1. Iterative Aggregation Disaggregation Nicole Typaldos Missouri State University

2. Graph Web Graph Matrix Pagerankvector Process of Webpage ranking

3. Google’s page ranking algorithm

4. 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…

5. Example Set Up

6. Example Continued

7. Example Continued

8. The Google Matrix • Where • ℮ is a vector of ones • U is an arbitrary probabilistic vector • a is the vector for correcting dangling nodes > 0

9. Example Continued

10. Different Approaches • Power Method • Linear Systems • Iterative Aggregation Disaggregation (IAD)

11. 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

12. Rearranging H

13. 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:

14. 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=

15. 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

16. Combined methods • How to compute • Iterative Aggregation Disaggregation combined with: • Power Method • Linear Systems

17. With Power Method • = Ã • Ã is a full matrix • = =

18. With Power Method • Try to exploit the sparsity of H • solving = Ã • Exploiting dangling nodes:

19. With Power Method • Try to exploit the sparsity of H • Solving = Ã • Exploiting dangling nodes:

20. With Linear Systems • = Ã • After multiplication write as: • Since is unknown, make it arbitrary then adjust

21. With Linear Systems • Algorithm (dangling nodes) • Give an initial guess and a tolerance є • Repeat until • Solve • Adjust

22. 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.

23. Iterative Aggregation Disaggregation Nicole Typaldos Missouri State University