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Ranking systems

Ranking systems






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Ranking systems Ranking systems Presentation Transcript

  • The Page Rank Axioms Based on Ranking Systems: The PageRank Axioms , by Alon Altman and Moshe Tennenholtz. Presented by Aron Matskin
    • 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
  • Talking Points
    • Ranking and reputation in general
    • Connections to the Internet world
    • PageRank web ranking system
    • PageRank representation theorem
  • Ranking: What
    • Abilities
    • Choices
    • Reputation
    • Quality
      • Quality of information
    • Popularity
    • Good looks
    • What not?
  • Ranking: How
    • Voting
    • Reputation systems
    • Peer review
    • Performance reviews
    • Sporting competition
    • Intuitive or ad-hoc
  • Ranking Systems’ Properties
    • Ad-hoc or systematic
    • Centralized or distributed
    • Feedback or indicator -based
    • Peer , “second-party” , or third-party
    • Update period
    • Volatility
    • Other?
  • Agents Ranking Themselves
    • Community reputation
    • Professional associations
    • Peer review
    • Performance reviews (in part)
    • Web page ranking
  • 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!
  • Ranking Systems: Analysis
    • Empirical
      • Because theories often lack
    • Theoretical
      • Because theoreticians need to eat, too
      • Provides valuable insight
  • 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)
  • 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.
  • 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
  • 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
  • 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
  • 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.
  • 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
  • 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)
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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’
  • 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
  • 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
  • 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 = =
  • 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
  • DEL: Proof a c b d c b d a VBC
  • DEL: Proof c b d a VBC c b d a
  • DEL: Proof ISO,PRO c b d a c b d a
  • DEL: Proof PRO c d a c b d a
  • DEL: Proof PRO c d a c d a
  • DEL: Proof VBC c d a c d a
  • DEL: Proof VBC c d a a c d
  • DEL for Self-Edge
    • It can also be shown that DEL holds for self-edges:
    a a
  • Useful Properties: DELETE
    • Nodes in P( x ) have no other outgoing edges
    • x has no other edges
    x = = = =
  • DELETE: Proof x = = = = COL x y
  • DELETE: Proof PRO x y
  • 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
  • DUPLICATE: Proof c b d a c b d a VBC
  • DUPLICATE: Proof c b d a VBC c b d a
  • DUPLICATE: Proof c b d a COL c b d a
  • DUPLICATE: Proof c b d a ISO,PRO c b d a
  • DUPLICATE: Proof c b d a COL -1 c b d a
  • DUPLICATE: Proof VBC -1 c b d a c b d a
  • 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
  • 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
  • Step 1: Insert Nodes
    • By DEL the relative ranking is preserved
    b c a d b c a d
  • Step 2: Choose Node to Remove b c a d
  • Step 3: Remove “self-edges” b c a d
  • Step 4: Duplicate Predecessors b c a d
  • Step 5: DELETE the Node b c d
  • Step 5: DELETE the Extras
    • There still are nodes to delete: back to Step 2
    b c d
  • Step 2: Choose Node to Remove
    • Steps 3,4 - no changes
    b c d
  • Step 5: DELETE the Node b d
  • Step 6: DELETE the Extras
    • No original nodes to remove: proceed to Step 7
    b d
  • Step 7: Balance by Duplication
    • This is our G’
    b d
  • Step 8: Equalize by Reverse DEL b d By ISO b=d. By DEL and SE: in G’ b<d.
  • Example for a and d b c a d b c a d
  • After Removal of c b a d
  • Duplicate Predecessors of b b a d
  • DELETE b a d
  • DELETE Extras a d
  • Before Balancing a d
  • After Balancing a d Conclusion: a<d.
  • What about a and b ? b a d
  • What about a and b ? b a d
  • What about a and b ? b a
  • What about a and b ? b a
  • What about a and b ? b a
  • What about a and b ? b a Conclusion: a=b.
  • 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
  • The End c b d a ½ 0 0 ½ 0 1 0 1 0 a b c a b c