1. Google’s PageRank By Zack Kenz

2. Outline Intro to web searching Review of Linear Algebra Weather example Basics of PageRank Solving the Google Matrix Calculating the PageRank Wrapping up

3. Some Search Engine History Early basis of searching was on page content only Bonuses for word placement Paying for placement Natural language searches (Think: Ask Jeeves) Meta search engines

4. Why Google? No one exploited the link structure of the internet Relatively easy to exploit content-based engines with concealed text Adaptive to a growing internet Simpler, faster

5. PageRank, According to Google “PageRank relies on the uniquely democratic nature of the web by using its vast link structure as an indicator of an individual page's value. In essence, Google interprets a link from page A to page B as a vote, by page A, for page B.” “Google looks at considerably more than the sheer volume of votes, or links a page receives; for example, it also analyzes the page that casts the vote. Votes cast by pages that are themselves ‘important’ weigh more heavily and help to make other pages ‘important.’”

6. Linear Algebra Terms Row Stochastic Matrix Eigenvector: A nonzero vector x such that Ax=λx for a scalar λ Eigenvalue: A scalar λ that gives a nontrivial solution x for Ax=λx Dominant Eigenvalue (eigenvector)

7. Tomorrow’s Weather Example on the board

8. Scoring Web Pages Random web surfer Goal: Assign a score to over 25 billion web pages, store the scores Score based on the probability of going to a particular page

9. Surf’s Up!

10. Hyperlink Matrix

12. Dangling Nodes

15. Web Link Surfer Matrix

16. One More Fix Need to account for the fact that a surfer can type in URLs instead of using links Add in a personalization vector, When multiplied by a column vector of ones, we get an additional personalization matrix

17. One More Fix Need to account for the fact that a surfer can type in URLs instead of using links Add in a personalization vector, When multiplied by a column vector of ones, we get an additional personalization matrix

18. Google Matrix Recall is a damping factor, usually.85

19. The True Google Matrix?

20. Solution of the Google Matrix Since the Google matrix is row stochastic, it has an eigenvalue of λ=1 λ=1 is biggest and not repeated Let be the corresponding eigenvector The eigensystem has a unique solution for, then, is a row probability vector

21. Solution of the Google Matrix Since the Google matrix is row stochastic, it has an eigenvalue of λ=1 λ=1 is biggest and not repeated Let be the corresponding eigenvector The eigensystem has a unique solution for, then, is a row probability vector contains every page’s PageRank

22. Computing Scores: The Linear Algebra Way Recall

23. λ = 1 is the dominant eigenvalue of G and is the dominant left eigenvector As a result the power method applied to G converges to the PageRank vector Given a starting vector like, the power method calculates successive iterates until a stopping condition is reached Computing Scores: The Power Method

24. Speeding Things Up

25. Wrapping Up: The Overall Page Scoring PageRank is still only a portion of what determines the order of search results Results are based off of many factors, especially page content

26. Wrapping Up: Improving PageRank Avoiding link spamming – tweak the personalization vector and α Power method convergence algorithms Dummy node

27. Questions?

28. Rebecca S Wills. Google’s PageRank: The Math Behind the Search Engine. Department of Mathematics, North Carolina State University. 1 May 2006. Amy N. Langville and Carl D. Meyer. Fiddling with PageRank. Department of Mathematics, North Carolina State University. 15 August 2003 http://www.searchenginehistory.com/ http://www.google.com/technology/ and http://www.google.com http://www.google.com/technology/http://www.google.com David C Lay. Linear Algebra and Its Applications, 3ed. Pearson Education: 2003. Dr. Biebighauser http://eperformance.co.uk/uploaded_images/google%20beta-786468.jpg http://webmechanics.uoregon.edu/Images/Surf%20web.jpg http://www.modmyifone.com/iphone_wallpapers/file.php?n=282&w=l http://www.smashingmagazine.com/images/pagerank/google- pagerank.jpg http://www.smashingmagazine.com/images/pagerank/google- pagerank.jpg Sources