puzzlef/pagerank-adjust-tolerance-function: Comparing the effect of using different functions for convergence check, with PageRank

Part of the report Adjusting PageRank parameters and Comparing results.

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Comparing the effect of using different functions for convergence check, with PageRank (pull, CSR).

This experiment was for comparing the performance between:

1. Find pagerank using L1 norm for convergence check.
2. Find pagerank using L2 norm for convergence check.
3. Find pagerank using L∞ norm for convergence check.

It is observed that a number of error functions are in use for checking convergence of PageRank computation. Although L1 norm is commonly used for convergence check, it appears nvGraph uses L2 norm instead. Another person in stackoverflow seems to suggest the use of per-vertex tolerance comparison, which is essentially the L∞ norm. The L1 norm ||E||₁ between two (rank) vectors r and s is calculated as Σ|rₙ - sₙ|, or as the sum of absolute errors. The L2 norm ||E||₂ is calculated as √Σ|rₙ - sₙ|2, or as the square-root of the sum of squared errors (euclidean distance between the two vectors). The L∞ norm ||E||ᵢ is calculated as max(|rₙ - sₙ|), or as the maximum of absolute errors.

This experiment was for comparing the performance between PageRank computation with L1, L2 and L∞ norms as convergence check, for damping factor α = 0.85, and tolerance τ = 10⁻⁶. 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 commonteleport contribution, α is the damping factor, rₙ is theprevious 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 fromany vertex in the graph due to the damping factor (1-α)/N, andteleport 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 execution time of each test case is measured using std::chrono::high_performance_timer. This is done 5 times for each test case, and timings are averaged (AM). The iterations taken with each test case is also 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.

From the results it is clear that PageRank computation with L∞ norm as convergence check is the fastest , quickly followed by L2 norm, and finally L1 norm. Thus, when comparing two or more approaches for an iterative algorithm, it is important to ensure that all of them use the same error function as convergence check (and the same parameter values). This would help ensure a level ground for a good relative performance comparison.

Also note in below charts that PageRank computation with L∞ norm as convergence check completes in a single iteration for all the road networks (ending with _osm). This is likely because it is calculated as ||E||ᵢ = max(|rₙ - sₙ|), and depending upon the order (number of vertices) N of the graph (those graphs are quite large), the maximum rank change for any single vertex does not exceed the tolerance τ value of 10⁻⁶.

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)
# [00457.972 ms; 063 iters.] [0.0000e+00 err.] pagerankL1Norm
# [00319.613 ms; 044 iters.] [3.6018e-05 err.] pagerankL2Norm
# [00298.079 ms; 041 iters.] [6.1306e-05 err.] pagerankLiNorm
#
# ...
#
# Loading graph /home/subhajit/data/soc-LiveJournal1.mtx ...
# order: 4847571 size: 68993773 {}
# order: 4847571 size: 68993773 {} (transposeWithDegree)
# [14178.771 ms; 051 iters.] [0.0000e+00 err.] pagerankL1Norm
# [06703.683 ms; 024 iters.] [4.4204e-04 err.] pagerankL2Norm
# [04735.651 ms; 017 iters.] [1.7652e-03 err.] pagerankLiNorm
#
# ...

<br> <br>

• RAPIDS nvGraph NVIDIA graph library
• How to check for Page Rank convergence?
• L0 Norm, L1 Norm, L2 Norm & L-Infinity Norm
• PageRank Algorithm, Mining massive Datasets (CS246), Stanford University
• Weighted Geometric Mean Selected for SPECviewperf® Composite Numbers
• SuiteSparse Matrix Collection

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