Nicole Typaldos

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Iterative Aggregation Disaggregation

• Nicole Typaldos
• Missouri State University

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Process of Webpage ranking
Graph
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Googles page ranking algorithm

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Conditioning the matrix H

• Definitions
• Reducible if there exist a permutation matrix
  Pnxn and an integer 1 r n-1 such that
• otherwise if a matrix is not reducible then it is
  irreducible
• Primitive if and only if Ak gt0 for some
  k1,2,3

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Example Set Up
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Example Continued
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Example Continued
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The Google Matrix
gt 0

• Where
• e is a vector of ones
• U is an arbitrary probabilistic vector
• a is the vector for correcting dangling nodes

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Example Continued
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Different Approaches

• Power Method
• Linear Systems
• Iterative Aggregation Disaggregation (IAD)

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Linear Systems andDangling Nodes

• Simplify computation by arranging dangling nodes
  of H in the lower rows
• Rewrite by reordering dangling nodes
• Where is a square matrix that represents
  links between nondangling nodes to nondangling
  nodes is a square matrix representing
  links to dangling nodes

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Rearranging H
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Exact aggregation disaggregation

• Theorem
• If G transition matrix for an irreducible Markov
  chain with stochastic complement
• is the stationary dist of S, and
  is the stationary distribution of A then the
  stationary of G is given by

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Approximate aggregation disaggregation

• Problem Computing S and is too
  difficult
• and too expensive. So,
• Ã
• Where A and à differ only by one row
• Rewrite as
• Ã

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Approximate aggregation disaggregation

• Algorithm
• Select an arbitrary probabilistic vector
• and a tolerance ?
• For k 1,2,
• Find the stationary distribution of
• Set
• Let
• If then stop
• Otherwise

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Combined methods

• How to compute
• Iterative Aggregation Disaggregation
• combined with
• Power Method
• Linear Systems

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With Power Method

• Ã
• Ã is a full matrix

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With Power Method

• Try to exploit the sparsity of H
• solving Ã
• Exploiting dangling nodes

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With Power Method

• Try to exploit the sparsity of H
• Solving Ã
• Exploiting dangling nodes

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With Linear Systems

• Ã
• After multiplication write as
• Since is unknown, make it arbitrary then
  adjust

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With Linear Systems

• Algorithm (dangling nodes)
• Give an initial guess and a tolerance ?
• Repeat until
• Solve
• Adjust

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References

• Berry, Michael W. and Murray Browne.
  Understanding Search Engines Mathematical
  Modeling and Text Retrieval. Philadelphia, PA
  Society for Industrial and Applied Mathematics,
  2005.
• Langville, Amy N. and Carl D. Meyer. Google's
  PageRank and Beyond The Science of Search Engine
  Rankings. Princeton, New Jersey Princeton
  University Press, 2006.
• "Updating Markov Chains with an eye on Google's
  PageRank." Society for Industrial and Applied
  mathematics (2006) 968-987.
• Rebaza, Jorge. "Ranking Web Pages." Mth 580 Notes
  (2008) 97-153.
•  

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Iterative Aggregation Disaggregation

• Nicole Typaldos
• Missouri State University