1. Presented By: - Chandrika B N
The PageRank Citation Ranking: Bringing Order to the Web Lawrence Page ,  Sergey Brin ,  Rajeev Motwani ,  Terry Winograd January 29th , 1998 Stanford InfoLab
Presented By:
- Chandrika B N

2. Agenda Technology Overview Introduction Link Structure of the Web
Simplified PageRank
Eigenvalue and Eigenvector
PageRank Definition
Random Surfer Model
Dangling Links
PageRank Implementation
Convergence
Searching with PageRAnk
Personalized PageRank
Application
Conclusion

3. Technology Overview Recognized the need for a new kind of server setup
Linked PC’s to quickly find each query’s answers
This resulted in: Faster Response Time
Greater Scalability
Lower costs
Google uses more than 200 signals (including PageRank algorithm) to determine which pages are important
Google then performs hypertext-matching
- Google Corporate Information

4. Life of a Google Query
- Google Corporate Information

5. The mechanism Web Crawler: Finds and retrieves pages on the web
Repository: web pages are compressed and stored here
Indexer: each index entry has a list of documents in which the term appears and the location within the text where it occurs

6. Introduction WWW is very large and heterogeneous
The web pages are extremely diverse
Problem:
How can the most relevant pages be ranked at the top?
Answer:
Take advantage of the link structure of the Web to produce ranking of every web page known as PageRank

7. Link Structure of the Web
A and B are Backlinks of C
Every page has some number of forward links (outedges) and backlinks (inedges)
We can never know all the backlinks of a page, but we know all of its forward links
Generally, highly linked pages are more “important”

8. A page is important if important pages refer to it
PageRank
PageRank - a method for computing a ranking for every web page based on the graph of the web
A page has high rank if the sum of the ranks of its backlinks is high
Page has many backlinks
Page has a few highly ranked backlinks
Page rank is a link analysis algorithm that assigns a numerical weight that represents how important a page is on the web
The web is democratic i.e., pages vote for pages
Google interprets a link from page A to page B as a vote, by page A, for page B.
It also analyses the page that cast the vote.
A page is important if important pages refer to it

9. Simple Ranking Function:
u: web page
Bu: backlinks
Nu = |Fu| number of links from u
c: factor used for normalization
Simplified PageRank Calculation
The PageRanks form a probability distribution over web pages, so the sum of all web pages’ PageRanks will be one

10. Eigenvalue and Eigenvector
Eigenvalues and Eigenvectors are properties of a matrix
In general, a matrix acts on a vector by changing both its magnitude and direction
However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged – Eigenvector
A matrix acts on an eigenvector by multiplying its magnitude by a factor called the Eigenvalue
Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation
In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x

11. Given a square matrix A, the eigenvalue eq can be expressed as
The eigenvector equation for A can be written as
Example
A =
λ is the eigenvalue
Solving this eq we get λ = 1 and λ = 3

12. Considering first the eigenvalue λ = 3, we have
After matrix-multiplication
This can be represented as 2 linear equations:
2x + y = 3x and x + 2y = 3y
The equations can be reduced to x = y
We can choose any value for x. Taking x=1, we get y=1
Eigenvector with eigenvalue 3
Eigenvector with eigenvalue 1

13. Computing PageRank given a Directed Graph
The Transition matrix A =
We get the eigenvalue λ = 1
Calculating the eigenvector

14. On substituting we get,
so the vector u is of the form
Choose v to be the unique eigenvector with the sum of all entries equal to 1
PageRank vector

15. Calculating the PageRank
Finding the Eigenvalue and Eigenvector Let Au,v = 1/Nu , if there is an edge from u to v 0, otherwise If R is a vector over the web pages, then R = cAR where , R: eigenvector of A c: eigenvalue
Problem: Rank Sink
Consider two web pages that point to each other but to no other page
Suppose there is some web page which points to one of them, then
During iteration, this loop will accumulate rank but will never distribute any rank
This forms a trap called the RANK SINK. This can be overcome by introducing a Rank Source

16. PageRank Definition:
Let E(u) be some vector over the Web pages that corresponds to a source of rank. Then, the PageRank of a set of Web pages is an assignment, R’, to the Web pages which satisfies
such that c is maximized and ||R’||1 = 1 (||R’||1 denotes the L1 norm of R’).
PageRank of document v
that links to u
Vector of web pages that the Surfer randomly jumps to
PageRank of document u
Normalization
factor
Number of outlinks from document v

17. Computing PageRank Loop: while S: any vector over the web pages
Calculate the Ri+1 vector using Ri
Calculate the normalizing factor
Find the vector Ri+1 using d
Find the norm of the difference of 2 vectors
while
Loop until convergence

18. Random Surfer Model
The “Random surfer” simply keeps clicking on successive links at random
A Real Web Surfer will unlikely continue in a loop forever
The surfer periodically “gets bored” and jumps to another random page

19. Dangling Links Links that point to any page with no outgoing links
They do not affect the ranking of any other page directly
Problem: It is not clear where their weight should be distributed
Solution: They can be removed from the system until all the PageRanks are calculated

20. PageRank Implementation
Convert each URL into a unique integer ID
Sort the link structure by ID
Remove the dangling links
Make an initial assignment of ranks
Iteratively compute PageRank until Convergence
Add the dangling links back
Recompute the rankings
NOTE: After adding the dangling links back, we need to iterate as many times as was required to remove the dangling links

21. Convergence PR (322 Million Links): 52 iterations
Scaling factor is roughly linear in logn

22. Convergence The web is an expander-like graph
A graph is said to be an expander if:
Every subset of nodes S has a neighborhood that is larger than some factor α times |S|
α is called the expansion factor
A graph has a good expansion factor if and only if the largest eigenvalue is sufficiently larger than the second-largest eigenvalue

23. Searching with PageRank
Two search engines:
Title-based search engine
Full text search engine
Searches only the “Titles”
Finds all the web pages whose titles contain all the query words
Sorts the results by PageRank
Very simple and cheap to implement
Title match ensures high precision, and PageRank ensures high quality
Called Google
Examines all the words in every stored document and also performs PageRank (Rank Merging)

24. Title-based search for University

25. Personalized PageRank
Important component of PageRank calculation is E
A vector over the web pages (used as source of rank)
Powerful parameter to adjust the page ranks
E vector corresponds to the distribution of web pages that a random surfer periodically jumps to
Having an E vector that is uniform over all the web pages results in some web pages with many related links receiving an overly high rank eg: copyright page or forums
General Search over the internet
Instead in Personalized PageRank E consists of a single web page

26. Applications Estimating Web Traffic
On analyzing the statistics, it was found that there are some sites that have a very high usage, but low PageRank.
eg: Links to pirated software
PageRank as Backlink Predictor
The goal is to try to crawl the pages in as close to the optimal order as possible i.e., in the order of their rank.
PageRank is a better predictor than citation counting
User Navigation: The PageRank Proxy
The user receives some information about the link before they click on it
This proxy can help users decide which links are more likely to be interesting

27. Conclusion
PageRank is a global ranking of all web pages base of their location in the Web’s graph structure
PageRank uses information which is external to the Web pages – backlinks
Backlinks from important pages are more significant than backlinks from average pages
The structure of the Web graph is very useful for information retrieval tasks.

28. References
L. Page, S. Brin, R. Motwani, T. Winograd. The PageRank Citation Ranking: Bringing Order to the Web, 1998
L. Page and S. Brin. The anatomy of a large-scale hypertextual web search engine, 1998
THE $25,000,000,000 EIGENVECTOR THE LINEAR ALGEBRA BEHIND GOOGLE by KURT BRYAN AND TANYA LEISE
Google Corporate Information: