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=2
1
1
1
1
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
½01
½00
a b c
a
b
c
PageRank: Some Math
Find a solution of the equation:
AG 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 r0
Let ri+1 = AG ri
Then the sequence rk 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
aMyRank(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
gf
j
i
he = 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 (r1,…,rv,…,rN) be the rank vector of G, then (r1,…,rv+1/k,…,rN) 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 rank2. |P(x)| = |S(x)|3. x is the only successor of each of its
predecessors
x
=
=
Useful Properties: DEL
1. |P(b)|=|S(b)|=12. There is no direct edge between a and c3. a and c are otherwise unrestricted
a
cb
d
a
c
d
DEL: Proof
a
cb
d
cb
d
a
VBC
DEL: Proof
cb
d
a
VBCcb
d
a
DEL: Proof
ISO,PROcb
d
a
cb
d
a
DEL: Proof
PROc
d
a
cb
d
a
DEL: Proof
PROc
d
a
c
d
a
DEL: Proof
VBCc
d
a
c
d
a
DEL: Proof
VBCc
d
a a
c
d
DEL for Self-Edge
It can also be shown that DEL holds for self-edges:
a a
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
a b c
a
b
cd
⅓00½
⅓00½
⅓000
0110d
d
3
3
1
4
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
cd
Step 5: DELETE the Extras
There still are nodes to delete: back to Step 2
b
cd
Step 2: Choose Node to Remove
Steps 3,4 - no changes
b
cd
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
dBy 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
ba
d
Duplicate Predecessors of b
ba
d
DELETE b
a
d
DELETE Extras
a
d
Before Balancing
a
d
After Balancing
a
dConclusion: a<d.
What about a and b?
ba
d
What about a and b?
ba
d
What about a and b?
ba
What about a and b?
ba
What about a and b?
ba
What about a and b?
ba
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 Endc
b
d
a
½0
0½0
101
0a b
c
a
b
c
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