1. PageRank Google : its search listings always seemed deliver the “good stuff” up front. 1 2 Part of the magic behind it is its PageRank Algorithm PageRank™ algorithm, developed by Google’s founders, Larry Page and Sergey Brin, when they were graduate students at Stanford University.

2. PageRank (Basic Idea) Importance Score: rating for webpage importance Importance score Suppose the web of interest contains n pages Each page indexed by an integer k 3 42 1 The importance score of page k on the web Indicate that page m is more important than page j Example:

3. PageRank (Basic Idea) 3 42 1 Number of links This approach ignore : a link from an important page Simple approach: Do they have same importance. Page 1 has link from page 3 (Page 3 has the maximum score)

4. PageRank (Basic Idea) 3 42 1 Let’s compute the score of page j as the sum of the scores of all pages linking to page j. Do they have same importance. Page 1 has link from page 3 (Page 3 has the maximum score) Just as in election:  a link to page k becomes a vote for page k’s importance  we don’t want a single indivisiual to gain influence merely by casting multiple votes

5. PageRank (Basic Idea) 3 42 1 Just as in election:  a link to page k becomes a vote for page k’s importance  we don’t want a single indivisiual to gain influence merely by casting multiple votes

6. PageRank (Basic Idea) 3 42 1 Problem: In linear algebra language: Find an eigenvector for a matrix A associated with eigenvalue Note that Page 3 is not the most important page

7. PageRank (Basic Idea) 3 42 1 In Numerical linear algebra language: Find an efficient computational algorithm to compute eigenvectors Google's PageRank is an eigenvector of a matrix of order 2.7 billion (May 2002) (a google blog post claimed in 2008) Difficulties & Features There are more than one linearly independent eigenvectors It is recomputed about once a month and does not involve any of the actual content of Web pages or of any individual query.

8. PageRank (Basic Idea) 3 42 1 In Numerical linear algebra language: Find an efficient computational algorithm to compute eigenvectors one way to compute the eigenvector x would be to start with a good approximate solution, such as the PageRanks from the previous month, and simply repeat the assignment statement (In Numerical: Continuation Method) The matrix A is sparse (tons of zeros) Difficulties & Features The matrix A. Its elements are all strictly between zero and one and its column sums are all equal to one. (Markov chain)