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The Page Rank Axioms Based on  Ranking Systems: The PageRank Axioms ,   by Alon Altman and Moshe Tennenholtz. Presented by Aron Matskin
[object Object],[object Object],[object Object],[object Object]
Talking Points ,[object Object],[object Object],[object Object],[object Object]
Ranking: What ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Ranking: How ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Ranking Systems’ Properties ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Agents Ranking Themselves ,[object Object],[object Object],[object Object],[object Object],[object Object]
Ranking: Problems and Issues ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Ranking Systems: Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object]
Social Choice Theory ,[object Object],[object Object],[object Object]
PageRank Method ,[object Object],[object Object]
PageRank: Intuition ,[object Object],[object Object],[object Object],[object Object],b=2 c=1 a=2 1 1 1 1
PageRank as Random Walk ,[object Object],[object Object]
PageRank: Some Math ,[object Object],b c a a b c a b c G A G ½ 0 0 ½ 0 1 0 1 0
PageRank: Some Math ,[object Object],A G   r = r ,[object Object],[object Object],The solution r is the rank vector.
Calculating PageRank ,[object Object],[object Object],[object Object],[object Object]
PageRank: The Good News ,[object Object],[object Object],[object Object],[object Object],[object Object]
PageRank: The Bad News ,[object Object],[object Object],[object Object],[object Object],[object Object]
The Representation Theorem ,[object Object],[object Object],[object Object],[object Object]
Ranking Systems Defined ,[object Object]
Ranking Systems: Example ,[object Object],G = MyRank(G): c = a < b PageRank(G): c < a = b b c a
Axiom 1: Isomorphism (ISO) ,[object Object],[object Object],b e a g f j i h e = f = g = h = i = j a = b
Axiom 2: Self Edge (SE) ,[object Object],[object Object],[object Object]
Axiom 3: Vote by Committee (VBC) a c b a c b ,[object Object],[object Object]
Axiom 4: Collapsing (COL) b a b ,[object Object],[object Object],[object Object]
Axiom 5: Proxy (PRO) ,[object Object],[object Object],[object Object],x = =
Useful Properties: DEL ,[object Object],[object Object],[object Object],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 ,[object Object],a a
Useful Properties: DELETE ,[object Object],[object Object],x = = = =
DELETE: Proof x = = = = COL x y
DELETE: Proof PRO x y
Useful Properties: DUPLICATE ,[object Object],[object Object],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 ,[object Object],[object Object],[object Object]
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 ,[object Object],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 ,[object Object],b c d
Step 2: Choose Node to Remove ,[object Object],b c d
Step 5: DELETE the Node b d
Step 6: DELETE the Extras ,[object Object],b d
Step 7: Balance by Duplication ,[object Object],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 ,[object Object]
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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Ranking systems

Editor's Notes

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