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Papers presentation

2) Ranking Systems: The PageRank Axioms- Alon Altman- Moshe Tennenholtz

1) Popularity, Novelty and Attention - Fang Wu- Bernardo A. Huberman

Presented by Michael Aubourg

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Ranking Systems: The PageRank Axioms

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Roadmap

1) Introduction

2) Page ranking

3) The axioms

4) Properties implied by these axioms

5) Completeness

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1) Introduction

Today, PR is the most famous ranking alorithm.

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The ranking of agents based on other agents input is fundamental to multi-agent systems.

More specifically, ranking systems are the keystone of e-commerce and Internet technologies.

1) IntroductionExamples :

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1) Introduction Here, the paper bridges the gap between page ranking algorithms

and the theory of social choice by suggesting the axiomatic approach

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It presents a set of simple axioms that are satisfied by PageRank and :

any page ranking algorithm that does satisfy them must = PageRank

1) Introduction

Major problem :

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→ How to study the rationale of using a particular page ranking algorithm ?

How to identify or differentiate algorithms ?

1) Introduction

How to treat Internet ?

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→ As a graph.

Nodes = pages = agents

Edges = links originating = preferences from the node

Graph theory Internet reality Social choice theory

parallelism

1) Introduction

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→ Hence, the page ranking problem becomes a problem of social choice.

But …new feature of the page ranking setting:

Set of agents = Set of alternatives

→ We will have to consider transitive effect.

1) Introduction

The paper introduce a representation theorem for PageRank.

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Definition : Given a particular algorithm A, it satisfies many properties. The goal is to find a small set of axioms satisfied by A, and which has the additional feature that every algorithm that satisfies these properties must coincide with A.

1) Introduction

Main result :

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The paper looked for simple axioms one may require a page ranking to satisfy.- The PR does satisfy these axioms- Any page ranking algorithm that does satisfy these axioms MUST coincide with PR.

2) Page ranking

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Directed graph: G=(V,E) where V = set of nodes

E = set of ordered pairs of vertices

Strongly connected graph: for every pair of vertices, we can go from one to the other

2) Page ranking

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Ordering, ranking system, successors, and predecessors are easy and intuitive concepts. I won’t define them again.

The PageRank matrix : G=({v1,v2,…,vn},E)

[AG]i,j = if (vj,vi) E∈ 0 otherwise

Where is the successor set of vj

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| ( ) |G jS v

( )G jS v

2) Page ranking

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PageRank is the stationary limit probability distribution reached in a random walk in a graph, where we start at random.

The previous matrix A, does capture this random walk created by the PR procedure.

2) Page ranking

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R��������������R��������������

PageRank PRG(vi) of a vertex vi :

Is defined as PRG(vi)=Ri where R is the unique solution of the system AG. = with R1 = 1and G=({v1,v2,…,vn},E)

3) The axioms

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The idea is to search for simple axioms we wish the page ranking system to satisfy

They should be graph-theoretical and ordinal axioms

3) The axioms

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1) Isomorphism

2) Self edge

3) Vote by committee

4) Collapsing

5) Proxy

1) Isomorphism

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This requirement is very basic : It means that the ranking procedure shouldn’t depend on the way we name the vertices.

2) Self edge

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This axiom is also intuitive. It tells that if a≥b in graph G, where in G a does not link to itself, then, if all that we add to G is a link from a to itself, a>b

→This point is questionable in general case.

3) Vote by committee

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If page a links to page b and c, then the relative ranking of all pages should be the same as in the case where the direct links from a to b and c are replaced by links from a to a new set of pages which link to b and c.

4) Collapsing

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If there is a pair of pages, A and B, where both A and B link to the same set of pages, but the sets of pages that link to A and B are disjoint, then if we collapse {A,B} into {A}, where all links to B become now links to A, then the relative ranking of all pages, excluding A and B should remain as before.

5) Proxy

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If there is a set of k pages, all having the same importance, which link to A, where A itself links to k pages, then if we drop A and connect directly in a 1-1 fashion,the pages which linked to A to the pages that A linked to, then the relative ranking of all pages excluding A, should remain the same.

Which of these axioms are not reasonable ? Any comment so far ?

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1) Isomorphism ?

2) Self edge ?

3) Vote by committee ?

4) Collapsing ?

5) Proxy ?

At this point, we can check that the PageRank system satisfies the 5 axioms.

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4) Properties implied by these axioms

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1) Weak deletion property

2) Strong deletion property

3) Edge duplication property

Weak deletion property

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Strong deletion property

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F has the strong deletion property if for every vertex set V , for every vertex v V , for all v∈ 1, v2 V \ {v}, and ∈for every graph G = (V,E) GV s.t. S(v) = {s∈ 1, s2, . . . , st}, P(v) = {pij|j = 1, . . . , t; i = 0, . . . ,m}, S(pij) = {v} for all j {1, . . . t} and i {0, . . . ,m}, and p∈ ∈ ij ≈ pik for all i {0, . . . ,m} and j, k {1, . . . t}: ∈ ∈Let G0 = Delete(G, v, {(s1, {pi1|i =0, . . .m}), . . . (st, {pit|i = 0, . . .m})}). Then, v1 ≤ v2 v1 ≤ v2.

FG

FG 0

FG

Edge duplication property

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When our axioms are satisfied then this operator does not change the relative ranking of the pages, excluding the ones which havebeen duplicated

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1) Isomorphism

2) Self edge

3) Vote by committee

4) Collapsing

5) Proxy

1) W. deletion property

2) S. deletion property

3) Edge duplication property

4) Properties implied by these axioms

5) Completeness

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We can now show that our axiom fully characterize the PageRank system

Theorem : A ranking system F satisfies isomorphism, self edge, vote by committee, collapsing, and proxy if and only if F is the PageRank ranking system.

5) Completeness

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5) Completeness

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5) Completeness

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Discussion :

Please submit all your comments now !

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Thank you