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Ranking systems


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Ranking systems

  1. 1. The Page Rank Axioms Based on Ranking Systems: The PageRank Axioms , by Alon Altman and Moshe Tennenholtz. Presented by Aron Matskin
  2. 2. <ul><li>Judge and be prepared to be judged. </li></ul><ul><li>Ayn Rand </li></ul><ul><li>רבי שמעון אומר שלשה כתרים הם : כתר תורה , וכתר כהונה , וכתר מלכות ; וכתר שם טוב עולה על גביהן . </li></ul><ul><li>פירקי אבות </li></ul>
  3. 3. Talking Points <ul><li>Ranking and reputation in general </li></ul><ul><li>Connections to the Internet world </li></ul><ul><li>PageRank web ranking system </li></ul><ul><li>PageRank representation theorem </li></ul>
  4. 4. Ranking: What <ul><li>Abilities </li></ul><ul><li>Choices </li></ul><ul><li>Reputation </li></ul><ul><li>Quality </li></ul><ul><ul><li>Quality of information </li></ul></ul><ul><li>Popularity </li></ul><ul><li>Good looks </li></ul><ul><li>What not? </li></ul>
  5. 5. Ranking: How <ul><li>Voting </li></ul><ul><li>Reputation systems </li></ul><ul><li>Peer review </li></ul><ul><li>Performance reviews </li></ul><ul><li>Sporting competition </li></ul><ul><li>Intuitive or ad-hoc </li></ul>
  6. 6. Ranking Systems’ Properties <ul><li>Ad-hoc or systematic </li></ul><ul><li>Centralized or distributed </li></ul><ul><li>Feedback or indicator -based </li></ul><ul><li>Peer , “second-party” , or third-party </li></ul><ul><li>Update period </li></ul><ul><li>Volatility </li></ul><ul><li>Other? </li></ul>
  7. 7. Agents Ranking Themselves <ul><li>Community reputation </li></ul><ul><li>Professional associations </li></ul><ul><li>Peer review </li></ul><ul><li>Performance reviews (in part) </li></ul><ul><li>Web page ranking </li></ul>
  8. 8. Ranking: Problems and Issues <ul><li>Eliciting information </li></ul><ul><li>Information aggregation </li></ul><ul><li>Information distribution </li></ul><ul><li>Truthfulness </li></ul><ul><ul><li>Strategic considerations </li></ul></ul><ul><ul><li>Fear of retribution / expectation of kick-backs </li></ul></ul><ul><ul><li>Coalition formation </li></ul></ul><ul><li>Agent identification (pseudonym problem) </li></ul><ul><li>Need analysis! </li></ul>
  9. 9. Ranking Systems: Analysis <ul><li>Empirical </li></ul><ul><ul><li>Because theories often lack </li></ul></ul><ul><li>Theoretical </li></ul><ul><ul><li>Because theoreticians need to eat, too </li></ul></ul><ul><ul><li>Provides valuable insight </li></ul></ul>
  10. 10. Social Choice Theory <ul><li>Two approaches: </li></ul><ul><ul><li>Normative – from properties to implementations. Example: Arrow’s Impossibility Theorem </li></ul></ul><ul><ul><li>Descriptive – from implementation to properties. The Holy Grail: representation theorems (uniqueness results) </li></ul></ul>
  11. 11. PageRank Method <ul><li>A method for computing a popularity (or importance) ranking for every web page based on the graph of the web. </li></ul><ul><li>Has applications in search, browsing, and traffic estimation. </li></ul>
  12. 12. PageRank: Intuition <ul><li>Internet pages form a directed graph </li></ul><ul><li>Node’s popularity measure is a positive real number. The higher number represents higher popularity. Let’s call it weight </li></ul><ul><li>Node’s weight is distributed equally among nodes it links to </li></ul><ul><li>We look for a stationary solution: the sum of weights a page receives from its backlinks is equal to its weight </li></ul>b=2 c=1 a=2 1 1 1 1
  13. 13. PageRank as Random Walk <ul><li>Suppose you land on a random page and proceed by clicking on hyper-links uniformly randomly </li></ul><ul><li>Then the (normalized) rank of a page is the probability of visiting it </li></ul>
  14. 14. PageRank: Some Math <ul><li>Represent the graph as a matrix: </li></ul>b c a a b c a b c G A G ½ 0 0 ½ 0 1 0 1 0
  15. 15. PageRank: Some Math <ul><li>Find a solution of the equation: </li></ul>A G r = r <ul><li>Under the assumption that the graph is strongly connected there is only one normalized solution </li></ul><ul><li>The assumption is not used by the real PageRank algorithm which uses workarounds to overcome it </li></ul>The solution r is the rank vector.
  16. 16. Calculating PageRank <ul><li>Take any non-zero vector r 0 </li></ul><ul><li>Let r i+1 = A G r i </li></ul><ul><li>Then the sequence r k converges to r </li></ul><ul><li>Since the Internet graph is an expander, the convergence is very fast: O(log n) steps to reach given precision </li></ul>
  17. 17. PageRank: The Good News <ul><li>Intuitive </li></ul><ul><li>Relatively easy to calculate </li></ul><ul><li>Hard to manipulate </li></ul><ul><li>Great for common case searches </li></ul><ul><li>May be used to assess quality of information (assuming popularity ≈ trust) </li></ul>
  18. 18. PageRank: The Bad News <ul><li>PageRank is proprietary to </li></ul><ul><ul><li>Webmasters can’t manipulate it, but can </li></ul></ul><ul><ul><li>Every change in the algorithm is good for someone and is bad for someone else </li></ul></ul><ul><li>Popular become more popular </li></ul><ul><li>Popularity ≠ quality of information </li></ul>
  19. 19. The Representation Theorem <ul><li>We next present a set of axioms (i.e. properties) for ranking procedures </li></ul><ul><li>Some of the axioms are more intuitive then others, but all are satisfied by PageRank </li></ul><ul><li>We then show that PageRank is the only ranking algorithm that satisfies the axioms </li></ul><ul><li>We try to be informal, but convincing </li></ul>
  20. 20. Ranking Systems Defined <ul><li>A ranking system F is a functional that maps every finite strongly connected directed graph (SCDG) G=(V,E) into a reflexive, transitive, complete, and anti-symmetric binary relation ≤ on V </li></ul>
  21. 21. Ranking Systems: Example <ul><li>MyRank ranks vertices in G in ascending order of the number of incoming links </li></ul>G = MyRank(G): c = a < b PageRank(G): c < a = b b c a
  22. 22. Axiom 1: Isomorphism (ISO) <ul><li>F satisfies ISO iff it is independent of vertex names </li></ul><ul><ul><li>Consequence: symmetric vertices have the same rank </li></ul></ul>b e a g f j i h e = f = g = h = i = j a = b
  23. 23. Axiom 2: Self Edge (SE) <ul><li>Node v has a self-edge (v,v) in G’, but does not in G. Otherwise G and G’ are identical. F satisfies SE iff for all u,w ≠ v: </li></ul><ul><li>(u ≤ v  u <’ v) and (u ≤ w  u ≤’ w) </li></ul><ul><li>PageRank satisfies SE: Suppose v has k outgoing edges in G. Let (r 1 ,…,r v ,…,r N ) be the rank vector of G, then (r 1 ,…,r v + 1/k ,…,r N ) is the rank vector of G’ </li></ul>
  24. 24. Axiom 3: Vote by Committee (VBC) a c b a c b <ul><li>In the example page a links only to b and c , but there may be more successors of a </li></ul><ul><li>Incoming links of a and all other links of the successors of a remain the same </li></ul>
  25. 25. Axiom 4: Collapsing (COL) b a b <ul><li>The sets of predecessors of a and b are disjoint </li></ul><ul><li>Pages a and b must not link to each other or have self-links </li></ul><ul><li>The sets of successors of a and b coincide </li></ul>
  26. 26. Axiom 5: Proxy (PRO) <ul><li>All predecessors of x have the same rank </li></ul><ul><li>|P( x )| = |S( x )| </li></ul><ul><li>x is the only successor of each of its predecessors </li></ul>x = =
  27. 27. Useful Properties: DEL <ul><li>|P( b )|=|S( b )|=1 </li></ul><ul><li>There is no direct edge between a and c </li></ul><ul><li>a and c are otherwise unrestricted </li></ul>a c b d a c d
  28. 28. DEL: Proof a c b d c b d a VBC
  29. 29. DEL: Proof c b d a VBC c b d a
  30. 30. DEL: Proof ISO,PRO c b d a c b d a
  31. 31. DEL: Proof PRO c d a c b d a
  32. 32. DEL: Proof PRO c d a c d a
  33. 33. DEL: Proof VBC c d a c d a
  34. 34. DEL: Proof VBC c d a a c d
  35. 35. DEL for Self-Edge <ul><li>It can also be shown that DEL holds for self-edges: </li></ul>a a
  36. 36. Useful Properties: DELETE <ul><li>Nodes in P( x ) have no other outgoing edges </li></ul><ul><li>x has no other edges </li></ul>x = = = =
  37. 37. DELETE: Proof x = = = = COL x y
  38. 38. DELETE: Proof PRO x y
  39. 39. Useful Properties: DUPLICATE <ul><li>All successors of a are duplicated the same number of times </li></ul><ul><li>There are no edges from S( a ) to S( a ) </li></ul>c b d a c b d a
  40. 40. DUPLICATE: Proof c b d a c b d a VBC
  41. 41. DUPLICATE: Proof c b d a VBC c b d a
  42. 42. DUPLICATE: Proof c b d a COL c b d a
  43. 43. DUPLICATE: Proof c b d a ISO,PRO c b d a
  44. 44. DUPLICATE: Proof c b d a COL -1 c b d a
  45. 45. DUPLICATE: Proof VBC -1 c b d a c b d a
  46. 46. The Representation Theorem Proof <ul><li>Given a SCDG G=(V,E) and a,b in V, we eliminate all other nodes in G while preserving the relative ranking of a and b </li></ul><ul><li>In the resulting graph G’ the relative ranking of a and b given by the axioms can be uniquely determined. Therefore the axioms rank any SCDG uniquely </li></ul><ul><li>It follows that all ranking systems satisfying the axioms coincide </li></ul>
  47. 47. Proof by Example on b and d b c a a b c a b c G A G d d d R G a b c d 0 1 1 0 0 0 0 ⅓ ½ 0 0 ⅓ ½ 0 0 ⅓ 4 1 3 3
  48. 48. Step 1: Insert Nodes <ul><li>By DEL the relative ranking is preserved </li></ul>b c a d b c a d
  49. 49. Step 2: Choose Node to Remove b c a d
  50. 50. Step 3: Remove “self-edges” b c a d
  51. 51. Step 4: Duplicate Predecessors b c a d
  52. 52. Step 5: DELETE the Node b c d
  53. 53. Step 5: DELETE the Extras <ul><li>There still are nodes to delete: back to Step 2 </li></ul>b c d
  54. 54. Step 2: Choose Node to Remove <ul><li>Steps 3,4 - no changes </li></ul>b c d
  55. 55. Step 5: DELETE the Node b d
  56. 56. Step 6: DELETE the Extras <ul><li>No original nodes to remove: proceed to Step 7 </li></ul>b d
  57. 57. Step 7: Balance by Duplication <ul><li>This is our G’ </li></ul>b d
  58. 58. Step 8: Equalize by Reverse DEL b d By ISO b=d. By DEL and SE: in G’ b<d.
  59. 59. Example for a and d b c a d b c a d
  60. 60. After Removal of c b a d
  61. 61. Duplicate Predecessors of b b a d
  62. 62. DELETE b a d
  63. 63. DELETE Extras a d
  64. 64. Before Balancing a d
  65. 65. After Balancing a d Conclusion: a<d.
  66. 66. What about a and b ? b a d
  67. 67. What about a and b ? b a d
  68. 68. What about a and b ? b a
  69. 69. What about a and b ? b a
  70. 70. What about a and b ? b a
  71. 71. What about a and b ? b a Conclusion: a=b.
  72. 72. Concluding Remarks <ul><li>‘ Representation theorems isolate the “essence” of particular ranking systems, and provide means for the evaluation (and potential comparison) of such systems ’ – Alon & Tennenholtz </li></ul>
  73. 73. The End c b d a ½ 0 0 ½ 0 1 0 1 0 a b c a b c