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=

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 010 ½ 01 ½ 00 abc a b c

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 b c a MyRank(G): c = a < b PageRank(G): c < a = b

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 1.In the example page a links only to b and c, but there may be more successors of a 2.Incoming links of a and all other links of the successors of a remain the same

Axiom 4: Collapsing (COL) b a b 1.The sets of predecessors of a and b are disjoint 2.Pages a and b must not link to each other or have self-links 3.The sets of successors of a and b coincide

Axiom 5: Proxy (PRO) 1.All predecessors of x have the same rank 2.|P(x)| = |S(x)| 3. x is the only successor of each of its predecessors x = =

Useful Properties: DEL 1.|P(b)|=|S(b)|=1 2.There is no direct edge between a and c 3. 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: aa

Useful Properties: DELETE 1.Nodes in P(x) have no other outgoing edges 2. x has no other edges x = = = =

DELETE: Proof x = = = = COL x y

DELETE: Proof PRO x y

Useful Properties: DUPLICATE 1.All successors of a are duplicated the same number of times 2.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 abc a b c d ⅓ 00 ½ ⅓ 00 ½ ⅓ d d a b c d

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

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 b d This is our G ’

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

b a d

b a

b a

b a

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 00 ½ a bc a b c