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The Linear Algebra Behind Google
by
SINDHUJARANI.V
Roll No. [10212338]
() May 1, 2012 1 / 34
Outline of the Talk
⇒ Basic working of the Google.
⇒ Page Rank Algorithm.
⇒ Solving the Page Rank algorithm by eigen system.
⇒ Solving the Page Rank algorithm by linear system.
() May 1, 2012 2 / 34
Introdution
⇒ The Internet can be seen as a large directed graph, where the Web pagesthemselves represent vertices’s, and their links as edges.
⇒ The page rank algorithm ranking the back links of the vertices’s.
⇒ Which vertices’s having more back links getting more important.
() May 1, 2012 3 / 34
Example:Figure 1
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() May 1, 2012 4 / 34
⇒ we have the vote for the page x1 = 2, x2 = 1, x3 = 3, and x4 = 2. So thatpage 3 is the most important, pages 1 and 4 are second important, and page 2 isleast important.
Drawback:
⇒ Not all votes are equally important. A vote from a page with low importanceshould be less important than the vote from the more importance page.
⇒ To avoid this, each vote′s importance is divided by the number of differentvotes a page casts.
() May 1, 2012 5 / 34
Matrix Model
⇒ The new formate consider as a matrix which is in the form of
Aij =
{
1/Nj if Pj links to Pi
0 otherwise, (1)
where Nj is the number of out links from page Pj .
Recursive form:
⇒The recursive form per page is defined as
ri =∑
j∈Li
rj/Nj , (2)
where ri is the page rank of page Pi , Nj is the number of out links from page Pj
and Li are the pages that link to page Pi .
() May 1, 2012 6 / 34
⇒ Let′s apply this approach to figure 1. For page 1, the recursive form asx1=
x31 + x4
2 , for the page 2 x2=x13 , x3=
x13 + x2
2 + x42 and x4=
x13 + x2
2 .
⇒ Now, these linear equations can be written Ax = x , where x=[x1, x2, x3, x4]T
and in the matrix form as
A =
0 0 1 1/21/3 0 0 01/3 1/2 0 1/21/3 1/2 0 0
,
which transforms the web ranking problem into the “standard”problem of findingan eigenvector for a square matrix.
⇒ In this case, we obtain x1 ≃ 0.387, x2 ≃ 0.129, x3 ≃ 0.290, and x4 ≃ 0.194,where page 1 getting rank 1, page 2 getting rank 3, page 3 getting rank 2, andpage 4 getting rank 4.
() May 1, 2012 7 / 34
Speciality of the matrix A
Definition:
A square matrix is called a column stochastic matrix, if all of its entries arenonnegative and the entries in each column sum to 1.
⇒ A is called a column stochastic matrix.
⇒ A has 1 as an eigenvalue.
⇒ A has left eigenvector, which sum is equal to 1.
() May 1, 2012 8 / 34
Difficulty arise when using the formula 2
⇒ Stuck at a page.
⇒ Nonuniquness rankings.
⇒ Stuck in a subgraph.
() May 1, 2012 9 / 34
Stuck at a page
Definition:
A node that has no out links is called dangling node.
⇒ If graph has dangling node then the link matrix has a column of zeros to thatnode. so we cannot get the column stochastic matrix.
⇒To modify the link matrix as a column stochastic matrix, replace all zeros with1nin all the zero column, where n is the dimension of the matrix.
⇒ Now the matrix as
A = A+1
neTd (3)
where e is a row vector of ones, and d is a row vector, defined as
dj =
{
1 if Nj = 0
0 otherwise(4)
() May 1, 2012 10 / 34
Example:Figure 2
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() May 1, 2012 11 / 34
For our figure 2, we have d = [1, 0, 0, · · ·0]. Thus
A = A+1
neTd
=
0 12
13 0 0 0
0 0 13 0 0 0
0 12 0 0 0 0
0 0 13 0 1/2 1
0 0 0 12
12 0
0 0 0 12 0 0
+
16 0 0 0 0 016 0 0 0 0 016 0 0 0 0 016 0 0 0 0 016 0 0 0 0 016 0 0 0 0 0
=
16
12
13 0 0 0
16 0 1
3 0 0 016
12 0 0 0 0
16 0 1
3 0 12 1
16 0 0 1
212 0
16 0 0 1
2 0 0
With the creation of matrix A, we have a column stochastic matrix.
() May 1, 2012 12 / 34
Nonuniquness ranking
⇒ For our rankings, it is desirable that the dimension of V1(A) (corresponding tothe eigenvalue 1) equal′s 1, so that there is a nonzero eigenvector x with
∑
i xi=1which can be for page ranking.
⇒ It is not always true in general.
() May 1, 2012 13 / 34
Example:Figure 3
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() May 1, 2012 14 / 34
⇒ The link matrix of, figure 3 as
A =
0 1 0 0 01 0 0 0 00 0 0 1 1/20 0 1 0 1/20 0 0 0 0
.
We find here that V1(A) is two-dimensional. One possible pair of basis vectors isx = [1/2, 1/2, 0, 0, 0]T and y = [0, 0, 1/2, 1/2, 0]T.
⇒ We know that any linear combination of these two vectors yields anothervector in V1(A). so we will face the problem in the ranking.
() May 1, 2012 15 / 34
Overcome the dim(V1(A))
⇒To solving this problem we are modifying the equation (2).This analysis thatfollows, basically a special case of the Perron Frobenius theorem.
Perron Frobenius theorem:
Let B be an n × n matrix with nonnegative real entries. Then we have thefollowing:
1. B has a nonnegative real eigenvalue. The largest such eigenvalue λ(B),dominates the absolute values of all other eigenvalues of B. The dominationis strict if the entries of B are strictly positive.
2. If B has strictly positive entries, then λ(B) is a simple positive eigenvalue,and the corresponding eigenvector can be normalized to have strictly positiveentries.
3. If B has an eigenvector v with strictly positive entries, then thecorresponding eigenvalue λ is λ(B).
() May 1, 2012 16 / 34
A Modification of the Link Matrix A
⇒ For an n page web with no dangling nodes, We will replace the matrix A withthe matrix
A = αA+ (1− α)S . (5)
⇒ For an n page web with dangling nodes, We will replace the matrix A with thematrix
A = αA+ (1− α)S . (6)
where 0 ≤ α ≤ 1 is called a damping factor and S denote an n× n matrix with allentries 1
n. The matrix S is column stochastic, and also V1(S) is one dimensional.
() May 1, 2012 17 / 34
Speciality of the matrix A
1. All the entries Aij satisfy 0 ≤ Aij ≤ 1.
2. Each of the columns sum to one,∑
i Aij = 1 for all j.
3. If the value of α = 0, then we have the original problem as A = A.
4. If the value of α = 1, then we have the problem as A = S .
() May 1, 2012 18 / 34
Random walker
⇒The random walker starts from a random page, and then selects one of the outlinks from the page in a random fashion.
⇒The page rank of a specific page can now be viewed as the asymptoticprobability that the walker is present at the page.
⇒This is possible, as the walker is more likely to randomly wander to pages withmany votes (lots of in links), giving him a large probability of ending up in suchpages.
() May 1, 2012 19 / 34
Stuck in a subgraph
⇒There is still one possible pitfall in the ranking. The walker wander into asubsection of the complete graph that does not link to any outside pages.
⇒The link matrix for this model reducible matrix.
⇒We therefore want the matrix to be irreducible, which making sure he cannotget stuck in a subgraph. This irreducibility is called “teleportation”which meansability to jump with a small probability from any page in the link structure to anyother page. This can mathematically be described for page with no danglingnodes as:
A = αA+ (1− α)1
neTe. (7)
For page with dangling nodes as:
A = αA+ (1− α)1
neT e (8)
where e is a row vector of ones, and α is a damping factor.
() May 1, 2012 20 / 34
Example:Figure 4
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() May 1, 2012 21 / 34
⇒ The link matrix for figure 4 using equation (8) as,
A = αA+ (1− α)1
neT e =
16
1112
1960
160
160
160
16
160
1960
160
160
160
16
1112
160
160
160
160
16
160
1960
160
715
1112
16
160
160
715
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160
16
160
160
715
160
160
.
Here α set to 0.85. Here the matrix A is a column stochastic matrix.
⇒ When adding (1− α) 1neTe gives an equal chance of jumping to all pages.
() May 1, 2012 22 / 34
Analysis of the matrix A
Definition:
A matrix A is positive if Aij > 0 for all i and j .
⇒ If A is positive and column stochastic, then any eigenvector in V1(A) has allpositive or all negative components.
⇒ If A is positive and column stochastic, then V1(A) has dimension 1.
() May 1, 2012 23 / 34
Solution Methods for Solving the Page rank Problem
⇒The page rank is the same as finding the eigenvector corresponding to thelargest eigenvalue of the matrix A.
⇒To solve this we need an iterative method that works well for large sparsematrices.
⇒ There are two methods for solving Page rank Problem:
1. Eigen system problem. (The power method)
2. Linear system problem. (Jacobi method,Gauss-Seidel method,SORmethod,etc..)
() May 1, 2012 24 / 34
The power method
⇒The Power method is a simple method for finding the largest eigenvalue andcorresponding eigenvector of a matrix.
⇒It can be used when there is a dominant eigenvalue of A.
⇒ Consider iterates of the power method applied to A as
Ax (k−1) = αAx (k−1) + α1
neTdx (k−1) + (1 − α)
1
neTex (k−1) = x (k),
where x (k−1) is a probability vector, and thus ex (k−1) = 1.
() May 1, 2012 25 / 34
Convergence of the power method
⇒Rescale the power method at each iteration by xk =Axk−1
‖Axk−1‖, where ‖ · ‖ can be
any vector norm.
⇒Every positive column stochastic matrix A has a unique vector x with positivecomponents such that Ax = x with ‖x‖1 = 1. The vector x can be computed asx=limk→∞ Akx0 for any initial guess x0 with positive components such that‖x0‖1 = 1.
⇒The rate of convergence may be shown to be linear for the Power method is|λ2/λ1|.
() May 1, 2012 26 / 34
Linear system problem
⇒ We begin by formulating the page rank problem as a linear system.
⇒The eigen system problem Ax = αAx + (1− α) 1neTex = x can be rewritten as,
(I − αA)x = (1− α)1
neT =: b. (9)
⇒Let we split the matrix (I − αA) as,
(I − αA) = A = (L+ D + U), (10)
where D is the diagonal matrix and, L and U are strict lower triangular and strictupper triangular respectively.
() May 1, 2012 27 / 34
Properties of (I − αA)
1. (I − αA) is an M-matrix.
2. (I − αA) is nonsingular.
3. The row sums of (I − αA) are 1− α.
4. ‖I − αA‖∞ = 1+ α, provided at least one nondangling node exists.
5. Since (I − αA) is an M-matrix, (I − αA)−1 ≥ 0.
6. The row sums of (I − αA)−1 is 11−α
. Therefore ‖(I − αA)−1‖∞ = 11−α
.
7. Thus, the condition number κ∞(I − αA) = 1+α
1−α.
Definition:
A real matrix A that has Aij ≤ 0 when i 6= j and aii ≥ 0 for all i .A can beexpressed as A = sI − B, where s > 0 and B ≥ 0. when s > ρ(B), A is called anM matrix. M matrix can be either singular or nonsingular.
() May 1, 2012 28 / 34
Jacobi method
⇒The Jacobi method can be applied to Google matrix (10)
(L+ D + U)x = b
Dxk = b − (L+ U)xk−1
xk = D−1[b − (L+ U)xk−1],
where D is invertible matrix.
() May 1, 2012 29 / 34
Gauss Seidel method
⇒The Gauss Seidel method can be applied to Google matrix (10)
(L+ D + U)x = b
(L+ D)xk = b − Uxk−1
xk = (L+ D)−1[b − Uxk−1],
where (L+ D) is invertible matrix.
⇒The Gauss seidel method converges much faster than the Power and Jacobimethods.
⇒The disadvantage is very hard to parallelize.
() May 1, 2012 30 / 34
SOR method
⇒The SOR method can be applied to Google matrix (10)
ω(L+ D + U)x = ωb
(ωL+ D)xk = ω(b − Uxk−1) + (1− ω)Dxk−1
xk = (ωL+ D)−1[ω(b − Uxk−1) + (1− ω)Dxk−1],
where 1 ≤ ω ≤ 2. when ω=1, this method return to the Gauss seidel. Here(ωL+ D) is invertible matrix.
⇒ The cost of SOR method per iteration is more expansive and less efficient inparallel computing for huge matrix system.
() May 1, 2012 31 / 34
Plot for number of the iteration required for
convergence by different method
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure: Jacobi method
1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure: Gauss-seidal
1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure: SOR method
1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure: Power method
() May 1, 2012 32 / 34
Conclusion
⇒We are discussed mathematical idea used in Google search engine.
⇒Investigated various problem arises when computing the Google matrix (Linkmatrix).
⇒Taken, an example of large matrix representation of the Internet, and developedcomputing the Page rank using different methods.
() May 1, 2012 33 / 34
Bibliography
Erik Andersson and Per-Anders Ekstrom.Investigating google’s pagerank algorithm.(2):1–9, 2004.
Pavel Berkhin.A survey on pagerank computing.(1):88–89, 13-07-2005.
Kurt Bryan and Tanya Leise.The 25,000,000,000 eigenvector: The linear algebra behind google.SIAM Review, (3), 2006.
Amy N. Langville and Carl D. Meyer.Deeper inside pagerank.2004.
() May 1, 2012 34 / 34