Convergence of PageRank and HITS AlgorithmsVictor Boyarshinov

Eric Anderson

12/5/02

Outline

Algorithms Convergence Graph data and a bad graph Results

PageRank Algorithm

initialize ranks R0while (not converged)

for each vertex i

end

end

iBj j

kk N

jRddiR

)()1()(1

HITS Algorithm

initialize authority and hub weights, x0 and y0

while (not converged)

for each vertex i

end

end

iFjkk jxiy )()(1

iBjkk jyix )()(1

Convergence

Many sensible options:Maximum change between iterationsSum of changes between iterationsChange 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

Size Iterations

1000 4

2000 5

4000 5

8000 5

16000 6

Size Iterations

1000 4

2000 5

4000 5

8000 5

16000 6

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