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

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

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

Life of a Google Query - Google Corporate Information

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

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

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”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

Title-based search for University

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

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

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.

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: http://www.google.com/corporate/tech.html http://en.wikipedia.org/wiki/PageRank http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace http://www.googleguide.com/google_works.html http://www.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture3/lecture3.html http://pr.efactory.de/