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A topological approach to spectral clustering
The rankability of weighted data from pairwise comparisons
1. | Department of Computer Science and Software Engineering, California Polytechnic State University, San Luis Obispo, CA, USA |
2. | Department of Mathematics and Computer Science, Davidson College, Davidson, NC, USA |
3. | Mathematics Department, College of Charleston, Charleston, SC, USA |
In prior work [
Rankability paper [
References:
[1] |
T. Achterberg,
Scip: Solving constraint integer programs, Mathematical Programming Computation, 1 (2009), 1-41.
doi: 10.1007/s12532-008-0001-1. |
[2] |
N. Ailon, M. Charikar and A. Newman, Aggregating inconsistent information: Ranking and clustering, Journal of the ACM, 55 (2008), 27 pp.
doi: 10.1145/1411509.1411513. |
[3] |
I. Ali, W. D. Cook and M. Kress,
On the minimum violations ranking of a tournament, Management Science, 32 (1986), 660-672.
doi: 10.1287/mnsc.32.6.660. |
[4] |
P. Anderson, T. Chartier and A. Langville, The rankability of data, SIAM Journal on the Mathematics of Data Science, (2019), 121–143.
doi: 10.1137/18M1183595. |
[5] |
P. Anderson, T. Chartier, A. Langville and K. Pedings-Behling, IGARDS technical report, 2019. Available from: https://igards.github.io/research/IGARDS_Technical_Report_November_2019.pdf. Google Scholar |
[6] |
P. Anderson, T. Chartier, A. Langville and K. Pedings-Behling, Revisiting rankability of unweighted data technical report, 2020. Available from: https://igards.github.io/research/Revisiting_Rankability_Unweighted_Data.pdf. Google Scholar |
[7] |
R. D. Armstrong, W. D. Cook and L. M. Seiford,
Priority ranking and consensus formation: The case of ties, Management Science, 28 (1982), 638-645.
doi: 10.1287/mnsc.28.6.638. |
[8] |
J. Brenner and P. Keating, Chaos kills, ESPN, The Magazine, (2016), 20–23. Google Scholar |
[9] |
S. Brin and L. Page, The anatomy of a large-scale hypertextual web search engine, Computer Networks and ISDN Systems, 33 (1998), 107–117.
doi: 10.1016/S0169-7552(98)00110-X. |
[10] |
S. Brin, L. Page, R. Motwami and T. Winograd, The PageRank citation ranking: Bringing order to the web, Tech. Report 1999-0120, Computer Science Department, Stanford University, 1999. Google Scholar |
[11] |
T. R. Cameron, A. N. Langville and H. C. Smith,
On the graph Laplacian and the rankability of data, Linear Algebra and its Applications, 588 (2020), 81-100.
doi: 10.1016/j.laa.2019.11.026. |
[12] |
C. R. Cassady, L. M. Maillart and S. Salman,
Ranking sports teams: A customizable quadratic assignment approach, INFORMS: Interfaces, 35 (2005), 497-510.
doi: 10.1287/inte.1050.0171. |
[13] |
T. P. Chartier, E. Kreutzer, A. N. Langville and K. E. Pedings, Sensitivity of ranking vectors, SIAM Journal on Scientific Computing, 33 (2011), 1077–1102.
doi: 10.1137/090772745. |
[14] |
B. J. Coleman,
Minimizing game score violations in college football rankings, INFORMS: Interfaces, 35 (2005), 483-496.
doi: 10.1287/inte.1050.0172. |
[15] |
W. D. Cook and L. M. Seiford,
Priority ranking and consensus formation, Management Science, 24 (1978), 1721-1732.
doi: 10.1287/mnsc.24.16.1721. |
[16] |
T. G. Dietterich,
Approximate statistical tests for comparing supervised classification learning algorithms, Neural Computation, 10 (1998), 1895-1923.
doi: 10.1162/089976698300017197. |
[17] |
S. Dutta, S. H. Jacobson and J. J. Sauppe,
Identifying ncaa tournament upsets using balance optimization subset selection, Journal of Quantitative Analysis in Sports, 13 (2017), 79-93.
|
[18] |
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979. |
[19] |
F. M. Harper and J. A. Konstan,
The movielens datasets: History and context, Acm Transactions on Interactive Intelligent Systems (TIIS), 5 (2015), 1-19.
doi: 10.1145/2827872. |
[20] |
X. Jiang, L.-H. Lim, Y. Yao and Y. Ye,
Statistical ranking and combinatorial Hodge theory, Mathematical Programming, 127 (2011), 203-244.
doi: 10.1007/s10107-010-0419-x. |
[21] |
P. Keating, personal communication. Google Scholar |
[22] |
Y. Kondo,
Triangulation of input-output tables based on mixed integer programs for inter-temporal and inter-regional comparison of production structures, Journal of Economic Structures, 3 (2014), 1-19.
doi: 10.1186/2193-2409-3-2. |
[23] |
A. N. Langville and C. D. Meyer, Who's #1? The Science of Rating and Ranking Items, Princeton University Press, Princeton, 2012.
![]() |
[24] |
A. N. Langville, K. Pedings and Y. Yamamoto, A minimum violations ranking method, Optimization and Engineering, (2011), 1–22.
doi: 10.1007/s11081-011-9135-5. |
[25] |
A. S. Lee and D. R. Shier, A method for ranking teams based on simple paths, UMAP Journal, 39 (2018), 353-371. Google Scholar |
[26] |
W. W. Leontief, Input-Output Economics, 2nd edition, Oxford University Press, 1986.
doi: 10.1038/scientificamerican1051-15. |
[27] |
M. J. Lopez and G. J. Matthews,
Building an NCAA men's basketball predictive model and quantifying its success, Journal of Quantitative Analysis in Sports, 11 (2015), 5-12.
doi: 10.1515/jqas-2014-0058. |
[28] |
R. Marti and G. Reinelt, The linear ordering problem: Exact and heuristic methods in combinatorial optimization, AMS, 2011.
doi: 10.1007/978-3-642-16729-4. |
[29] |
S. Mehrotra and Y. Ye, Finding an interior point in the optimal face of linear programs, 62 (1993), 497–515.
doi: 10.1007/BF01585180. |
[30] |
J. Park, On minimum violations ranking in paired comparisons, 2005. Google Scholar |
[31] |
G. Reinelt, The Linear Ordering Problem: Algorithms and Applications, Heldermann Verlag, 1985. |
[32] |
V. N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, Berlin, Heidelberg, 1995.
doi: 10.1007/978-1-4757-2440-0. |
[33] |
S. Wartenberg, How many people will fill out March Madness brackets?, The Columbus Dispatch, (2015). Google Scholar |
show all references
References:
[1] |
T. Achterberg,
Scip: Solving constraint integer programs, Mathematical Programming Computation, 1 (2009), 1-41.
doi: 10.1007/s12532-008-0001-1. |
[2] |
N. Ailon, M. Charikar and A. Newman, Aggregating inconsistent information: Ranking and clustering, Journal of the ACM, 55 (2008), 27 pp.
doi: 10.1145/1411509.1411513. |
[3] |
I. Ali, W. D. Cook and M. Kress,
On the minimum violations ranking of a tournament, Management Science, 32 (1986), 660-672.
doi: 10.1287/mnsc.32.6.660. |
[4] |
P. Anderson, T. Chartier and A. Langville, The rankability of data, SIAM Journal on the Mathematics of Data Science, (2019), 121–143.
doi: 10.1137/18M1183595. |
[5] |
P. Anderson, T. Chartier, A. Langville and K. Pedings-Behling, IGARDS technical report, 2019. Available from: https://igards.github.io/research/IGARDS_Technical_Report_November_2019.pdf. Google Scholar |
[6] |
P. Anderson, T. Chartier, A. Langville and K. Pedings-Behling, Revisiting rankability of unweighted data technical report, 2020. Available from: https://igards.github.io/research/Revisiting_Rankability_Unweighted_Data.pdf. Google Scholar |
[7] |
R. D. Armstrong, W. D. Cook and L. M. Seiford,
Priority ranking and consensus formation: The case of ties, Management Science, 28 (1982), 638-645.
doi: 10.1287/mnsc.28.6.638. |
[8] |
J. Brenner and P. Keating, Chaos kills, ESPN, The Magazine, (2016), 20–23. Google Scholar |
[9] |
S. Brin and L. Page, The anatomy of a large-scale hypertextual web search engine, Computer Networks and ISDN Systems, 33 (1998), 107–117.
doi: 10.1016/S0169-7552(98)00110-X. |
[10] |
S. Brin, L. Page, R. Motwami and T. Winograd, The PageRank citation ranking: Bringing order to the web, Tech. Report 1999-0120, Computer Science Department, Stanford University, 1999. Google Scholar |
[11] |
T. R. Cameron, A. N. Langville and H. C. Smith,
On the graph Laplacian and the rankability of data, Linear Algebra and its Applications, 588 (2020), 81-100.
doi: 10.1016/j.laa.2019.11.026. |
[12] |
C. R. Cassady, L. M. Maillart and S. Salman,
Ranking sports teams: A customizable quadratic assignment approach, INFORMS: Interfaces, 35 (2005), 497-510.
doi: 10.1287/inte.1050.0171. |
[13] |
T. P. Chartier, E. Kreutzer, A. N. Langville and K. E. Pedings, Sensitivity of ranking vectors, SIAM Journal on Scientific Computing, 33 (2011), 1077–1102.
doi: 10.1137/090772745. |
[14] |
B. J. Coleman,
Minimizing game score violations in college football rankings, INFORMS: Interfaces, 35 (2005), 483-496.
doi: 10.1287/inte.1050.0172. |
[15] |
W. D. Cook and L. M. Seiford,
Priority ranking and consensus formation, Management Science, 24 (1978), 1721-1732.
doi: 10.1287/mnsc.24.16.1721. |
[16] |
T. G. Dietterich,
Approximate statistical tests for comparing supervised classification learning algorithms, Neural Computation, 10 (1998), 1895-1923.
doi: 10.1162/089976698300017197. |
[17] |
S. Dutta, S. H. Jacobson and J. J. Sauppe,
Identifying ncaa tournament upsets using balance optimization subset selection, Journal of Quantitative Analysis in Sports, 13 (2017), 79-93.
|
[18] |
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979. |
[19] |
F. M. Harper and J. A. Konstan,
The movielens datasets: History and context, Acm Transactions on Interactive Intelligent Systems (TIIS), 5 (2015), 1-19.
doi: 10.1145/2827872. |
[20] |
X. Jiang, L.-H. Lim, Y. Yao and Y. Ye,
Statistical ranking and combinatorial Hodge theory, Mathematical Programming, 127 (2011), 203-244.
doi: 10.1007/s10107-010-0419-x. |
[21] |
P. Keating, personal communication. Google Scholar |
[22] |
Y. Kondo,
Triangulation of input-output tables based on mixed integer programs for inter-temporal and inter-regional comparison of production structures, Journal of Economic Structures, 3 (2014), 1-19.
doi: 10.1186/2193-2409-3-2. |
[23] |
A. N. Langville and C. D. Meyer, Who's #1? The Science of Rating and Ranking Items, Princeton University Press, Princeton, 2012.
![]() |
[24] |
A. N. Langville, K. Pedings and Y. Yamamoto, A minimum violations ranking method, Optimization and Engineering, (2011), 1–22.
doi: 10.1007/s11081-011-9135-5. |
[25] |
A. S. Lee and D. R. Shier, A method for ranking teams based on simple paths, UMAP Journal, 39 (2018), 353-371. Google Scholar |
[26] |
W. W. Leontief, Input-Output Economics, 2nd edition, Oxford University Press, 1986.
doi: 10.1038/scientificamerican1051-15. |
[27] |
M. J. Lopez and G. J. Matthews,
Building an NCAA men's basketball predictive model and quantifying its success, Journal of Quantitative Analysis in Sports, 11 (2015), 5-12.
doi: 10.1515/jqas-2014-0058. |
[28] |
R. Marti and G. Reinelt, The linear ordering problem: Exact and heuristic methods in combinatorial optimization, AMS, 2011.
doi: 10.1007/978-3-642-16729-4. |
[29] |
S. Mehrotra and Y. Ye, Finding an interior point in the optimal face of linear programs, 62 (1993), 497–515.
doi: 10.1007/BF01585180. |
[30] |
J. Park, On minimum violations ranking in paired comparisons, 2005. Google Scholar |
[31] |
G. Reinelt, The Linear Ordering Problem: Algorithms and Applications, Heldermann Verlag, 1985. |
[32] |
V. N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, Berlin, Heidelberg, 1995.
doi: 10.1007/978-1-4757-2440-0. |
[33] |
S. Wartenberg, How many people will fill out March Madness brackets?, The Columbus Dispatch, (2015). Google Scholar |









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Dominance Method | Parameters | Method | MAE | |
1 | Direct+Indirect | dt=0, st=1, wi=1 | Hillside | 3.148221 |
2 | Direct+Indirect | dt=0, st=0, wi=1 | Hillside | 3.148221 |
3 | Direct+Indirect | dt=1, st=0, wi=1 | LOP | 3.172793 |
4 | Direct+Indirect | dt=1, st=1, wi=1 | LOP | 3.172793 |
5 | Direct+Indirect | dt=0, st=0, wi=0.25 | Hillside | 3.213978 |
6 | Direct+Indirect | dt=0, st=1, wi=0.25 | Hillside | 3.213978 |
7 | Direct | dt=2 | Hillside | 3.309455 |
8 | Direct+Indirect | dt=2, st=1, wi=1 | LOP | 3.311588 |
9 | Direct+Indirect | dt=2, st=0, wi=1 | LOP | 3.311588 |
10 | Direct+Indirect | dt=2, st=2, wi=0.5 | Hillside | 3.331698 |
Dominance Method | Parameters | Method | MAE | |
1 | Direct+Indirect | dt=0, st=1, wi=1 | Hillside | 3.148221 |
2 | Direct+Indirect | dt=0, st=0, wi=1 | Hillside | 3.148221 |
3 | Direct+Indirect | dt=1, st=0, wi=1 | LOP | 3.172793 |
4 | Direct+Indirect | dt=1, st=1, wi=1 | LOP | 3.172793 |
5 | Direct+Indirect | dt=0, st=0, wi=0.25 | Hillside | 3.213978 |
6 | Direct+Indirect | dt=0, st=1, wi=0.25 | Hillside | 3.213978 |
7 | Direct | dt=2 | Hillside | 3.309455 |
8 | Direct+Indirect | dt=2, st=1, wi=1 | LOP | 3.311588 |
9 | Direct+Indirect | dt=2, st=0, wi=1 | LOP | 3.311588 |
10 | Direct+Indirect | dt=2, st=2, wi=0.5 | Hillside | 3.331698 |
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