Post on 25-Feb-2016
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ITERATIVE AGGREGATION DISAGGREGATION
Web Graph
Matrix Page rank vector
PROCESS OF WEBPAGE RANKING
Graph
GOOGLE’S PAGE RANKING
ALGORITHM
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 >0 for some k=1,2,3…
EXAMPLE SET UP
EXAMPLE CONTINUED
EXAMPLE CONTINUED
THE GOOGLE MATRIX
Where
℮ is a vector of ones
U is an arbitrary probabilistic vector
a is the vector for correcting dangling nodes
> 0
EXAMPLE CONTINUED
DIFFERENT APPROACHES
Power Method
Linear Systems
Iterative Aggregation Disaggregation (IAD)
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
REARRANGING H
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:
EXACT AGGREGATION
DISAGGREGATION
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=
APPROXIMATE AGGREGATION
DISAGGREGATION Algorithm
Select an arbitrary probabilistic vector
and a tolerance є
For k = 1,2, … Find the stationary distribution of
Set
Let
If then stop
Otherwise
COMBINED METHODS
How to compute
Iterative Aggregation Disaggregation
combined with:
Power Method
Linear Systems
WITH POWER METHOD
= Ã
à is a full matrix
=
=
WITH POWER METHOD
Try to exploit the sparsity of H
solving = Ã
Exploiting dangling nodes:
WITH POWER METHOD
Try to exploit the sparsity of H
Solving = Ã
Exploiting dangling nodes:
WITH LINEAR SYSTEMS
= Ã
After multiplication write as:
Since is unknown, make it arbitrary then adjust
WITH LINEAR SYSTEMS
Algorithm (dangling nodes)
Give an initial guess and a tolerance є
Repeat until
Solve
Adjust
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.
ITERATIVE AGGREGATION DISAGGREGATION