puzzlef/pagerank-adjust-damping-factor: For Adjusting PageRank parameters and Comparing results [v1]

Part of the report Adjusting PageRank parameters and Comparing results.

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Comparing the effect of using different values of damping factor, with PageRank (pull, CSR).

Adjustment of the damping factor α is a delicate balancing act. For smaller values of α, the convergence is fast, but the link structure of the graph used to determine ranks is less true. Slightly different values for α can produce very different rank vectors. Moreover, as α → 1, convergence slows down drastically, and sensitivity issues begin to surface.

For this experiment, the damping factor α (which is usually 0.85) is varied from 0.50 to 1.00 in steps of 0.05. This is in order to compare the performance variation with each damping factor. The calculated error is the L1 norm with respect to default PageRank (α = 0.85). The PageRank algorithm used here is the standard power-iteration (pull) based PageRank. The rank of a vertex in an iteration is calculated as c₀ + αΣrₙ/dₙ, where c₀ is the common teleport contribution, α is the damping factor, rₙ is the previous rank of vertex with an incoming edge, dₙ is the out-degree of the incoming-edge vertex, and N is the total number of vertices in the graph. The common teleport contribution c₀, calculated as (1-α)/N + αΣrₙ/N , includes the contribution due to a teleport from any vertex in the graph due to the damping factor (1-α)/N, and teleport from dangling vertices (with no outgoing edges) in the graph αΣrₙ/N. This is because a random surfer jumps to a random page upon visiting a page with no links, in order to avoid the rank-sink effect.

All seventeen graphs used in this experiment are stored in the MatrixMarket (.mtx) file format, and obtained from the SuiteSparse Matrix Collection. These include: web-Stanford, web-BerkStan, web-Google, web-NotreDame, soc-Slashdot0811, soc-Slashdot0902, soc-Epinions1, coAuthorsDBLP, coAuthorsCiteseer, soc-LiveJournal1, coPapersCiteseer, coPapersDBLP, indochina-2004, italy_osm, great-britain_osm, germany_osm, asia_osm. The experiment is implemented in C++, and compiled using GCC 9 with optimization level 3 (-O3). The system used is a Dell PowerEdge R740 Rack server with two Intel Xeon Silver 4116 CPUs @ 2.10GHz, 128GB DIMM DDR4 Synchronous Registered (Buffered) 2666 MHz (8x16GB) DRAM, and running CentOS Linux release 7.9.2009 (Core). The iterations taken with each test case is measured. 500 is the maximum iterations allowed. 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.

Results indicate that increasing the damping factor α beyond 0.85 significantly increases convergence time , and lowering it below 0.85 decreases convergence time. As the damping factor α increases linearly, the iterations needed for PageRank computation increases almost exponentially. On average, using a damping factor α = 0.95 increases iterations needed by 190% (~2.9x), and using a damping factor α = 0.75 decreases it by 41% (~0.6x), compared to damping factor α = 0.85. Note that a higher damping factor implies that a random surfer follows links with higher probability (and jumps to a random page with lower probability).

All outputs are saved in out and a small part of the output is listed here. Some charts are also included below, generated from sheets. The input data used for this experiment is available at "graphs" (for small ones), and the SuiteSparse Matrix Collection. This experiment was done with guidance from Prof. Dip Sankar Banerjee and Prof. Kishore Kothapalli.

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$ g++ -O3 main.cxx
$ ./a.out ~/data/min-1DeadEnd.mtx
$ ./a.out ~/data/min-2SCC.mtx
$ ...

# ...
#
# Loading graph /home/subhajit/data/web-Stanford.mtx ...
# order: 281903 size: 2312497 {}
# order: 281903 size: 2312497 {} (transposeWithDegree)
# [00405.436 ms; 063 iters.] [0.0000e+00 err.] pagerank
# [03199.666 ms; 500 iters.] [1.1731e+00 err.] pagerank [damping=1.00]
# [01233.223 ms; 192 iters.] [3.3279e-01 err.] pagerank [damping=0.95]
# [00617.424 ms; 096 iters.] [1.3471e-01 err.] pagerank [damping=0.90]
# [00405.537 ms; 063 iters.] [2.1264e-07 err.] pagerank [damping=0.85]
# [00302.939 ms; 047 iters.] [1.0787e-01 err.] pagerank [damping=0.80]
# [00238.925 ms; 037 iters.] [2.0008e-01 err.] pagerank [damping=0.75]
# [00193.744 ms; 030 iters.] [2.8214e-01 err.] pagerank [damping=0.70]
# [00161.573 ms; 025 iters.] [3.5654e-01 err.] pagerank [damping=0.65]
# [00135.776 ms; 021 iters.] [4.2481e-01 err.] pagerank [damping=0.60]
# [00116.598 ms; 018 iters.] [4.8881e-01 err.] pagerank [damping=0.55]
# [00103.777 ms; 016 iters.] [5.5010e-01 err.] pagerank [damping=0.50]
#
# ...
#
# Loading graph /home/subhajit/data/soc-LiveJournal1.mtx ...
# order: 4847571 size: 68993773 {}
# order: 4847571 size: 68993773 {} (transposeWithDegree)
# [11693.781 ms; 051 iters.] [0.0000e+00 err.] pagerank
# [121446.203 ms; 500 iters.] [8.6730e-01 err.] pagerank [damping=1.00]
# [35313.477 ms; 152 iters.] [1.9719e-01 err.] pagerank [damping=0.95]
# [17886.010 ms; 077 iters.] [8.6216e-02 err.] pagerank [damping=0.90]
# [11846.550 ms; 051 iters.] [1.1429e-07 err.] pagerank [damping=0.85]
# [08593.597 ms; 037 iters.] [7.3479e-02 err.] pagerank [damping=0.80]
# [06973.961 ms; 030 iters.] [1.3968e-01 err.] pagerank [damping=0.75]
# [06038.274 ms; 026 iters.] [1.9829e-01 err.] pagerank [damping=0.70]
# [05117.046 ms; 022 iters.] [2.5813e-01 err.] pagerank [damping=0.65]
# [04184.997 ms; 018 iters.] [3.1309e-01 err.] pagerank [damping=0.60]
# [03957.167 ms; 017 iters.] [3.5808e-01 err.] pagerank [damping=0.55]
# [03489.042 ms; 015 iters.] [3.9641e-01 err.] pagerank [damping=0.50]
#
# ...

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• Deeper Inside PageRank :: Amy N. Langville, Carl D. Meyer
• PageRank Algorithm, Mining massive Datasets (CS246), Stanford University
• SuiteSparse Matrix Collection

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