C++ for Scientists - Technische Universität Dresden

14.4. GOOGLE’S PAGE RANK 275
Taking the new refinements into account we can compute the importance score xk of a page k
as follows:
xk = � xj
(14.4)
nj
j∈Lk
Where Lk denotes the set of pages with a link to page k. Consider the simple example of Figure
14.2. Using the formula (14.4) we get the following equations for the importance scores of the
pages in this example:
x1 = x3 + x4
2
x2 = x1
3
x3 = x1
3
x4 = x1
3
+ x2
2
+ x2
2
+ x4
2
These linear equations can be written as Ax = x where x = [x1 x2 x3 x4] T and
⎡
⎤
A =
⎢
⎣
0 0 1 1
2
1
3 0 0 0
1
3
1
3
1 1
2 0 2
1
2 0 0
This transforms the web ranking problem into the standard problem of finding an eigenvector
x with eigenvalue 1 for the square matrix A. This eigenvector can be found iteratively using
the power method with a threshold τ:
The power method converges to the eigenvector corresponding to the dominant eigenvalue λ1.
1
2
3
4
5
6
1. v (0) = some vector with � v (0) �= 1
2. Repeat for k=1,2, . . . :
2.1. Apply A: w = Av (k−1) .
2.2. Normalize: v (k) = w/ � w �.
3. Until �v (k−1) − v (k) � < τ
The matrix A is called a column stochastic matrix, since it is a square matrix with positive
entries and the entries in each column sum to one. In the case of a column stochastic matrix,
this dominant eigenvalue is 1.
14.4.1 Software
Write a generic function:
template
void power iteration( V& v, Function & f, double tau ) ;
that computes the power iteration algorithm 5 for a matrix A with starting vector v. The
resulting eigenvector should be stored in v. The argument f is a functor that returns the result
of the matrix vector product. Also write documentation and specify the conceptual constraints
for the arguments.
⎥
⎦ .