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

HITS Algorithm initialize authority and hub weights, x 0 and y 0 while (not converged) for each vertex i end

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

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

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

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?

Some PageRank Results SizeIterations SizeIterations

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

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