CS286r

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CS286r
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Bravo Obama !
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Papers presentation

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

• 2) Ranking Systems The PageRank Axioms
• Alon Altman
• Moshe Tennenholtz

Presented by Michael Aubourg
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• Please ask your questions and make your comments
  during the presentation
• ? More interactive

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Ranking Systems The PageRank Axioms
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Roadmap

• Introduction
• Page ranking
• The axioms
• Properties implied by these axioms
• Completeness

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1) Introduction

• Today, PR is the most famous ranking alorithm.

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.
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1) Introduction

• Examples

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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

It presents a set of simple axioms that are
satisfied by PageRank and
any page ranking algorithm that does satisfy
them must PageRank
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1) Introduction

• Major problem

? How to study the rationale of using a
particular page ranking algorithm ?
How to identify or differentiate algorithms ?
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1) Introduction

• How to treat Internet ?

? As a graph.
Nodes pages agents
Edges links originating preferences
from the node
Graph theory
Internet reality
Social choice theory
parallelism
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1) Introduction
? 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.
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1) Introduction

• The paper introduce a representation theorem for
  PageRank.

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.
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1) Introduction

• Main result

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.
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2) Page ranking
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
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2) Page ranking
Ordering, ranking system, successors, and
predecessors are easy and intuitive concepts. I
wont define them again.
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2) Page ranking
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.
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2) Page ranking
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3) The axioms
The idea is to search for simple axioms we wish
the page ranking system to satisfy
They should be graph-theoretical and ordinal
axioms
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3) The axioms
1) Isomorphism
2) Self edge
3) Vote by committee
4) Collapsing
5) Proxy
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1) Isomorphism
This requirement is very basic It means that
the ranking procedure shouldnt depend on the way
we name the vertices.
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2) Self edge
This axiom is also intuitive. It tells that if
ab 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, agtb ?This point is
questionable in general case.
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3) Vote by committee
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.
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4) Collapsing
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.
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5) Proxy
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.
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Which of these axioms are not reasonable ? Any
comment so far ?
1) Isomorphism ?
2) Self edge ?
3) Vote by committee ?
4) Collapsing ?
5) Proxy ?
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At this point, we can check that the PageRank
system satisfies the 5 axioms.
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4) Properties implied by these axioms
1) Weak deletion property
2) Strong deletion property
3) Edge duplication property
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Weak deletion property
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Strong deletion property
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Edge duplication property
When our axioms are satisfied then this operator
does not change the relative ranking of the
pages, excluding the ones which have been
duplicated
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4) Properties implied by these axioms
1) Isomorphism
1) W. deletion property
2) Self edge
2) S. deletion property
3) Vote by committee
4) Collapsing
3) Edge duplication property
5) Proxy
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5) Completeness
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.
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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