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Ranking systems

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  • Booby Fischer was #49 on PCA ratings list in 1994, although he had not played for 20 years
  • Transcript

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

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