PageRank Algorithm

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PageRank Algorithm

• Directed Graphs, probability, and
• Markov Chains

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N 0
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N 1
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N 2
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N 3
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N 4
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N 5
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N 6
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N 7
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N 8
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Ranking 6 5 4 1 3 8 7 2
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Goal

• Our goal is find a probabilistic distribution of
  rankings that accurately represent the network
  topology.

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Link Structure

A
B
C
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Adjacency Matrix

• A B C
• A
• B
• C

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Link Structure

A
B
C
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Adjacency Matrix

• A B C
• A 0 1 1
• B
• C

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Link Structure

A
B
C
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Adjacency Matrix

• A B C
• A 0 1 1
• B 0 0 1
• C

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Link Structure

A
B
C
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Adjacency Matrix

• A B C
• A 0 1 1
• B 0 0 1
• C 1 0 0

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Binary to Probabilistic

• We need to know if a page links to another page.
• As well as the relative importance of that
  particular link.
• So, we transform (or, normalize) the binary
  adjacency matrix to a probabilistic or stochastic
  matrix.

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Normalization

• A B C
• A 0 1 1 2 Links
• B 0 0 1
• C 1 0 0

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Normalization

• A B C
• A 0 ½ ½ 2 Links
• B 0 0 1
• C 1 0 0

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Normalization

• A B C
• A 0 ½ ½
• B 0 0 1 1 link
• C 1 0 0

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Normalization

• A B C
• A 0 ½ ½
• B 0 0 1
• C 1 0 0

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Stochastic Matrix

• A B C
• A 0 ½ ½
• B 0 0 1
• C 1 0 0

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The Stationary State

• R(A) R(C)
• R(B) ½ R(A)
• R(C) ½ R(A) R(B)
• R(A) R(B) R(C) 1

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Markov

• ½R(A)
• ½R(A)
• R(C) R(B)

A
B
C
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• R(A) R(C)
• R(B) ½ R(A)
• R(C) ½ R(A) R(B)
• ½ R(A)
• ½ R(A)
• R(C) R(B)

A
B
C
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Markov Chain

• We can think of this probability matrix
  intuitively as the path of the random surfer.
• Then, we measure importance, or Page Rank, R,
  by the probabilities the random surfer will end
  up at this page at any particular point in time.

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Stationary State
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Stationary State

• 0.2
• 0.2 0.4 0.2

A 0.4
B 0.2
C 0.4
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The Setup

• Let R(u) be the rank of page u.
• nu Number of outlinks from u.
• So for vs linked to u
• R(u)

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The Problem

• Let R(u) be the rank of page u.
• nu Number of outlinks from u.
• So for vs linked to u
• R(u)
• N equations and N unknowns is very different when
  N is HUGE (O(106))!

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Power Method

• We want RT RTP
• Notice RT is the (left) eigenvector associated
  with the eigenvalue 1.
• It is well-known that the dominant eigenvalue in
  a stochastic matrix is 1.

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Power Method

• So we iterate
• XTm 1 XTmP
• Here, X is any vector, and m is the number of
  iterations.
• This method converges to the dominant eigenvector
  of P, regardless of x0

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Power Method

• If (for all i) ?i lt 1, then
• XTm P RT where RT P RT
• This is the stationary state of the Markov Chain.
• In our case Page Rank!

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Whats Going On Here?
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Three Disjoint Subsets!
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Reducible Matrix
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Uniqueness

• Note A reducible matrix can have more than one
  eigenvector associated to the dominant
  eigenvalue.
• So, the power method fails to converge to a
  unique vector.
• We need irreducibility.

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Irreducibility

• Irreducible Stochastic matrix means there exists
  a path from any i to any J.
• For each pair i, J, there exists an M such that
• (PM)iJ ? 0
• Here, the problem is with bidirectional
  connectivity.

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Disjoint Subsets

• With an undirected graph, irreducible is
  equivalent to disjoint non-empty subsets.

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Solution to Disjoint Subsets

• Here it is possible to rank each of these
  subsets.
• But then we have the issue of meshing these
  disjoint rankings back together in a meaningful
  way.
• And

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The Sink

• Since we are dealing with a directed graph, we
  also have to be concerned with the elusive
  sink.
• Much more difficult to pinpoint.

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The Sink
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The Issue

• The question of irreducibility is very difficult
  to answer when a matrix contains thousands or
  even millions of rows and columns.
• A sink could be thousands, millions itself.

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The Sink

• We experimented with several methods to solve or
  alleviate this sink problem.
• Add a master node which connects to and from all
  the others.
• This alters structure, but not significantly.

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Master Node
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Alternative Method

• Alternatively, we perturb the whole matrix.
• Possibility of visiting a page by means other
  than a link (e.g. word of mouth, advertisements,
  etc.)
• Base rank, if a page has no inlinks.
• Again, we alter the structure. But, again not
  significantly.

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Perturbation
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Perturbation

• We now have
• 0 lt a lt 1
• R(u) a (1- a)
• This is often chosen as 0.85.

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Justification

• By doing this, our matrix is now everywhere
  connected!
• Stochastic and irreducible.
• So we can use the famous Power Method

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Furthermore

• Continued research
• Comparison analysis of methods to handle
• Dead ends (always exist on boundary)
• Sinks
• Convergence analysis
• Master Node
• Perturbation
• Implementaion and Results Analysis
• Analyzing micro and macro network topology
• Exploring implementation methods

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THEEND
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Stochastics and the Dead End

• What if a node links to no other nodes,
• ie we have a row of all zeros.
• THE MATRIX IS NOT STOCHASTIC
• We call these problem links, or Dead ends in
  our model.
• Some are actual Dead ends.
• But, Dead ends always exist at the boundary of
  our data gathering.

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The Dead End
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Solutions

• There are several methods to solve this problem
• 1) Remove the Deadends
• Alteration of incoming data.
• We see it does not drastically affect the page
  rank, in particular of the higher ranked pages.
  These are the most important for the rank
  relevance.

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• 2) Alter the Deadends
• Alteration of Processing
• Add a master node.
• Equally redistribute this PR to ever other page.
• Again we change the matrix, but we see it, again,
  doesnt affect the top pages.

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Redistribution
1 1 1 0
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Adjacency Matrix
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Adjacency Matrix
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Continuing
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Adjacency Matrix
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Adjacency Matrix
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Adjacency Matrix
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Matrix Format

• We have matrixA
• But, we want
• Pij

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Normalization
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Normalization
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Normalization
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Normalization
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The Stochastic Matrix
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The Stationary State
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Mathematically Speaking

• Let Ai be the binary vector of outlinks from page
  i
• Ai ( ai1 , ai2, , aiN )
• Ai 1 AiJ
• P

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• Pi (pi1 , piN)
• So, P i1 PiJ 1
• We now have a row stochastic probability matrix
  (ie we are normal)
• Unless, of course, a page (node) points to no
  others
• Ai Pi 0

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Stochastics

• Let wiT (1/N)
• where i 1, , N
• Also, let
• di 0 if i is not a dead end.
• 1 if i is a dead end.
• W dwT
• ? S W P
• is a stochastic matrix

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Mathematically Speaking

• The stochastic matrix S becomes
• G a S ( 1 a) D
• Where D ewT
• e lt1,1,,1,1gt
• wiT lt1/N,,1/Ngt as before

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Stochastics and the Dead End

• We call these problem, degenerate links, or Dead
  ends in our random walk model.
• Some are actual Dead ends.
• But, Dead ends always exist at the boundary of
  our data gathering.

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Convergence

• It has been proven that in the case of a
  reducible matrix S with at least 2 irreducible
  subclasses, we actually have ?2s 1. Moreover,
  the eigenvalues of G become ?1 1 gt ?2 a ? ?3
  ? ..
• Here a dominates the convergence of the matrix!
• Note a 1, no guaranteed convergence.

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The Dead End
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Solutions

• There are several methods to solve this problem
• 1) Remove the Deadends
• Alteration of incoming data.
• We see it does not drastically affect the page
  rank, in particular of the higher ranked pages.
  These are the most important for the rank
  relevance.

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• 2) Alter the Deadends
• Alteration of Processing
• Equally redistribute this PR to ever other page.
• Again we change the matrix, but we see it, again,
  doesnt affect the top pages.

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1 1 1 1
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Stochastics

• Let wiT (1/N)
• where i 1, , N
• Also, let
• di 0 if i is not a dead end.
• 1 if i is a dead end.
• W dwT
• ? S W P
• is a stochastic matrix

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A Stationary State

• 0.2
• 0.2 0.4 0.2

A 0.4
B 0.2
C 0.4
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• R(A) 0.5 R(B) 0.5 R(C)
• R(B) R(C)
• R(C) R(A)

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Revisited

• Let R(u) be the rank of page u.
• nu Number of outlinks from u.
• So for vs linked to u
• R(u)

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Explanation

• 0 0.5 0.5
• (R(A) R(B) R(C)) 0 0 1 (R(A) R(B) R(C))
• 1 0 0

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A Stationary State

• 0 0.5 0.5
• (0.4 0.2 0.4) 0 0 1 (0.4 0.2 0.4)
• 1 0 0

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Implementation

• Now, of course G is dense.
• But, the sparse nature of the original adjacency
  matrix can still be exploited.

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Manipulation

• Since G aP adwT (1-a)ewT
• If we multiply both sides by xT, we get
• xTG axT P xT ((1-a)e ad) wT
• axT P (1-a a( xT1 - xTP1wT)

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Manipulation

• Simply let xmT P ymT , then
• Let xm1T a ymT 1- a a( xmT 1 -
  ymT1)wT
• This is equivalent to xmT G xm1T, where we can
  utilize the properties of a stochastic
  irreducible matrix with one recurrent class.

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• This is a relatively easy and inexpensive
  calculation at each iteration in the method.
  (Fortunately, since we are dealing with 600000 or
  higher dimensions)

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A Stationary State

• 0.2
• 0.2 0.4 0.2

A 0.4
B 0.2
C 0.4
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Explanation

• 0 0.5 0.5
• (R(A) R(B) R(C)) 0 0 1 (R(A) R(B) R(C))
• 1 0 0

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A Stationary State

• 0 0.5 0.5
• (0.4 0.2 0.4) 0 0 1 (0.4 0.2 0.4)
• 1 0 0

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Displacement

• 0 0.5 0.5 1/3 1/3 1/3
• 0.85 0 0 1 0.15 1/3 1/3
  1/3
• 1 0 0 1/3 1/3 1/3
• 0.05 0.475 0.475
• 0.05 0.05 0.9
• 0.9 0.05 0.05