![4
Simple PageRank
Pi
is a page
r: page → [0,1] is the PageRank function
BPi
is the set of pages pointing to Pi (backlinks)
|Pj
| is the number of outlinks pointing from Pj
Calculate by iterating:](https://image.slidesharecdn.com/b2be5449-bebf-4ab8-a754-1c5cdefc0298-160522200857/85/The-Maths-behind-Web-search-engines-4-320.jpg)
The Maths behind Web search engines
- 1. 1 The Maths behind Web search engines (PageRank) 2010 Dante Vsevolod Zubov
- 2. 2 Web search in a nutshell Storing pages Use depth first search to discover new pages (crawling) Ranking results Human generated ranking − Yahoo! Directory Automated ranking − Text − Meta data (title, keywords etc.)
- 3. 3 Motivation for the PageRank prioritization of results is crucial PageRank uses the link structure of the Web to rank pages by their “importance” named after Larry Page (co-founder of Google)
- 4. 4 Simple PageRank Pi is a page r: page → [0,1] is the PageRank function BPi is the set of pages pointing to Pi (backlinks) |Pj | is the number of outlinks pointing from Pj Calculate by iterating:
- 5. 5 Simple PageRank Will r converge in all cases? What about pages with no outlinks?
- 6. 6 Random Surfer model • on any page, random surfer will follow one of the outlinks at random with some probability d (damping factor, usually taken as 0.85); • random surfer will get bored and select some page from entire Web at random with probability (1-d) ; • if the page does not have outlinks then random surfer is “teleported” to some random page on the Web.
- 7. 7 Adjusted PageRank BPi now includes all of the sink pages r a discrete probability distribution
- 9. 9 Matrix representation The link structure of the Web can be represented by an adjacency matrix Define
- 10. 10 Matrix representation The matrix M on the right is column normalized version of the adjacency matrix we saw earlier. l (Pi , Pj )= 0 if Pj does not link to Pi l (Pi , Pj )= 1/|Pj | if Pj links to Pi
- 11. 11 existence and uniqueness of R Let E be the N by N matrix with all its elements equal to 1 then ER = 1. Call the matrix in the middle M' then R is a 1-eigenvector of M' M' is a stochastic matrix By Perron–Frobenius theorem R does in fact exist and is unique
- 12. 12 computation of R Algebraic method Iterative method repeat iteration until
- 13. 13 Questions? References: Brin, S. and Page, L. (1998) The Anatomy of a Large-Scale Hypertextual Web Search Engine. Google's PageRank and Beyond, Langville & Meyer (2006), Chapter 4 http://en.wikipedia.org/wiki/PageRank (11/2010)