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Convergence of PageRank and HITS Algorithms

Convergence of PageRank and HITS Algorithms. Victor Boyarshinov Eric Anderson 12/5/02. Outline. Algorithms Convergence Graph data and a bad graph Results. PageRank Algorithm. initialize ranks R 0 while (not converged) for each vertex i end end. HITS Algorithm.

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Convergence of PageRank and HITS Algorithms

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  1. Convergence of PageRank and HITS Algorithms Victor Boyarshinov Eric Anderson 12/5/02

  2. Outline • Algorithms • Convergence • Graph data and a bad graph • Results

  3. PageRank Algorithm initialize ranks R0 while (not converged) for each vertex i end end

  4. HITS Algorithm initialize authority and hub weights, x0 and y0 while (not converged) for each vertex i end end

  5. Convergence • Many sensible options: • Maximum change between iterations • Sum of changes between iterations • Change of top q% of weights • Choice: sum of changes

  6. Performance of PageRank • Converges in O(log(n)) iterations on expander graphs • Motivation: propagation depends on diameter • Iterations are expensive • Constant in order could have a large influence

  7. Graph Data • Synthetic data • Erdös-Rényi model • Chose to keep mean out-degree constant • Standard mean out-degree: 10 • Size on the order of thousands of vertices

  8. Bad Graph • Constructed from two random graphs of equal size • Single link and backlink from one cluster to the other • Idea: bottleneck slows propagation • Hypothesis: iterations will grow like diameter: twice that of each cluster • Check: O(2*log(n/2)) iterations?

  9. Some PageRank Results

  10. Summary of PageRank results • Hypothesis failed completely • Changing edge probability changes iterations, but not comparative performance • Seemingly impossible to stump PageRank

  11. Conclusion • PageRank is stable • HITS is stable • Nearly doubling the diameter has no noticeable effect on convergence

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