Convergence of Page Rank and HITS Algorithms Victor

Convergence of Page. Rank and HITS Algorithms Victor Boyarshinov Eric Anderson 12/5/02

Outline Algorithms ¢ Convergence ¢ Graph data and a bad graph ¢ Results ¢

Page. Rank 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 l Sum of changes between iterations l Change of top q% of weights l ¢ Choice: sum of changes

Performance of Page. Rank 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 Page. Rank Results Size Iterations 1000 4 2000 5 4000 5 8000 5 16000 6

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

Conclusion Page. Rank is stable ¢ HITS is stable ¢ Nearly doubling the diameter has no noticeable effect on convergence ¢