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1 COMP 4601 Page Rank 1

2 Motivation Remember, we were interested in giving back the most relevant documents to a user. Importance is measured by reference as well as content. Think of this like academic paper citations; the more citations, the more important the paper. 2

3 The Problem Having people visit your site is the basis of economic value in the Web. The more time spent on your site, the more valuable it is. So, given high value word; e.g., Bieber, we want to attract users to our page by using valuable terms. So, the temptation is to SPAM. 3

4 SPAM Crawler Page with real content Page returned to a browser SPAM Page 4

5 The SPAM Problem To a crawler, return content which says all the right things; e.g., have the word Bieber a large number of times, place it in important locations such as title, and headings. Indexed page scores highly for Bieber but when browser requests page return your advertising or irrelevant material. 5

6 Early Search Engines Early search engines were driven purely by content of the page. No citation-based dependence. Modern search engines use link-based measures to prevent spam; if no-one links to you page it will never score highly. 6

7 Aside: Anchor Text Remember, we said that titles were important. Think of anchor text: anchor text is descriptive of the content of a page. Furthermore, the anchor text is provided on a linked page, not the page itself. <a href=" of the ACM.</a> Anchor text 7

8 The Gap Clearly, there s a gap between terms on a page and how users describe that page. That s why alt text and anchor text is useful in indexing. Is it infallible? NO! We should use machine learning to determine the importance of terms. More on machine learning later. 8

9 Background History: Proposed by Sergey Brin and Lawrence Page in 1998 at Stanford. Algorithm of the first generation of Google Search Engine. The Anatomy of a Large-Scale Hypertextual Web Search Engine. Target: Measure importance of Web page based on link structure alone. Assign each node numerical score between 0 and 1: PageRank. Rank Web pages based on PageRank values.

10 Page Rank Linkage represents interest or confidence in content. If I link to a page, I m indicating an interest in it. If others link to the page that s even better! My contribution to the importance of a node depends upon my importance. 10

11 Consider the following A B C Imagine walking randomly. We ll visit some nodes more often than others. This will depend upon the number of incoming links. D 11

12 But no out links? Hmm, there s a problem. What if a node has no out-links? Introduce a teleport operation. This is equivalent to typing a URL into the browser. So, for N nodes, the teleport operation occurs with probability 1/N. 12

13 The PageRank Score The score relies on: With a node with links, I teleport with probability α (0 < α < 1) I follow an outlink with probability 1-α So, what s α? Typical value might be

14 Calculating Page Rank The random walk through the graph forms what is known as a Markov chain. A Markov chain is a discrete-time stochastic process. 14

15 The question is: What are the steady state probabilities of being in a particular state. It can be shown that an ergodic Markov chain converges, given enough time. Ergodic means: Probability of transition doesn t depend on where I ve been ( memoryless ) All states are accessible 15

16 The question is: This is what we want to calculate: it s the PageRank score. 16

17 Back to the surfer Random surfing maps onto a Markov chain. So for an N page web, we have an NxN matrix. Use the adjacency matrix, formulation for the graph. (Aside, there s a 1 if there s an edge, 0 otherwise) 17

18 Remaking the Adjacency Matrix If a row of the adjacency matrix has no 1's, then replace each element by 1/N. For all other rows proceed as follows 18

19 Remaking the Adjacency Matrix Divide each 1 in by the number of 1's in its row. Thus, if there is a row with three 1's, then each of them is replaced by 1/3. Multiply the resulting matrix by 1-α. Add α/n to every entry of the resulting matrix, to obtain our probability matrix. 19

20 Exercise on PageRank Consider a Web graph with three nodes 1, 2, and 3. The links are as follows: 1->2, 3->2, 2->1, 2->3. Write down the transition probability matrices P for the surfers walk with teleporting, with the value of teleport probability α= A = (1- α)* α* ½ 0 ½ /3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 = Each 1 divided by the number of ones in this row 1/6 2/3 1/6 5/12 1/6 5/12 1/6 2/3 1/6

21 Calculating PageRank Let steady state probabilities be a vector π. Then: π P = π This is true for each iteration, so : π P 2 = π, π P 3 = π, To solve, then, we just keep multiplying until the difference between the values of π at 1 iteration and the next are very 21 small.

22 An Example Say our surfer starts at page 1 (top left corner of A) Starts at: (1, 0, 0) 22

23 An Example After 1 iteration After 2 iterations 23

24 Converges? YES! Steady state probabilities; these are the PageRank scores. 24

25 Properties PageRank scores sum to 1. Well, they are probabilities When calculating PageRank this is good to check 25

26 Another Example 26

27 The Transition Matrix 27

28 The Transition Matrix Ranking: d 6, d 3, d 4, d 2, d 0, d 1,d 5 28

29 Summary: PageRank Computation Target Solve the steady-state probability vector π, which is the PageRank of the corresponding Web page. πp=λ π, λ is 1 for stochastic matrix. Method Power iteration. Given an initial probability distribution vector x 0 x 0 *P=x 1, x 1 *P=x 2 Until the probability distribution converges. (Variation in the computed values are below some predetermined threshold.)

30 Reference Google Seminal paper: JAMA: A Java matrix package 30

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