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