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
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2)
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5)
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 =
Edges =
Graph theory
pages
links originating
from the node
Internet reality
= agents
= preferences
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 won’t define them again.
The PageRank matrix : G=({v1,v2,…,vn},E)
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[AG]i,j = | S (v ) | if (vj,vi) ∈ E
G
j
0
otherwise
Where SG (v j ) is the successor set of vj
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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
PageRank PRG(vi) of a vertex vi :
Is defined as PRG(vi)=Ri where R is the
unique solution of the system AG. R = R
with R1 = 1and G=({v1,v2,…,vn},E)
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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 shouldn’t
depend on the way we name the vertices.
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2) Self edge
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.
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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
F has the strong deletion property if for every vertex set
V , for every vertex v ∈ V , for all v1, v2 ∈ V \ {v}, and for
every graph G = (V,E) ∈ GV s.t. S(v) = {s1, s2, . . . , st},
P(v) = {pij|j = 1, . . . , t; i = 0, . . . ,m}, S(pij) = {v} for all j ∈
F
{1, . . . t} and i ∈ {0, . . . ,m}, and pij ≈ G pik
for all i ∈{0, . . . ,m} and j, k ∈ {1, . . . t}:
Let G0 = Delete(G, v, {(s1, {pi1|i =0, . . .m}), . . . (st, {pit|i
F
= 0, . . .m})}). Then, v1 ≤GF v2  v1 ≤ G0v2.
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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
2) Self edge
1) W. deletion property
3) Vote by committee 2) S. deletion property
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
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