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Effect of stepwise adjustment of
Damping factor upon PageRank
Subhajit Sahu1
, Kishore Kothapalli1
, Dip Sankar Banerjee2
1
International Institute of Information Technology, Hyderabad
2
Indian Institute of Technology, Jodhpur
Abstract — The effect of adjusting damping factor α, from a small initial value α0 to
the final desired αf value, upon then iterations needed for PageRank computation is
observed. Adjustment of the damping factor is done in one or more steps. Results
show no improvement in performance over a fixed damping factor based PageRank.
Index terms — PageRank algorithm, Step-wise adjustment, Damping factor.
1. Introduction
Web graphs are by default reducible, and thus the convergence rate of the
power-iteration method is the rate at which αk
→ 0, where α is the damping
factor, and k is the iteration count. An estimate of the number of iterations
needed to converge to a tolerance τ is log10 τ / log10 α [1]. For τ = 10-6
and α =
0.85, it can take roughly 85 iterations to converge. For α = 0.95, and α = 0.75,
with the same tolerance τ = 10-6
, it takes roughly 269 and 48 iterations
respectively. For τ = 10-9
, and τ = 10-3
, with the same damping factor α =
0.85, it takes roughly 128 and 43 iterations respectively. Thus, adjusting the
damping factor or the tolerance parameters of the PageRank algorithm can
have a significant effect on the convergence rate.
2. Method
The idea behind this experiment was to adjust the damping factor α in steps,
to see if it might help reduce PageRank computation time. The PageRank
computation first starts with a small initial damping factor α = α0. After the
2
ranks have converged, the damping factor α is updated to the next damping
factor step, say α1 and PageRank computation is continued again. This is
done until the final desired value of αf is reached. For example, the
computation starts initially with α = α0 = 0.5, lets ranks converge quickly, and
then switches to α = αf = 0.85 and continues PageRank computation until it
converges. This single-step change is attempted with the initial (fast
converge) damping factor α0 from 0.1 to 0.8. Similar to this, two-step,
three-step, and four-step changes are also attempted. With a two-step
approach, a midpoint between the initial damping value α0 and αf = 0.85 is
selected as well for the second set of iterations. Similarly, three-step and
four-step approaches use two and three midpoints respectively.
3. Experimental setup
A small sample graph is used in this experiment, which is stored in the
MatrixMarket (.mtx) file format. The experiment is implemented in Node.js,
and executed on a personal laptop. Only the iteration count of each test case
is measured. The tolerance τ = 10-5
is used for all test cases. Statistics of
each test case is printed to standard output (stdout), and redirected to a log
file, which is then processed with a script to generate a CSV file, with each
row representing the details of a single test case. This CSV file is imported
into Google Sheets, and necessary tables are set up with the help of the
FILTER function to create the charts.
4. Results
From the results, as shown in figure 4.1, it is clear that modifying the
damping factor α in steps is not a good idea. The standard fixed damping
factor PageRank, with α = 0.85, converges in 35 iterations. Using a
single-step approach increases the total number of iterations required, by at
least 4 iterations (with an initial damping factor α0 = 0.1). Increasing α0
further increases the total number of iterations needed for computation.
Switching to a multi-step approach also increases the number of iterations
needed for convergence. The two-step, three-step, and four-step approaches
3
require a total of atleast 49, 60, and 71 iterations respectively. Again,
increasing α0 continues to increase the total number of iterations needed for
computation. A possible explanation for this effect is that the ranks for
different values of the damping factor α are significantly different, and
switching to a different damping factor α after each step mostly leads to
recomputation.
Figure 4.1: Iterations required for PageRank computation, when damping factor α is
adjusted in 1-4 steps, starting with an initial small damping factor α0
(damping_start). 0-step is the fixed damping factor PageRank, with α = 0.85.
5. Conclusion
Adjusting the damping factor α in steps is not a good idea, most likely
because ranks obtained for even nearby damping factors are significantly
different. Fixed damping factor PageRank continues to be the best approach.
The links to source code, along with data sheets and charts, for adjusting
damping factor in steps [2] is included in references.
4
References
[1] A. Langville and C. Meyer, “Deeper Inside PageRank,” Internet Math.,
vol. 1, no. 3, pp. 335–380, Jan. 2004, doi:
10.1080/15427951.2004.10129091.
[2] S. Sahu, “puzzlef/pagerank-adjust-damping-factor-stepwise.js:
Experimenting PageRank improvement by adjusting damping factor (α)
between iterations.”
https://github.com/puzzlef/pagerank-adjust-damping-factor-stepwise.js
(accessed Aug. 06, 2021).

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Effect of stepwise adjustment of Damping factor upon PageRank : REPORT

  • 1. 1 Effect of stepwise adjustment of Damping factor upon PageRank Subhajit Sahu1 , Kishore Kothapalli1 , Dip Sankar Banerjee2 1 International Institute of Information Technology, Hyderabad 2 Indian Institute of Technology, Jodhpur Abstract — The effect of adjusting damping factor α, from a small initial value α0 to the final desired αf value, upon then iterations needed for PageRank computation is observed. Adjustment of the damping factor is done in one or more steps. Results show no improvement in performance over a fixed damping factor based PageRank. Index terms — PageRank algorithm, Step-wise adjustment, Damping factor. 1. Introduction Web graphs are by default reducible, and thus the convergence rate of the power-iteration method is the rate at which αk → 0, where α is the damping factor, and k is the iteration count. An estimate of the number of iterations needed to converge to a tolerance τ is log10 τ / log10 α [1]. For τ = 10-6 and α = 0.85, it can take roughly 85 iterations to converge. For α = 0.95, and α = 0.75, with the same tolerance τ = 10-6 , it takes roughly 269 and 48 iterations respectively. For τ = 10-9 , and τ = 10-3 , with the same damping factor α = 0.85, it takes roughly 128 and 43 iterations respectively. Thus, adjusting the damping factor or the tolerance parameters of the PageRank algorithm can have a significant effect on the convergence rate. 2. Method The idea behind this experiment was to adjust the damping factor α in steps, to see if it might help reduce PageRank computation time. The PageRank computation first starts with a small initial damping factor α = α0. After the
  • 2. 2 ranks have converged, the damping factor α is updated to the next damping factor step, say α1 and PageRank computation is continued again. This is done until the final desired value of αf is reached. For example, the computation starts initially with α = α0 = 0.5, lets ranks converge quickly, and then switches to α = αf = 0.85 and continues PageRank computation until it converges. This single-step change is attempted with the initial (fast converge) damping factor α0 from 0.1 to 0.8. Similar to this, two-step, three-step, and four-step changes are also attempted. With a two-step approach, a midpoint between the initial damping value α0 and αf = 0.85 is selected as well for the second set of iterations. Similarly, three-step and four-step approaches use two and three midpoints respectively. 3. Experimental setup A small sample graph is used in this experiment, which is stored in the MatrixMarket (.mtx) file format. The experiment is implemented in Node.js, and executed on a personal laptop. Only the iteration count of each test case is measured. The tolerance τ = 10-5 is used for all test cases. Statistics of each test case is printed to standard output (stdout), and redirected to a log file, which is then processed with a script to generate a CSV file, with each row representing the details of a single test case. This CSV file is imported into Google Sheets, and necessary tables are set up with the help of the FILTER function to create the charts. 4. Results From the results, as shown in figure 4.1, it is clear that modifying the damping factor α in steps is not a good idea. The standard fixed damping factor PageRank, with α = 0.85, converges in 35 iterations. Using a single-step approach increases the total number of iterations required, by at least 4 iterations (with an initial damping factor α0 = 0.1). Increasing α0 further increases the total number of iterations needed for computation. Switching to a multi-step approach also increases the number of iterations needed for convergence. The two-step, three-step, and four-step approaches
  • 3. 3 require a total of atleast 49, 60, and 71 iterations respectively. Again, increasing α0 continues to increase the total number of iterations needed for computation. A possible explanation for this effect is that the ranks for different values of the damping factor α are significantly different, and switching to a different damping factor α after each step mostly leads to recomputation. Figure 4.1: Iterations required for PageRank computation, when damping factor α is adjusted in 1-4 steps, starting with an initial small damping factor α0 (damping_start). 0-step is the fixed damping factor PageRank, with α = 0.85. 5. Conclusion Adjusting the damping factor α in steps is not a good idea, most likely because ranks obtained for even nearby damping factors are significantly different. Fixed damping factor PageRank continues to be the best approach. The links to source code, along with data sheets and charts, for adjusting damping factor in steps [2] is included in references.
  • 4. 4 References [1] A. Langville and C. Meyer, “Deeper Inside PageRank,” Internet Math., vol. 1, no. 3, pp. 335–380, Jan. 2004, doi: 10.1080/15427951.2004.10129091. [2] S. Sahu, “puzzlef/pagerank-adjust-damping-factor-stepwise.js: Experimenting PageRank improvement by adjusting damping factor (α) between iterations.” https://github.com/puzzlef/pagerank-adjust-damping-factor-stepwise.js (accessed Aug. 06, 2021).