The Page Rank Axioms

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. Abilities Choices Reputation Quality Quality of information Popularity Good looks What not? Ranking: What

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. 1 a=2 b=2 1 1 1 c=1 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

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: G A G a b c a a b b c c

15. PageRank: Some Math Find a solution of the equation: AGr = r The solution r is the rank vector. • 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

16. Calculating PageRank Take any non-zero vector r0 Let ri+1 = AGri Then the sequence rk 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. a b c Ranking Systems: Example • MyRank ranks vertices in G in ascending order of the number of incoming links MyRank(G): c = a < b PageRank(G): c < a = b G =

22. Axiom 1: Isomorphism (ISO) • F satisfies ISO iff it is independent of vertex names • Consequence: symmetric vertices have the same rank e = f = g = h = i = j a = b h e i b a f j g

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 (r1,…,rv,…,rN) be the rank vector of G, then (r1,…,rv+1/k,…,rN) is the rank vector of G’

24. Axiom 3: Vote by Committee (VBC) b b a a c c • 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) a b 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) = x = • All predecessors of x have the same rank • |P(x)| = |S(x)| • x is the only successor of each of its predecessors

27. Useful Properties: DEL • |P(b)|=|S(b)|=1 • There is no direct edge between a and c • a and c are otherwise unrestricted c c b a a d d

28. DEL: Proof c b VBC c b a a d d

29. DEL: Proof VBC c c b b a a d d

30. DEL: Proof ISO,PRO c c b b a a d d

31. DEL: Proof PRO c c b a a d d

32. DEL: Proof PRO c c a a d d

33. DEL: Proof VBC c c a a d d

34. DEL: Proof c VBC c a a d 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 = COL = x 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) b b c c a a d d

40. DUPLICATE: Proof b b VBC c c a a d d

41. DUPLICATE: Proof b b VBC c c a a d d

42. DUPLICATE: Proof b b COL c c a a d d

43. DUPLICATE: Proof b b ISO,PRO c c a a d d

44. DUPLICATE: Proof b b COL-1 c c a a d d

45. DUPLICATE: Proof b b VBC-1 c c a a d d

46. The Representation TheoremProof • 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 G A R G G d a b c a a a b b b c c c d d d

48. Step 1: Insert Nodes By DEL the relative ranking is preserved a b a b d c d c

49. Step 2: Choose Node to Remove a b d c

50. Step 3: Remove “self-edges” a b d c