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I TERATIVE A GGREGATION D ISAGGREGATION P ROCESS OF W EBPAGE RANKING Web Page rank vector Graph Matrix G OOGLE ’ S PAGE RANKING ALGORITHM C ONDITIONING 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… |λ1| > |λ2| Power Method Converges Irreducible Unique Dominant Eigenvector Primitive E XAMPLE S ET U P E XAMPLE C ONTINUED E XAMPLE C ONTINUED T HE G OOGLE M ATRIX >0 Where ℮ is a vector of ones U is an arbitrary probabilistic vector a is the vector for correcting dangling nodes E XAMPLE C ONTINUED D IFFERENT A PPROACHES Power Method Linear Systems Iterative Aggregation Disaggregation (IAD) L INEAR S YSTEMS AND D ANGLING N ODES 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 R EARRANGING H E XACT 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: A PPROXIMATE 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= A PPROXIMATE AGGREGATION DISAGGREGATION Algorithm Select an arbitrary probabilistic vector and a tolerance є For k = 1,2, … Find the stationary distribution Set Let If Otherwise then stop of C OMBINED METHODS How to compute Iterative Aggregation Disaggregation combined with: Power Method Linear Systems W ITH P OWER M ETHOD = à à is a full matrix = = W ITH P OWER M ETHOD Try to exploit the sparsity of H solving = à Exploiting dangling nodes: W ITH P OWER M ETHOD Try to exploit the sparsity of H Solving = à Exploiting dangling nodes: W ITH L INEAR S YSTEMS = à After multiplication write as: Since is unknown, make it arbitrary then adjust W ITH L INEAR S YSTEMS Algorithm (dangling nodes) Give an initial guess Repeat until Solve Adjust and a tolerance є R EFERENCES 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. I TERATIVE A GGREGATION D ISAGGREGATION