9. Additional Factors Influencing PageRank (continued)

Different Evaluation of
Links within a Document

Two of the criteria for the evaluation of links mentioned by Lawrence Page in his PageRank patent specifications are the visibilty of a link and its position within a document. Regarding the Random Surfer Model, those criteria reflect the probability for the random

surfer clicking on a link on a specific web page. In the original PageRank algorithm, this probability is given by the term (1/C(Ti)), whereby the probability is equal for each link on one page.

Assigning different probabilities to each link on a page can, for instance, be realized as follows:

	We take a look at a web consisting of three pages A, B anc C, where each of these pages has outbound links to both of the other pages. Links are weighted by two evaluation criteria X and Y. X represents the visibility of a link. X equals 1 if a link is not particularly emphasized, and 2 if the link is, for instance, bold or italic. Y represents the position of a link within a document. Y equals 1 if the link is on the lower half of the page, and 3 if the link is on the upper half of the page.

If we assume a multiplicative correlation between X and Y, the links in our example are evaluated as follows:

X(A,B) × Y(A,B) = 1 × 3 = 3
X(A,C) × Y(A,C) = 1 × 1 = 1
X(B,A) × Y(B,A) = 2 × 3 = 6
X(B,C) × Y(B,C) = 2 × 1 = 2
X(C,A) × Y(C,A) = 2 × 3 = 6
X(C,B) × Y(C,B) = 2 × 1 = 2

For the purpose of determinig the single factors L, the evaluated links must not simply be weighted by the number of outbound links on one page, but in fact by the total of evaluated links on the page. Thereby, we get the following weighting quotients Z(Ti) for the single pages Ti:

Z(A) = X(A,B) × Y(A,B) + X(A,C) × Y(A,C) = 4
Z(B) = X(B,A) × Y(B,A) + X(B,C) × Y(B,C) = 8
Z(C) = X(C,A) × Y(C,A) + X(C,B) × Y(C,B) = 8

The evaluation factors L(T1,T2) for a link which is pointing from page T1 to page T2 are hence given by

L(T1,T2) = X(T1,T2) × Y(T1,T2) / Z(T1)

Their values regarding our example are as follows:

L(A,B) = 0.75
L(A,C) = 0.25
L(B,A) = 0.75
L(B,C) = 0.25
L(C,A) = 0.75
L(C,B) = 0.25

At a damping factor d of 0.5, we get the following equations for the calculation of PageRank values:

PR(A) = 0.5 + 0.5 (0.75 PR(B) + 0.75 PR(C))
PR(B) = 0.5 + 0.5 (0.75 PR(A) + 0.25 PR(C))
PR(C) = 0.5 + 0.5 (0.25 PR(A) + 0.25 PR(B))

Solving these equations gives us the follwing PageRank values for our example:

PR(A) = 819/693
PR(B) = 721/693
PR(C) = 539/693

First of all, we see that page A has the highest PageRank of all three pages. This is caused by page A receiving the relatively higher evaluated link from page B as well as from page C.

Furthermore, we see that even by the evaluation of single links, the sum of the PageRank values of all pages equals 3 (2079/693) and thereby the total number of pages. So, the PageRank values computed by our modified PageRank algorithm can be used for the general ranking of web pages by Google without any normalisation being needed.

9. Additional Factors Influencing PageRank (continued)

This article reproduced with permission of eFactory.
© 2002 eFactory Internet-Agentur KG Online-Marketing - written by Markus Sobek
PageRank and Google are trademarks of Google Inc., Mountain ViewCA, USA.
PageRank is protected by US Patent 6,285,999.