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Does Google uses the theory from 1800’s to rank a Site ??
*Wednesday, February 7, 2007*

*Posted by sarasotapodcast in Blogroll, Feed, My Favorite Tags, Sarasota Podcast.*

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**Does Google uses the theory from 1800’s to rank a Site ??
**

While I was going thru the Google Algorithm and ranking theory, came up with some fascinating facts and interesting theories used to rank a website. Find all the related links and resources below

**Perron-Frobenius theorem:**

**Perron–Frobenius theorem**, named after Oskar Perron and Ferdinand Georg Frobenius, is a theorem in matrix theory about the eigenvalues and eigenvectors of a real positive n×n matrix.

Perron-Frobenius theorem has particular use in algebraic graph theory. The “underlying graph” of a nonnegative real matrix is the graph with vertices and arc ij if and only if . If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the generalized Perron-Frobenius theorem applies.

**Google page rank:**

PageRank relies on the uniquely democratic nature of the web by using its vast link structure as an indicator of an individual page’s value. In essence, Google interprets a link from page A to page B as a vote, by page A, for page B. But, Google looks at more than the sheer volume of votes, or links a page receives; it also analyzes the page that casts the vote. Votes cast by pages that are themselves “important” weigh more heavily and help to make other pages “important.

For more info: http://www.google.com/technology/

Please let me know your thoughts and comments, even thou Math is not my subject, definitely love to hear from people who are good at Math.

**References:**

- http://209.85.165.104/search?q=cache:PjNaBazSGRYJ:www.chachadays.org/Govan-ChaCha06.pdf+Perron%E2%80%93Frobenius+theorem+and+google&hl=en&ct=clnk&cd=8&gl=us
- http://www.gnxp.com/MT2/archives/001364.html
- http://pdfdl.oceighty.net/pdf2html.php?url=http://people.ccmr.cornell.edu/~ginsparg/INFO295/pr.pdf
**http://www9.org/w9cdrom/251/251.html**

##### Theorem 2.1 (Perron-Frobenius Theorem )

Source: http://www9.org/w9cdrom/251/251.html

If a n-dimensional matrix is positive or non-negative irreducible, then

- The spectrum radius of , , is also a latent root of ;
- There is a positive eigenvector of corresponding to ;
- The eigenfunction of has a single root , i.e., ,
- ��

Although the rank algorithms based on linkage structure break through the limitation of traditional IR technology, they still have some shortcomings:

- Only the authority metric has been taken into account;
- The iterative computation results in bad performance;
- It is difficult to deal with the overflow or underflow problem during iteration;
- Because of the “rank sink problem”, the iterative computation may not converge.