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# PageRank and Markov Chain

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A brief introduction to the methodology used by PageRank to rank the webpages.

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### PageRank and Markov Chain

1. 1. Markov Chains as methodology used by PageRank to rank the Web Pages on Internet. Sergio S. Guirreri - www.guirreri.host22.com Google Technology User Group (GTUG) of Palermo. 5th March 2010 Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 1 / 14
2. 2. Overview 1 Concepts on Markov-Chains. 2 The idea of the PageRank algorithm. 3 The PageRank algorithm. 4 Solving the PageRank algorithm. 5 Conclusions. 6 Bibliography. 7 Internet web sites. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 2 / 14
3. 3. Concepts on Markov-Chains. Stochastic Process and Markov-Chains. Let assume the following stochastic process {Xn; n = 0, 1, 2, . . . } with values in a set E, called the state space, while its elements are called state of the process. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14
4. 4. Concepts on Markov-Chains. Stochastic Process and Markov-Chains. Let assume the following stochastic process {Xn; n = 0, 1, 2, . . . } with values in a set E, called the state space, while its elements are called state of the process. Let assume the set E is ﬁnite or countable. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14
5. 5. Concepts on Markov-Chains. Stochastic Process and Markov-Chains. Let assume the following stochastic process {Xn; n = 0, 1, 2, . . . } with values in a set E, called the state space, while its elements are called state of the process. Let assume the set E is ﬁnite or countable. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14
6. 6. Concepts on Markov-Chains. Stochastic Process and Markov-Chains. Let assume the following stochastic process {Xn; n = 0, 1, 2, . . . } with values in a set E, called the state space, while its elements are called state of the process. Let assume the set E is ﬁnite or countable. Deﬁnition A Markov Chain is a stochastic process Xn that hold the following feature: Prob{Xn+1 = j|Xn = i, Xn−1 = in−1, . . . , X0 = i0} = = Prob{Xn+1 = j|Xn = i} = pij(n) where E is the state space set and j, i, in−1, . . . , i0 ∈ E, n ∈ N. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14
7. 7. Concepts on Markov-Chains. Stochastic Process and Markov-Chains. Let assume the following stochastic process {Xn; n = 0, 1, 2, . . . } with values in a set E, called the state space, while its elements are called state of the process. Let assume the set E is ﬁnite or countable. Deﬁnition A Markov Chain is a stochastic process Xn that hold the following feature: Prob{Xn+1 = j|Xn = i, Xn−1 = in−1, . . . , X0 = i0} = = Prob{Xn+1 = j|Xn = i} = pij(n) where E is the state space set and j, i, in−1, . . . , i0 ∈ E, n ∈ N. The transition probability matrix P of the process Xn is composed of pij, ∀i, j ∈ E. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 3 / 14
8. 8. The idea of the PageRank algorithm. PageRank’s idea. The idea behind the PageRank algorithm is similar to the idea of the impact factor index used to rank the Journals [Page et al.(1999)] [Brin and Page(1998)] [Langville et al.(2008)]. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 4 / 14
9. 9. The idea of the PageRank algorithm. PageRank’s idea. The idea behind the PageRank algorithm is similar to the idea of the impact factor index used to rank the Journals [Page et al.(1999)] [Brin and Page(1998)] [Langville et al.(2008)]. PageRank the impact factor of Internet. The impact factor of a journal is deﬁned as the average number of citations per recently published papers in that journal. By regarding each web page as a journal, this idea was then extended to measure the importance of the web page in the PageRank Algorithm. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 4 / 14
10. 10. The idea of the PageRank algorithm. Elements of the PageRank. To illustrate the PageRank algorithm I deﬁne the following variables [Ching and Ng(2006)]: let be N the total number of web pages in the web. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14
11. 11. The idea of the PageRank algorithm. Elements of the PageRank. To illustrate the PageRank algorithm I deﬁne the following variables [Ching and Ng(2006)]: let be N the total number of web pages in the web. let be k the outgoing links of web page j. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14
12. 12. The idea of the PageRank algorithm. Elements of the PageRank. To illustrate the PageRank algorithm I deﬁne the following variables [Ching and Ng(2006)]: let be N the total number of web pages in the web. let be k the outgoing links of web page j. let be Q the so called hyperlink matrix with elements: Qij =    1 k if web page i is an outgoing link of web page j; 0 otherwise; Qi,i > 0 ∀i. (1) Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14
13. 13. The idea of the PageRank algorithm. Elements of the PageRank. To illustrate the PageRank algorithm I deﬁne the following variables [Ching and Ng(2006)]: let be N the total number of web pages in the web. let be k the outgoing links of web page j. let be Q the so called hyperlink matrix with elements: Qij =    1 k if web page i is an outgoing link of web page j; 0 otherwise; Qi,i > 0 ∀i. (1) Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14
14. 14. The idea of the PageRank algorithm. Elements of the PageRank. To illustrate the PageRank algorithm I deﬁne the following variables [Ching and Ng(2006)]: let be N the total number of web pages in the web. let be k the outgoing links of web page j. let be Q the so called hyperlink matrix with elements: Qij =    1 k if web page i is an outgoing link of web page j; 0 otherwise; Qi,i > 0 ∀i. (1) The hyperlink matrix Q can be regarded as a transition probability matrix of a Markov chain. One may regard a surfer on the net as a random walker and the web pages as the states of the Markov chain. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 5 / 14
15. 15. The PageRank algorithm. The PageRank with irreducible Markov Chain. Assuming that the Markov chain is irreduciblea and aperiodicb then the steady-state probability distribution (p1, p2, . . . , pN )T of the states (web pages) exists. aA Markov chain is irreducible if all states communicate with each other. bA chain is periodic if there exists k > 1 such that the interval between two visits to some state s is always a multiple of k. Therefore a chain is aperiodic if k=1. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14
16. 16. The PageRank algorithm. The PageRank with irreducible Markov Chain. Assuming that the Markov chain is irreduciblea and aperiodicb then the steady-state probability distribution (p1, p2, . . . , pN )T of the states (web pages) exists. aA Markov chain is irreducible if all states communicate with each other. bA chain is periodic if there exists k > 1 such that the interval between two visits to some state s is always a multiple of k. Therefore a chain is aperiodic if k=1. The PageRank Each pi is the proportion of time that the surfer visiting the web page i. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14
17. 17. The PageRank algorithm. The PageRank with irreducible Markov Chain. Assuming that the Markov chain is irreduciblea and aperiodicb then the steady-state probability distribution (p1, p2, . . . , pN )T of the states (web pages) exists. aA Markov chain is irreducible if all states communicate with each other. bA chain is periodic if there exists k > 1 such that the interval between two visits to some state s is always a multiple of k. Therefore a chain is aperiodic if k=1. The PageRank Each pi is the proportion of time that the surfer visiting the web page i. The higher the value of pi is, the more important web page i will be. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14
18. 18. The PageRank algorithm. The PageRank with irreducible Markov Chain. Assuming that the Markov chain is irreduciblea and aperiodicb then the steady-state probability distribution (p1, p2, . . . , pN )T of the states (web pages) exists. aA Markov chain is irreducible if all states communicate with each other. bA chain is periodic if there exists k > 1 such that the interval between two visits to some state s is always a multiple of k. Therefore a chain is aperiodic if k=1. The PageRank Each pi is the proportion of time that the surfer visiting the web page i. The higher the value of pi is, the more important web page i will be. The PageRank of web page i is then deﬁned as pi. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 6 / 14
19. 19. The PageRank algorithm. The PageRank with reducible Markov Chain Since the matrix Q can be reducible to ensure that the steady-state probability exists and is unique the following matrix P must be considered: P = α     Q11 Q12 . . . Q1N Q21 Q22 . . . Q2N . . . . . . . . . . . . QN1 QN2 . . . QNN     + (1 − α) N     1 1 . . . 1 1 1 . . . 1 . . . . . . . . . . . . 1 1 . . . 1     (2) Where 0 < α < 1 and the most popular values of α are 0.85 and (1 − 1/N). Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 7 / 14
20. 20. The PageRank algorithm. The PageRank with reducible Markov Chain Since the matrix Q can be reducible to ensure that the steady-state probability exists and is unique the following matrix P must be considered: P = α     Q11 Q12 . . . Q1N Q21 Q22 . . . Q2N . . . . . . . . . . . . QN1 QN2 . . . QNN     + (1 − α) N     1 1 . . . 1 1 1 . . . 1 . . . . . . . . . . . . 1 1 . . . 1     (2) Where 0 < α < 1 and the most popular values of α are 0.85 and (1 − 1/N). Interpretation of PageRank The idea of the PageRank (2) is that, for a network of N web pages, each web page has an inherent importance of (1 − α)/N. If a page Pi has an importance of pi, then it will contribute an importance of α pi which is shared among the web pages that it points to. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 7 / 14
21. 21. The PageRank algorithm. The PageRank with reducible Markov Chain Solving the following linear system of equations subject to the normalization constraint one can obtain the importance of web page Pi :      p1 p2 ... pN      = α     Q11 Q12 . . . Q1N Q21 Q22 . . . Q2N . . . . . . . . . . . . QN1 QN2 . . . QNN          p1 p2 ... pN      + (1 − α) N      1 1 ... 1      (3) Since N i=1 pi = 1 the (3) can be rewritten as (p1, p2, . . . , pN )T = P(p1, p2, . . . , pN )T Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 8 / 14
22. 22. Solving the PageRank algorithm. The power method. The power method is an iterative method for solving the dominant eigenvalue and its corresponding eigenvectors of a matrix. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14
23. 23. Solving the PageRank algorithm. The power method. The power method is an iterative method for solving the dominant eigenvalue and its corresponding eigenvectors of a matrix. Given an n × n matrix A, the hypothesis of power method are: there is a single dominant eigenvalue. The eigenvalues can be sorted: |λ1| > |λ2| ≥ |λ3| ≥ . . . |λn| Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14
24. 24. Solving the PageRank algorithm. The power method. The power method is an iterative method for solving the dominant eigenvalue and its corresponding eigenvectors of a matrix. Given an n × n matrix A, the hypothesis of power method are: there is a single dominant eigenvalue. The eigenvalues can be sorted: |λ1| > |λ2| ≥ |λ3| ≥ . . . |λn| there is a linearly independent set of n eigenvectors: {u(1) , u(2) , . . . , u(n) } Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14
25. 25. Solving the PageRank algorithm. The power method. The power method is an iterative method for solving the dominant eigenvalue and its corresponding eigenvectors of a matrix. Given an n × n matrix A, the hypothesis of power method are: there is a single dominant eigenvalue. The eigenvalues can be sorted: |λ1| > |λ2| ≥ |λ3| ≥ . . . |λn| there is a linearly independent set of n eigenvectors: {u(1) , u(2) , . . . , u(n) } Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14
26. 26. Solving the PageRank algorithm. The power method. The power method is an iterative method for solving the dominant eigenvalue and its corresponding eigenvectors of a matrix. Given an n × n matrix A, the hypothesis of power method are: there is a single dominant eigenvalue. The eigenvalues can be sorted: |λ1| > |λ2| ≥ |λ3| ≥ . . . |λn| there is a linearly independent set of n eigenvectors: {u(1) , u(2) , . . . , u(n) } so that Au(i) = λiu(i) , i = 1, . . . , n. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 9 / 14
27. 27. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
28. 28. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) iterating the initial vector with the A matrix: Ak x(0) = a1Ak u(1) + a2Ak u(2) + · · · + anAk u(n) Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
29. 29. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) iterating the initial vector with the A matrix: Ak x(0) = a1Ak u(1) + a2Ak u(2) + · · · + anAk u(n) = a1λk 1u(1) + a2λk 2u(2) + · · · + anλk nu(n) . Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
30. 30. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) iterating the initial vector with the A matrix: Ak x(0) = a1Ak u(1) + a2Ak u(2) + · · · + anAk u(n) = a1λk 1u(1) + a2λk 2u(2) + · · · + anλk nu(n) . dividing by λk 1 Ak x(0) λk 1 = a1u(1) + a2 λ2 λ1 k u(2) + · · · + an λn λ1 k u(n) , Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
31. 31. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) iterating the initial vector with the A matrix: Ak x(0) = a1Ak u(1) + a2Ak u(2) + · · · + anAk u(n) = a1λk 1u(1) + a2λk 2u(2) + · · · + anλk nu(n) . dividing by λk 1 Ak x(0) λk 1 = a1u(1) + a2 λ2 λ1 k u(2) + · · · + an λn λ1 k u(n) , Since |λi| |λ1| < 1 → Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
32. 32. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) iterating the initial vector with the A matrix: Ak x(0) = a1Ak u(1) + a2Ak u(2) + · · · + anAk u(n) = a1λk 1u(1) + a2λk 2u(2) + · · · + anλk nu(n) . dividing by λk 1 Ak x(0) λk 1 = a1u(1) + a2 λ2 λ1 k u(2) + · · · + an λn λ1 k u(n) , Since |λi| |λ1| < 1 → lim k→∞ |λi|k |λ1|k = 0 → Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
33. 33. Solving the PageRank algorithm. The power method. The initial vector x0 can be wrote: x(0) = a1u(1) + a2u(2) + · · · + anu(n) iterating the initial vector with the A matrix: Ak x(0) = a1Ak u(1) + a2Ak u(2) + · · · + anAk u(n) = a1λk 1u(1) + a2λk 2u(2) + · · · + anλk nu(n) . dividing by λk 1 Ak x(0) λk 1 = a1u(1) + a2 λ2 λ1 k u(2) + · · · + an λn λ1 k u(n) , Since |λi| |λ1| < 1 → lim k→∞ |λi|k |λ1|k = 0 → Ak ≈ a1λk 1u(1) Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 10 / 14
34. 34. Conclusions. The power method and PageRank. Results. The matrix P of the PageRank algorithm is a stochastic matrix therefore the largest eigenvalue is 1. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14
35. 35. Conclusions. The power method and PageRank. Results. The matrix P of the PageRank algorithm is a stochastic matrix therefore the largest eigenvalue is 1. The convergence rate of the power method depends on the ratio of λ2 λ1 . Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14
36. 36. Conclusions. The power method and PageRank. Results. The matrix P of the PageRank algorithm is a stochastic matrix therefore the largest eigenvalue is 1. The convergence rate of the power method depends on the ratio of λ2 λ1 . It has been showed by [Haveliwala and Kamvar(2003)] that for the second largest eigenvalue of P, we have |λ2| ≤ α 0 ≤ α ≤ 1. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14
37. 37. Conclusions. The power method and PageRank. Results. The matrix P of the PageRank algorithm is a stochastic matrix therefore the largest eigenvalue is 1. The convergence rate of the power method depends on the ratio of λ2 λ1 . It has been showed by [Haveliwala and Kamvar(2003)] that for the second largest eigenvalue of P, we have |λ2| ≤ α 0 ≤ α ≤ 1. Since λ1 = 1 the converge rate depends on α. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14
38. 38. Conclusions. The power method and PageRank. Results. The matrix P of the PageRank algorithm is a stochastic matrix therefore the largest eigenvalue is 1. The convergence rate of the power method depends on the ratio of λ2 λ1 . It has been showed by [Haveliwala and Kamvar(2003)] that for the second largest eigenvalue of P, we have |λ2| ≤ α 0 ≤ α ≤ 1. Since λ1 = 1 the converge rate depends on α. The most popular value for α is 0.85. With this value it has been proved that the power method on web data set of over 80 million pages converges in about 50 iterations. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 11 / 14
39. 39. Conclusions. Really thanks to GTUG Palermo and see you to the next meeting! Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 12 / 14
40. 40. Bibliography. Bibliography. Brin, S. and Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Computer networks and ISDN systems, 30(1-7), 107–117. Ching, W. and Ng, M. (2006). Markov Chains: Models, Algoritms and Applications. Springer Science + Business Media, Inc. Haveliwala, T. and Kamvar, M. (2003). The second eigenvalue of the google matrix. Technical report, Stanford University. Langville, A., Meyer, C., and Fern´Andez, P. (2008). Google’s PageRank and beyond: the science of search engine rankings. The Mathematical Intelligencer, 30(1), 68–69. Page, L., Brin, S., Motwani, R., and Winograd, T. (1999). The PageRank Citation Ranking: Bringing Order to the Web. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 13 / 14
41. 41. Internet web sites. Internet web sites. Jon Atle Gulla (2007) - From Google Search to Semantic Exploration. - Norwegian University of Science Technology - www.slideshare.net/sveino/semantics-and-search?type=presentation Steven Levy (2010) - Exclusive: How Google’s Algorithm Rules the Web - Wired Magazine - www.wired.com/magazine/2010/02/ff_google_algorithm/ Ann Smarty (2009) - Let’s Try to Find All 200 Parameters in Google Algorithm - Search Engine Journal - www.searchenginejournal.com/200-parameters-in-google-algorithm/15457/. Sergio S. Guirreri - www.guirreri.host22.com (Google Technology User Group (GTUG) of Palermo.)Markov Chains as methodology used by PageRank to rank the Web Pages on Inte5th March 2010 14 / 14