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 R 0 while (not converged) for each vertex i end

4. HITS Algorithm initialize authority and hub weights, x 0 and y 0 while (not converged) for each vertex i 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 SizeIterations 10004 20005 40005 80005 160006 SizeIterations 10004 20005 40005 80005 160006

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