnected
subwebs.
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Linear Algebra behind Google Search
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Google’s approach: Mathematics
Definition
A matrix P is said to be positive if all elements of P are positive.
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Linear Algebra behind Google Search
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Google’s approach: Mathematics
Theorem
If a square matrix P is positive and column-stochastic, then any
eigenvector of P with eigenvalue 1 has all positive or negative
components.
Theorem
If a square matrix P is positive and column-stochastic, then the
eigenspace of P corresponding to the eigenvalue 1 has dimension 1.
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Linear Algebra behind Google Search
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Google’s approach: Mathematics
Properties of Google matrix
Let M be the Google matrix of a web without dangling nodes.
M is positive.
M is column stochastic.
1 is an eigenvalue of M.
The eigenspace of M corresponding to the eigenvalue 1 has
dimension 1.
Continued in next slide
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Linear Algebra behind Google Search
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Google’s approach: Mathematics
Properties of Google matrix (continued)
M has an eigenvector corresponding to the eigenvalue 1 with
all positive components.
M has a unique eigenvector x = [x1 x2 . . . xn]
corresponding to the eigenvalue 1 such that
xi > 0 for i = 1, 2, . . . , n.
x1 + x2 + · · · + xn = 1.
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Linear Algebra behind Google Search
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Computational scheme in
Google’s approach
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Linear Algebra behind Google Search
63. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Computational scheme
Notations:
Let W be a web with n pages and no dangling nodes.
Let A be the link matrix of the web W .
Let 1 − m be the damping factor.
Let u be the n-component column vector with all entries
equal to 1
n .
Let x(0) be some n-component column vector with positive
components and ||x(0)|| = 1.
Let q be the normalised importance score vector of the web
W .
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Computational scheme
The scheme:
Generate the sequence x(1), x(2), . . . of column vectors using the
following iteration scheme:
x(r+1)
= (1 − m)Ax(r)
+ mu.
Then
q = lim
r→∞
x(r)
.
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Linear Algebra behind Google Search
65. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Computational scheme: Example
Compute the importance score vector of web in Example 1 .
Notations:
n = 4
A =
0 0 1 1
2
1
3 0 0 0
1
3
1
2 0 1
2
1
3
1
2 0 0
m = 0.15
u = 1
4
1
4
1
4
1
4
T
.
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Computational scheme: Example
We choose x(0) = 1
4
1
4
1
4
1
4
T
.
In the next two slides we show the computations of x(1) and
x(2).
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Computational scheme: Example
The values of x(3), x(4), etc. are tabulated in the next slide. Note
that x(11) and x(12) are nearly identical. So further computations
won’t yield more accurate results.
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Computational scheme: Example
The importance scores of various pages in Example 1 are as given
below:
x1 = 0.3681, x2 = 0.1418, x3 = 0.2880, x4 = 0.2021.
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Computational scheme: Mathematics
Power method to find an eigenvector of a matrix G.
Start with an initial guess (initial approximation) x(0).
Generate successive approximations x(r) by the iteration
scheme
x(r)
= Gx(r−1)
,
or equivalently,
x(r)
= Gr
x(0)
.
For large r, the vector x(r) is a good approximation to an
eigenvector of G.
The power method produces successive approximations to the
eigenvector corresponding to the largest eigenvalue of G.
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Linear Algebra behind Google Search
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Computational scheme: Mathematics
Modified power method to find an eigenvector of a
matrix G.
Let x(r) = Gr x(0), for r = 1, 2, . . . .
x(r) may diverge to infinity or may decay to the zero vector.
A better iteration scheme is
x(r)
=
Gx(r−1)
||Gx(r−1)||
,
where || || is some vector norm.
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Linear Algebra behind Google Search
74. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Computational scheme: Mathematics
Power method applied to Google matrix
We apply the power method to compute the importance score
vector of a web.
Power method can be applied to compute the importance
score eigenvector only if 1 is the largest eigenvalue of the
Google matrix.
However, we can prove that the power method can be applied
to compute the importance score eigenvector without showing
that 1 is the greatest eigenvalue of the Google matrix.
See next few slides ...
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Linear Algebra behind Google Search
75. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Computational scheme: Mathematics
Power method applied to Google matrix
Let M be the Google matrix of a web. We have
M = (1 − m)A + mS.
Let x be a normalised column vector with positive components.
x(r+1)
= Mx(r)
= ((1 − m)A + mS)x(r)
= (1 − m)Ax(r)
+ mSx(r)
= (1 − m)Ax(r)
+ mu.
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Computational scheme: Mathematics
Definition
The 1-norm of a vector v is
||v||1 = |v1| + |v2| + · · · + |vn|.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
77. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Computational scheme: Mathematics
Theorem
Let P be a positive column-stochastic n × n real matrix and let V
be the subspace of Rn consisting of vectors v such that j vj = 0.
Then:
1 Pv ∈ V for any v ∈ V .
2 ||Pv||1 ≤ c||v||1 for any v ∈ V , where
c = max
1≤j≤n
|1 − 2 min
1≤i≤n
Pij | < 1.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
78. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
Computational scheme: Mathematics
Theorem
Every positive column-stochastic matrix P has a unique vector q
with positive components such that Pq = q with ||q||1 = 1. The
vector q can be computed as
q = lim
r→∞
Pr
x0
for any initial guess x0 with positive components such that
||x0||1 = 1.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
79. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
References
Kurt Brian and Tanya Leise, “The $25, 000, 000, 000
eigenvector: The linear algebra behind Google”, SIAM
Review, Vol.48, No.3, pp.568-581 (2005).
Amy N. Langville and Carl D. Meyer, ”Deeper Inside
PageRank”, 2004.
Hwai-Hui Fu, Dennis K.J. Lin and Hsien-Tang Tsai,
”Damping factor in Google page ranking”, Appl. Stochastic
Models Bus. Ind., 2006; 22:431444.
Christiane Rousseau and Yvan Saint-Aubin, Mathematics and
Technology (Chapter 9), Springer Undergraduate Texts in
Mathematics and Technology, 2008.
continued ...
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
80. Web Scores Approach 1 Approach 2 Approach 3 Dangling... Disconnected... Google’s approach Computational scheme
References (continued)
Monica Bianchini, Marco Gori, and Franco Scarselli, ”Inside
PageRank”, ACM Transactions on Internet Technology, Vol.
5, No. 1, February 2005, Pages 92128.
Sergey Brin and Lawrence Page, ”The Anatomy of a
Large-Scale Hypertextual Web Search Engine”, In Proceedings
of the 7th World Wide Web Conference (WWW7), 1998.
Dr. V.N. Krishnachandran
Linear Algebra behind Google Search
