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

8. 8
PageRank example

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)