Pagerank Paper ( the $25,000,000,000 Eigenvector )

PageRank paper (“The $25,000,000,000 eigenvector”)

1. Mike 21 min Introduction + Sec. 2.1 + Sec. 2.2.1 + Exercise 1.

2. Jian Zhu 18 min Sec. 2.2.2 + My posted notes on dangling nodes + Exercises 3, 4. For Exercise 3, compute all lambdas. Why are there more than one eigenvalue with |lambda| =1? For Exercise 4, conclude what happens to the maximum eigenvalue in the presence of dangling nodes.

Note : Material on dangling nodes can also be found in pp. 36, 37, and 80 of the photocopied pages of Langville—Meyer’s book “Google’s PageRank and beyond” and in the two other posted papers, “Deeper inside PageRank” and “Goodle’s PageRank: the math behind the search engine”. You may look into those sources, but do not really have to. If you do look, note that the correspondence of notations in the book and the PageRank paper is: 1) HT (book) = A (paper) (in general, all matrices or vectors in the book are transposes of the corresponding matrices or vectors in the paper); 2) e (book) is the column vector all of whose entries equal 1; 3) GT (book) = M (paper), provided that there are no dangling nodes; 4) alpha (book) = (1-m) (paper).

3. Tim 17 min Sec. 3.1. + Proposition 2. Where will the proof of Proposition 2 go wrong if M is not strictly positive? Also, mention what happens to the ranking as m tends to 1

4. Zuzana 16 min Proposition 3 + Lemma 3. Where in the proof of Lemma 3 is it required that M be strictly positive? Why is Lemma 3 important for the rest of the paper?

5. Matt D 19 min Exercises 7, 8, 10. topic 9, described at the end, will go here: Patrick 15 min

6. Joshuah 16 min Proposition 4. Emphasize that the main point of this Proposition is not the precise estimate of c, but the facts that if M is as stated in that Proposition, then all eigenvalues except lambda1 satisfy lambda_k < 1, for k=2, 3, … . 7. Cole 15 min Proposition 5 + Example 3 + the statement only of Theorem 4.7.1. in Langville—Meyer’s book (p. 46). (See the Note about the notations in topic 2.) The statement of the theorem must be presented in the notations of the PageRank paper, not Langville—Meyer’s book. Conclude with a punch line: As m increases, the power method will converge faster.

8. Matt F 16 min From the paragraph before Theorem 5 to the end of paper + Exercises 11, 14. Explain how Theorem 5 summarizes all the previous results in this paper (i.e., point to each statement it refers to and briefly remind who talked about it and what that person concluded). Discuss how Exercise 14 agrees with the statement of Theorem 4.7.1 in Langville—Meyer’s book. Also, what is the difference of this Exercise from Example 3, and why does it occur? State and explain the answer to Exercise 17.

9. (ideally, this topic should go between topics 5 and 6) Patrick Why is the power method used in PageRank calculations? See the first paragraph of Sec. 4 of the PageRank paper + pp. 40, 41 of Langville—Meyer’s book. (See the Note for topic 2 regarding the notations.) + Idea of the power method (see, e.g., the posted p. 239 of “Numerical analysis” by J. Leader) . How does the convergence of the power method depend on the ratio (|lambda2/lambda1|)? From you answer to the previous question, what will we need to prove about lambda2 of matrix M to ensure that the power method for the PageRank computation will converge?