Convergence of PageRank and HITS Algorithms

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(2log(n/2)) iterations?

9
Some PageRank Results
Size Iterations
1000 4
2000 5
4000 5
8000 5
16000 6
Size Iterations
1000 4
2000 5
4000 5
8000 5
16000 6
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