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The PageRank Citation Ranking: Bringing Order to the Web Lawrence Page, Sergey Brin, Rajeev Motwani, Terry Winograd January 29th, 1998 Stanford InfoLab Adaptive methods for the computation of PageRank Sepandar Kamvar, Taher Haveliwala, Gene Golub 2004 Presented By: Wang Hao March 8th, 2011 Agenda Technology Overview Introduction & Motivation Link Structure of the Web Simplified PageRank PageRank Definition How we can get PageRank Dangling Links PageRank Implementation Adaptive Methods for computation of PageRank Searching with PageRnnk Personalized PageRank Application Conclusion References Technology Overview Recognized the need for a new kind of server setup Linked PCs 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 Introduction & Motivation WWW is very large and heterogeneous The web pages are extremely diverse in terms of content, quality and structure Challenging for information retrieval on WWW Academic Citations link to other well known papers But they are peer reviewed and have quality control Web of academic documents are homogeneous in their quality, usage, citation & length Cont’d Most web pages link to web pages as well Quality measure of a web page is subjective to the user though Importance of a page is a quantity that isn’t intuitively possible to capture 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” PageRank Definition • 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 A page is important if important pages refer to it • PageRank 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. Cont’d Simple Ranking Function: u: web page Bu: backlinks Nu = |Fu| number of links from u c: factor used for normalization Simplified PageRank Calculation In principle, the PageRanks form a probability distribution over web pages, so the sum of all web pages’ PageRanks will be one Computing PageRank given a Directed Graph The transition matrix A = We get the eigenvalue λ = 1 Calculating the eigenvector Cont’d 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 How we can get PageRank It is a Markov chain. Set the probability distribution at time 0: X0 Set one-step transition probability matrix: A What we would like to get is the unique stationary distribution of the Markov chain: by successively iterating until convergence that This is the principal eigenvector of the matrix A, which is exactly the PageRank vector. Problem 1: Dangling Links Dangling links are links that point to any page with no outgoing links or pages not downloaded yet. Problem : how their weights should be distributed. Solution 1: they are removed from the system until all the PageRanks are calculated. Afterwards, they are added in without affecting things significantly. Problem 1: Dangling Links (cont’d) Solution 2 (presented in the second paper): Let v be a vector representing a uniform distribution over all nodes In terms of the random walk, the effect of D is to modify the transition probabilities so that a surfer visiting a dangling page randomly jumps to another page in the next time step, using the distribution given by v. Problem 2: Rank Sink Problem: Some pages form a loop that accumulates rank (rank sink) to the infinity. Solution: Random Surfer Model Jump to a random page based on some distribution E (rank source) Convergence and Random Walks : Why does it work? Irreducible Aperiodic Markov Chains with a Primitive transition probability matrix What are the issues all about? We need a transition matrix model that can guarantee convergence and does indeed converge to a unique stationary distribution vector. PageRank Expression: 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 PageRank of document v Vector of web PageRank of that links to u pages that the document u Surfer randomly jumps to u Normalization Number of outlinks factor from document v such that c is maximized and ||R’||1 = 1 (||R’||1 denotes the L1 norm of R’). Computing PageRank R0 S S: any vector over the web pages Loop: i1 R AR i Calculate the Ri+1 vector using Ri i1 Calculate the normalizing factor i 1 d RR 1 i i E RR Find the vector Ri+1 using d 1 d 1 RR i1 i Find the norm of the difference of 2 vectors while Loop until convergence 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 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 Convergence PR (322 Million Links): 52 iterations PR (161 Million Links): 45 iterations Scaling factor is roughly linear in logn Adaptive Methods for the computation of PageRank This paper presents two contributions: First, it shows that most pages in the web converge to their true PageRank quickly, while relatively few pages take much longer to converge. And it further shows that those slow-converging pages generally have high PageRank, and those pages that converge quickly generally have low PageRank. Experimental results supports the findings: Experimental results Adaptive Algorithms Second, the authors develop two algorithms, called Adaptive PageRank and Modified Adaptive PageRank, that exploit this observation to speed up the computation of PageRank by 18% and 28%, respectively. The main ideas of the all the proposed algorithms are the same, which is to speed up the computation of PageRank by reducing the cost (not computing the PageRank of converged pages at each iteration). Notations to be included: A: one-step transition probability matrix. x(k): probability distribution vector at time k. N = not yet converged; C = converged. Adaptive PageRank Filter-based Adaptive PageRank Reordering the matrix A at each iteration is expensive. Reducing the cost by introducing sparse (zero) entries. Filter-based Modified Adaptive PageRank Reducing redundant computation by not recomputing the components of the PageRanks of those pages in N due to links from those pages in C. Split A even further. Performance comparison Searching with PageRank • Two search engines: – Title-based search engine – Full text search engine • Title-based 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 • Full text search engine – Called Google – Examines all the words in every stored document and also performs PageRank (Rank Merging) – More precise but more complicated 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 e.g.: 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. e.g.: 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 according to an evaluation func. 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 based on their locations in the web 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 Sepandar Kamvar, Taher Haveliwala, Gene Golub, Adaptive methods for the computation of PageRank, Linear Algebra and its Applications 386, 2004 Published by Elsevier Inc., pp 51–65. 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/ Thank You! Q&A

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