The PageRank Axioms by bestt571

VIEWS: 16 PAGES: 41

PageRank is the Google ranking algorithm (ranking formula) is part of Google used to identify web pages for rank / importance of a method used to measure a website, Google the only criterion for good or bad. In a blend of identification, such as Title Keywords such as logo and all other relevant factors, Google PageRank to adjust results through, so that more "level / importance" of pages in search results get promoted in another site's ranking to improve search results relevance and quality.

More Info
									The PageRank Axioms
  Alon Altman and Moshe Tennennholtz

             Technion, IIT




                                       The PageRank Axioms – p.
Main Contribution
A representation theorem for PageRank, bridging the gap
between two well-known disciplines:
    Social choice – The theory of combining individual
    preferences into a social preference over alternatives.
    Ranking systems – Systems designated to rank
    agents based on mutual reports or links between the
    agents.




                                                   The PageRank Axioms – p.
Social Choice
   The classical social choice setting is comprised of:
      A set of agents.
      A set of alternatives.
      Each agent has a preference relation over the set
      of alternatives.
   A social welfare function is a mapping from the
   agents’ individual preferences into a social ranking
   over the alternatives.
   The goal of social choice theory is to produce
   “good” social welfare functions.


                                                The PageRank Axioms – p.
Ranking Systems
Ranking systems are widely used in practical
implementations:
    Reputation systems such as E-bay’s
    Journal influence measurement
    Internet page ranking:




                                               The PageRank Axioms – p.
Page Ranking
   The page ranking problem can be described as:
   Given a graph G = (V, E) representing the Internet,
   produce a ranking over V which represents the
   “power” or “relevance” of the vertices in G.
   Several solutions were suggested to the page ranking
   problem:
      Google’s PageRank — The most widely
      implemented technology behind the most
      popular search engine.
      Hubs&Authorities.


                                               The PageRank Axioms – p.
Page Ranking as Social Choice
Such problem can be viewed as a social choice problem:
    The set of agents is V .
    The set of alternatives coincides with the set of
    agents, and is also V .
    Each agent v ∈ V prefers the agents
    V = {v |(v, v ) ∈ E} it links to over the other
    agents.
    These individual preferences are to be combined to a
    social preference over V , which “represents” the
    individual preferences.

                                                The PageRank Axioms – p.
The Axiomatic Approach
   In the axiomatic approach, we try to find basic
   properties satisfied by page ranking systems.
   The axiomatic approach encompasses two distinct
   approaches:
       The normative approach, in which we study sets
       of axioms that intuitively should be satisfied by a
       ranking system; and
       The descriptive approach, in which we devise a
       set of axioms that are uniquely satisfied by a
       known ranking system.
   In this presentation, we apply the descriptive
   approach to PageRank.
                                                 The PageRank Axioms – p.
Previous Results
   The only work we know of suggesting an
   axiomatization of the PageRank is the one recently
   published by Palacois-Huerta and Volij(2004),
   dealing with relative influence of scientific journals.
   Our axiomatization differs in the fact that it captures
   the relative ranking using purely ordinal axioms, in
   the sense that there is no underlying numeric
   measure of power.
   This is opposed to the axioms suggested by
   Palacois-Huerta and Volij, which include arithmetic
   manipulation of numerical weights.


                                                  The PageRank Axioms – p.
PageRank
  The simplified version of PageRank we axiomatize
  here could be defined as ranking the vertices
  according to the stationary probabilities of a random
  walk on the graph.
  We assume the graph G = (V, E) to be strongly
  connected.
  Let AG be the following matrix:
                            1
                         |S(vj )|   (vj , vi ) ∈ E
            [AG ]i,j =
                         0          Otherwise

  where S(v) is the successor set of v.
                                                     The PageRank Axioms – p.
PageRank (cont.)
   The PageRank of a graph G is defined as the
   principal eigenvector of the matrix A.
   That is, the PageRank of G is the vector x satisfying
                         AG x = x
   The PageRank ranking system P R can now be
   defined as the ordering on V according to x:
                  v1   PR   v2 ⇔ x1 ≤ x2




                                                The PageRank Axioms – p. 1
The Axioms
Our representation theorem for PageRank requires the
following five axioms:
 1.   Isomorphism;
 2.   Self-Edge;
 3.   Vote by Committee;
 4.   Collapsing; and
 5.   Proxy




                                               The PageRank Axioms – p. 1
Axiom 1: Isomorphism
    This axiom states that the ranking must not rely on
    the names of the vertices, but only on the voting
    structure between the agents.
Formally:
    A ranking system F satisfies isomorphism if for
    every isomorphism function ϕ : V1 → V2 , and two
    isomorphic graphs G ∈ GV1 , ϕ(G) ∈ GV2 :
      F
      ϕ(G) = ϕ( F ).
                G




                                                 The PageRank Axioms – p. 1
Axiom 2: Self-Edge
    This axiom states that adding a self edge on v
    strengthens v, but does not change the ranking of
    other vertices.
Formally:
    SelfEdge(G, v) – The graph G with the self edge
    (v, v) added.
    Let F be a ranking system. F satisfies the self edge
    axiom if for every vertex set V and for every vertex
    v ∈ V and for every graph G = (V, E) ∈ GV
    s.t. (v, v) ∈ E, and for every v1 , v2 ∈ V \ {v}: Let
                /
    G = SelfEdge(G, v). If v1 F v then v F v1 ;
                                     G             G
    and v1 F v2 iff v1 F v2 .
               G           G                      The PageRank Axioms – p. 1
Axiom 3: Vote by Committee

            b                                      b

   a                             a

            c                                      c


   The Vote by Committee axiom captures the fact that
   an agent may vote indirectly via any number of
   intermediate agents, each of which vote to the
   agent’s original preferences.




                                              The PageRank Axioms – p. 1
Axiom 3: Vote by Committee (cont.)
Formal Definition:
Let F be a ranking system. F satisfies vote by committee
if for every vertex set V , for every vertex v ∈ V , for
every graph G = (V, E) ∈ GV , for every v1 , v2 ∈ V , and
for every m ∈ N: Let
G = (V ∪ {u1 , u2 , . . . , um }, E \ {(v, x)|x ∈
SG (v)} ∪ {(v, ui )|i = 1, . . . , m} ∪ {(ui , x)|x ∈
SG (v), i = 1, . . . , m}), where {u1 , u2 , . . . , um } ∩ V = ∅.
Then, v1 F v2 iff v1 F v2 .
            G               G




                                                         The PageRank Axioms – p. 1
Axiom 4: Collapsing

          a

                                            a

          b




   The collapsing axiom captures the fact that voters
   which have the same preferences may be collapsed
   to a single voter with the same preferences, voted by
   all the voters for both.
   We assume the voter sets for the collapsed vertices
   are disjoint and do not include a or b.
   This collapsing only change the rank of a.
                                                The PageRank Axioms – p. 1
Axiom 4: Collapsing (cont.)
Formal Definition:
Let F be a ranking system. F satisfies collapsing if for
every vertex set V , for every v, v ∈ V , for every
v1 , v2 ∈ V \ {v, v }, and for every graph
G = (V, E) ∈ GV for which SG (v) = SG (v ),
PG (v) ∩ PG (v ) = ∅, and [PG (v) ∪ PG (v )] ∩ {v, v } = ∅:
Let G = (V \ {v }, E \ {(v , x)|x ∈
SG (v )} \ {(x, v )|x ∈ PG (v )} ∪ {(x, v)|x ∈ PG (v )}).
Then, v1 F v2 iff v1 F v2 .
            G             G




                                                  The PageRank Axioms – p. 1
Axiom 5: Proxy

  =
              x
  =



      The proxy axiom captures the fact that n voters of
      equal rank who have voted via a proxy (another
      agent) for n alternatives, can achieve the same result
      by directly voting for one alternative each.



                                                    The PageRank Axioms – p. 1
Axiom 5: Proxy (cont.)
Formal Definition:
Let F be a ranking system. F satisfies proxy if for every
vertex set V , for every vertex v ∈ V , for every
v1 , v2 ∈ V \ {v}, and for every graph G = (V, E) ∈ GV
for which |PG (v)| = |SG (v)|, for all p ∈ PG (v):
SG (p) = {v}, and for all p, p ∈ PG (v): p p : Assume
PG (v) = {p1 , p2 , . . . , pm } and SG (v) = {s1 , s2 , . . . , sm }.
Let G = (V \ {v}, E \ {(x, v), (v, x)|x ∈
V } ∪ {(pi , si )|i ∈ {1, . . . , m}}). Then, v1 F v2 iff
                                                    G
v1 F v2 .
      G




                                                             The PageRank Axioms – p. 1
Soundness
Proposition 1 The PageRank ranking system P R
satisfies isomorphism, self edge, vote by committee,
collapsing, and proxy.


This proposition is proven by a simple application of
linear algebra.




                                                 The PageRank Axioms – p. 2
Completeness
In order to prove completeness, we will first show two
strong properties that are entailed by our five axioms:
    Weak Deletion;
    Strong Deletion; and
    Duplication




                                                 The PageRank Axioms – p. 2
Weak Deletion
   The Weak Deletion property allows us to remove a
   vertex that has both an in-degree and an out-degree
   of 1.
   Formally,
   Let V be a vertex set and let v ∈ V be a vertex. Let
   G = (V, E) ∈ GV be a graph where S(v) = {s}, P (v) = {p},
   and (s, p) ∈ E. We will use Del(G, v) to denote the graph
              /
   G = (V , E ) defined by:

              V   = V \ {v}
              E   = E \ {(p, v), (v, s)} ∪ {(p, s)}.


                                                       The PageRank Axioms – p. 2
Weak Deletion (cont.)
Now we can state the weak deletion property:
Let F be a ranking system. F has the weak deletion property if for
every vertex set V , for every vertex v ∈ V and for all vertices
v1 , v2 ∈ V \ {v}, and for every graph G = (V, E) ∈ GV s.t.
S(v) = {s}, P (v) = {p}, and (s, p) ∈ E: Let G = Del(G, v).
                                       /
Then, v1 F v2 iff v1 F v2 .
            G             G




                                                          The PageRank Axioms – p. 2
Strong Deletion

              =


              =


                    x
          =

          =




   The Strong Deletion property is a generalization of
   the proxy axiom, allowing removal of a vertex with
   m sets of t equal predecessors, and t successors.
   One element of each equal sets is set to point to each
   of the original successors.
   This change does not affect the relative rank of the
   remaining vertices.                           The PageRank Axioms – p. 2
Strong Deletion (cont.)
Formally, the strong deletion operator is defined:
Let V be a vertex set and let v ∈ V be a vertex. Let G = (V, E) ∈ GV be a
graph where S(v) = {s1 , s2 , . . . , st } and
P (v) = {pi |j = 1, . . . , t; i = 0, . . . , m}, and S(pi ) = {v} for all
              j                                           j
j ∈ {1, . . . t} and i ∈ {0, . . . , m}. We will use
Delete(G, v, {(s1 , {pi |i = 0, . . . m}), . . . , (st , {pi |i = 0, . . . m})}) to
                         1                                  t
denote the graph G = (V , E ) defined by:

        V    = V \ {v}
        E    = E \ {(pi , v), (v, sj )|i = 0, . . . , m; j = 1, . . . , t} ∪
                      j

                  ∪{(pi , sj )|i = 0, . . . , m; j = 1, . . . , t}.
                      j




                                                                         The PageRank Axioms – p. 2
Strong Deletion (cont.)
Now we can state the strong deletion property:
Let F be a ranking system. 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) = {pi |j = 1, . . . , t; i = 0, . . . , m}, S(pi ) = {v} for all j ∈ {1, . . . t}
            j                                        j
and i ∈ {0, . . . , m}, and pi F pi for all i ∈ {0, . . . , m} and
                               j   G k
j, k ∈ {1, . . . t}: Let
G = Delete(G, v, {(s1 , {pi |i = 0, . . . m}), . . . (st , {pi |i = 0, . . . m})}).
                                 1                             t
Then, v1 F v2 iff v1 F v2 .
             G              G




                                                                           The PageRank Axioms – p. 2
Duplication

                    b                     b



           a        c            a        c



                    d                     d




   The duplication property allows duplication of an
   agent’s successors by any factor.
   The new vertices have the same successors as the
   old.
   The relative ranking of all vertices except the
   duplicated successors does not change.
                                               The PageRank Axioms – p. 2
Duplication (cont.)
Formally, the duplication operator is defined:
Let V be a vertex set and let G = (V, E) ∈ GV be a graph. Let
S(v) = {s0 , s0 , . . . , s0 }. We will use Duplicate(G, v, m) to denote
          1 2              t
the graph G = (V , E ) defined by:

   V    = V ∪ {si |i = 1, . . . , m − 1; j = 1, . . . t}
                j

   E    = E ∪ {(v, si )|i = 1, . . . , m − 1; j = 1, . . . t} ∪
                    j

            ∪{(si , u)|i = 1, . . . , m − 1; j = 1, . . . t; u ∈ SG (s0 )}.
                j                                                     j




                                                                   The PageRank Axioms – p. 2
Duplication (cont.)
   Now we can state the duplication property:
   Let F be a ranking system. F has the edge duplication
   property if for every vertex set V , for all vertices v, v1 , v2 ∈ V ,
   for every m ∈ N, and for every graph G = (V, E) ∈ GV : Let
   S(v) = {s0 , s0 , . . . , s0 }, and let G = Duplicate(G, v, m).
              1 2             t
   Then, v1 F v2 iff v1 F v2 .
               G                 G




                                                               The PageRank Axioms – p. 2
Satisfication
These three properties are entailed by our axioms:
    Lemma 1 Let F be a ranking system that satisfies
    isomorphism, vote by committee, and proxy. Then, F
    has the edge duplication property.
    Lemma 2 Let F be a ranking system that satisfies
    collapsing and proxy. Then, F has the strong
    deletion property.
    Lemma 3 Let F be a ranking system that satisfies
    isomorphism, vote by committee, collapsing, and
    proxy. Then, F has the edge duplication property.


                                                 The PageRank Axioms – p. 3
Idea of Completeness Proof
   The completeness proof is a constructive one.
   The idea is to fix to vertices a and b, and then to
   manipulate the graph by applying the axioms and
   properties, preserving their relative ranking.
   Further manipulation is then applied to the graph
   using the self edge axiom to modify the relative
   ranking of a and b in one direction, until a and b can
   be proved of equal rank using the isomorphism
   axiom.
   As the relative ranking was manipulated in only one
   direction, we show that a and b must have a relative
   ranking opposite to the manipulation in any ranking
   system satisfying the axioms.
                                                 The PageRank Axioms – p. 3
Demonstration of Proof
                                                  a




                                        a
                                                      c




   Start with the input graph and       c
   two vertices a and b to be                 b

   compared.
                                    b
   Add a vertex on each edge.
   The relative ranking of a and              d

   b does not change because of         d
   the weak deletion property.
                                            The PageRank Axioms – p. 3
Demonstration of Proof (cont.)
   Select an original vertex
   except a and b (c in our                   a
   example), and delete all its
   self edges with a vertex on
   them.
                                        c
   This does not change the
   relative ranking of a and b
   due to the self-edge and weak
   deletion axioms.                 b


   Next, we use the duplication
   property to duplicate the pre-
   decessors of c by c’s out de-    d

   gree, without changing the
   relative ranking of a and b.             The PageRank Axioms – p. 3
Demonstration of Proof (cont.)
                                      a




   The isomorphism axiom
   guarantees that c satisfies the
   conditions of the strong               b

   deletion property.
   Thus, we can apply
   Delete(G, c).                          d


   Due to the strong deletion
   property, this does not change
   the relative ranking of a and b.
                                              The PageRank Axioms – p. 3
Demonstration of Proof (cont.)
   We apply the strong deletion
   property again to delete the
   new vertices that were                a

   successors of c.
   Again, this does not change
   the relative ranking of a and
   b.                                b


   Note that now again all suc-
   cessors and predecessors of
   the original vertices are new
                                             d
   vertices, and all successors
   and predecessors of the new
   vertices are original vertices.
                                                 The PageRank Axioms – p. 3
Demonstration of Proof (cont.)
   Repeat the previous steps, selecting a different
   vertex each time, until the only remaining original
   vertices are a and b:

                          a




                          b




                                                The PageRank Axioms – p. 3
Demonstration of Proof (cont.)
   Equalize the number of edges with vertices from a
   to b to the number of edges with vertices from b to a
   by duplicating a by the number edges with vertices
   from b to a and vice versa.
   In our example b is duplicated by 3.

                              a




                              b




                                                The PageRank Axioms – p. 3
Demonstration of Proof (cont.)
   Assume without loss of generality that b has fewer
   self edges with vertices than a.
   Add self edges with vertices to b, until a and b have
   the same number of self edges (with vertices).

                            a




                    b




                                                 The PageRank Axioms – p. 3
Completeness Proof – Conclusion
   Now, a b according to the isomorphism axiom.
   But, according to the self edge axiom we increased
   the relative rank of b compared to a, so we conclude
   that in the original graph, b a.
   This unique outcome is general, and thus the axioms
   guarantee a unique ranking, and thus exactly
   represent PageRank.




                                               The PageRank Axioms – p. 3
Concluding Remarks
   We have shown a first (graph-theoretic, ordinal)
   axiomatization of PageRank.
   This axiomatization bridges the gap between
   classical social choice theory, and the most widely
   used algorithm for ranking Internet pages.
   The completeness proof is constructive, and suggest
   a (grossly inefficient) algorithm for computing the
   relative PageRank of two nodes.




                                               The PageRank Axioms – p. 4
Future and Additional Work
   Descriptive Approach
      Hubs & Authorities
      PageRank with damping factor
   Normative Approach
      Impossibility results
   Incentive compatibility
      Impossibility result
   Negative votes



                                     The PageRank Axioms – p. 4

								
To top