In the GoogleMatrix I tried to understand the concept

of the PageRank algorithm that Google uses to list pages according to

their \’importance\’. So, if you want your webpage to come out first in

a certain search, you have to increase your PageRank-value (which

normally is a measure of webpages linking to your page) artificially. A

method to achieve this is by **link spamming**, that is if page A is

to webpage of which you want to increase the PageRank value, take a page

B (either under your control or that of a friend webmaster) and add a

dummy link page B -> page A. To find out the effect of this on the

PageRank and how the second eigenvalue of the GoogleMatrix is able to

detect such constructs let us set up a *micro-web* consisting of

just 3 pages with links 1->2 and 1->3. The corresponding GoogleMatrix

(with c=0.85 and v=(1/3,1/3,1/3) is

1/3 1/20 1/20 1/3 9/10 1/20 1/3 1/20 9/10

which has eigenvalues 1,0.85 and 0.28.

The eigenvector with eigenvalue 1 (the PageRank) is equal to (0.15,1,1)

so page 2 and page 3 are equally important to Google and if we scale

PageRank such that it adds up to 100% over all pages, the relative

importance values are 6,9%,46,5% and 46,5%. In this case the eigenvector

corresponding to the second eigenvalue 0.85 is (0,-1,1) and hence

detects the two leaf-nodes. Now, assume the owner of page 2 sets up a

link spam by creating page 4 and linking 4->3, then the corresponding

GoogleMatrix (with v=(1/4,1/4,1/4,1/4)) is

77/240 3/80 3/80 3/80 77/240 71/80 3/80 37/80 77/240 3/80 71/80 3/80 3/80 3/80 3/80 37/80

which has eigenvalues

1,0.85,0.425 and 0.283. The PageRank eigenvector with eigenvalue 1 is

in this case is (0.8,8.18,5.35,1) or in relative importance % we have

(4.9%,50.1%,32.7%,6.1%) and we see that the spammer achieved his/her

goal. The eigenvector corresponding to the second eigenvalue is

(0,-1,1,0) which again gives the leaf-nodes and the eigenvector of the

third eigenvalue is (0,-1,0,1) and detects the spam-construct.

## Comments