American Institute of Mathematical Sciences

March  2015, 10(1): 101-125. doi: 10.3934/nhm.2015.10.101

Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index)

 1 Institut de Matemàtica Multidisciplinària, Universitat Politècnica de València, 46022 Valencia, Spain 2 Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid), Spain 3 Departamento de Economía de la Empresa, Universidad Carlos III, 28903 Getafe (Madrid), Spain

Received  July 2014 Revised  December 2014 Published  February 2015

In this paper we extend the concept of Competitivity Graph to compare series of rankings with ties ( partial rankings). We extend the usual method used to compute Kendall's coefficient for two partial rankings to the concept of evolutive Kendall's coefficient for a series of partial rankings. The theoretical framework consists of a four-layer multiplex network. Regarding the treatment of ties, our approach allows to define a tie between two values when they are close enough, depending on a threshold. We show an application using data from the Spanish Stock Market; we analyse the series of rankings defined by $25$ companies that have contributed to the IBEX-35 return and volatility values over the period 2003 to 2013.
Citation: Francisco Pedroche, Regino Criado, Esther García, Miguel Romance, Victoria E. Sánchez. Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Networks and Heterogeneous Media, 2015, 10 (1) : 101-125. doi: 10.3934/nhm.2015.10.101
References:
 [1] A. Abhyankar, L. S. Copeland and W. Wong, Nonlinear dynamics in real-time equity market indices: Evidence from the United Kingdom, The Economic Journal, 105 (1995), 864-880. doi: 10.2307/2235155. [2] N. Ailon, M. Charikar and A. Newman, Aggregating inconsistent information: Ranking and clustering, Journal of the ACM, 55 (2008), Art. 23, 27 pp. doi: 10.1145/1411509.1411513. [3] J. Bar-Ilan, Comparing rankings of search results on the web, Information Processing and Management, 41 (2005), 1511-1519. doi: 10.1016/j.ipm.2005.03.008. [4] N. Betzler, R. Bredereck and R. Niedermeier, Partial kernelization for rank aggregation: Theory and experiments, in Parameterized and Exact Computation, Lecture Notes in Computer Sciences, 6478, Springer, Berlin, 2010, 26-37. doi: 10.1007/978-3-642-17493-3_5. [5] T. Biedl, F. J. Brandenburg and X. Deng, Crossings and permutations, Graph Drawing, Lecture Notes in Computer Sciences, 3843, Springer, Berlin, 2006, 1-12. doi: 10.1007/11618058_1. [6] S. Boccaletti, G. Bianconi, R. Criado, C. I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang and M. Zanin, The structure and dynamics of multilayer networks, Physics Reports, 544 (2014), 1-122. doi: 10.1016/j.physrep.2014.07.001. [7] R. A. Brealey, S. C. Myers and F. Allen, Principios de Finanzas Corporativas (in Spanish), $8^{th}$ edition, Mc Graw-Hill, 2006. [8] E. F. Brigham and P. R. Daves, International Financial Management, South-Western, 2002. [9] B. Carterette, On rank correlation and the distance between rankings, in Proc. of the 32nd Int. ACM Conf. Research and Development in Information Retrieval, ACM, New York, 2009, 436-443. doi: 10.1145/1571941.1572017. [10] W. W. Cohen, R. E. Schapire and Y. Singer, Learning to order things, Journal of Artificial Intelligence Research, 10 (1999), 243-270. doi: 10.1613/jair.587. [11] V. Conitzer, A. Davenport and Y. Heights, Improved bounds for computing Kemeny rankings, in Proceedings of The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference, 2006, 620-626. [12] W. D. Cook, M. Kress and L. M. Seiford, An axiomatic approach to distance on partial orderings, RAIRO, Recherche Operationelle/Operations Research, 20 (1986), 115-122. [13] W. D. Cook, Distance-based and ad hoc consensus models in ordinal preference ranking, European Journal of Operational Research, 172 (2006), 369-385. doi: 10.1016/j.ejor.2005.03.048. [14] R. Criado, E. García, F. Pedroche and M. Romance, A new method for comparing rankings through complex networks: Model and analysis of competitiveness of major European soccer leagues, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 043114. doi: 10.1063/1.4826446. [15] C. Dwork, R. Kumary, M. Naorz and D. Sivakumarx, Rank aggregation methods for the web, in Proc. 10th International Conference on World Wide Web, WWW'01, ACM, New York, 2001, 613-622. doi: 10.1145/371920.372165. [16] E. J. Emond and D. W. Mason, A new rank correlation coefficient with application to the consensus ranking problem, J. Multi-Crit. Decis. Anal., 11 (2002), 17-28. doi: 10.1002/mcda.313. [17] R. Fagin, R. Kumar, M. Mahdian, D. Sivakumar and E. Vee, Comparing partial rankings, SIAM J. Discrete Math., 20 (2006), 628-648. doi: 10.1137/05063088X. [18] E. Fama and K. French, Common risk factors in the returns of bonds and stocks, Journal of Financial Economics, 33 (1993), 3-56. doi: 10.1016/0304-405X(93)90023-5. [19] M. Golumbic, D. Rotem and J. Urrutia, Comparability graphs and intersection graphs, Discrete Mathematics, 43 (1983), 37-46. doi: 10.1016/0012-365X(83)90019-5. [20] J. Gómez-Gardeñes, I. Reinares, A. Arenas and L. M. Floria, Evolution of cooperation in multiplex networks, Sci. Rep., 2 (2012), 1-6. doi: 10.1038/srep00620. [21] S. Gomez, A. Díaz-Guilera, J. Gómez-Gardeñes, C.J. Pérez-Vicente, Y. Moreno and A. Arenas, Diffusion dynamics on multiplex networks, Physical Review Letters, 110 (2013), 028701. doi: 10.1103/PhysRevLett.110.028701. [22] I. F. Ilyas, G. Beskales and M. A. Soliman, A survey of top-kquery processing techniques in relational database systems, ACM Computing Surveys, 40 (2008), p11. doi: 10.1145/1391729.1391730. [23] , Invertia, The Economical Website of Telefónica (Spain). Available from: http://www.invertia.com/mercados/bolsa/indices/ibex-35/acciones-ib011ibex35. [24] M. Karpinski and W. Schudy, Faster algorithms for feedback arc set tournament, Kemeny rank aggregation and betweenness tournament, in Algorithms and computation. Part I, Lecture Notes in Computer Science, 6506, Springer, Berlin, 2010, 3-14. doi: 10.1007/978-3-642-17517-6_3. [25] J. G. Kemeny and J. L. Snell, Mathematical Models in the Social Sciences, $2^{nd}$ edition, The MIT Press, Cambridge, 1978. [26] M. G. Kendall, A new measure of rank correlation, Biometrika, 30 (1938), 81-93. doi: 10.1093/biomet/30.1-2.81. [27] M. G. Kendall and B. Babington Smith, The problem of $m$ rankings, Ann. Math. Statist., 10 (1939), 275-287. doi: 10.1214/aoms/1177732186. [28] D. Kratsch, R. M. Mcconnell, K. Mehlhorn and J. P. Spinrad, Certifying Algorithms for recognizing interval graphs and permutation graphs, SIAM J. Comput., 36 (2006), 326-353. doi: 10.1137/S0097539703437855. [29] R. Kumar and S. Vassilvitskii, Generalized distances between rankings, in Proc. 19th International Conference on World Wide Web, WWW'10, ACM, New York, 2010, 571-580. doi: 10.1145/1772690.1772749. [30] A. N. Langville and C. D. Meyer, Who's #1?: The Science of Rating and Ranking, $1^{st}$ edition, Princeton University Press, New Jersey, 2012. [31] , Madrid Stock Market, Official Annual Reports about Madrid and Spanish Stock Markets (in Spanish). Available from: http://www.bolsamadrid.es/esp/BMadrid/Protector/Protector3.aspx. [32] E. E. Peters, Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility, Wiley, Somerset, NJ, 1991. [33] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canadian Journal of Mathematics, 23 (1971), 160-175. doi: 10.4153/CJM-1971-016-5. [34] R. Savit, When random is not random: An introduction to chaos in market prices, The Journal of Futures Markets, 8 (1988), 271-290. doi: 10.1002/fut.3990080303. [35] C. S. Signorino and J. M. Ritter, Tau-b or not tau-b: Measuring the similarity of foreign policy positions, International Studies Quarterly, 43 (1999), 115-144. doi: 10.1111/0020-8833.00113. [36] K. Stefanidis, G. Koutrika and E. Pitoura, A survey on representation, composition and application of preferences in database systems, ACM Transactions on Database Systems, 36 (2011), p19. doi: 10.1145/2000824.2000829. [37] M. Szell, R. Lambiotte and S. Thurner, Multirelational organization of large-scale social networks in an online world, Proceedings of the National Academy of Sciences, 107 (2010), 13636-13641. doi: 10.1073/pnas.1004008107.

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References:
 [1] A. Abhyankar, L. S. Copeland and W. Wong, Nonlinear dynamics in real-time equity market indices: Evidence from the United Kingdom, The Economic Journal, 105 (1995), 864-880. doi: 10.2307/2235155. [2] N. Ailon, M. Charikar and A. Newman, Aggregating inconsistent information: Ranking and clustering, Journal of the ACM, 55 (2008), Art. 23, 27 pp. doi: 10.1145/1411509.1411513. [3] J. Bar-Ilan, Comparing rankings of search results on the web, Information Processing and Management, 41 (2005), 1511-1519. doi: 10.1016/j.ipm.2005.03.008. [4] N. Betzler, R. Bredereck and R. Niedermeier, Partial kernelization for rank aggregation: Theory and experiments, in Parameterized and Exact Computation, Lecture Notes in Computer Sciences, 6478, Springer, Berlin, 2010, 26-37. doi: 10.1007/978-3-642-17493-3_5. [5] T. Biedl, F. J. Brandenburg and X. Deng, Crossings and permutations, Graph Drawing, Lecture Notes in Computer Sciences, 3843, Springer, Berlin, 2006, 1-12. doi: 10.1007/11618058_1. [6] S. Boccaletti, G. Bianconi, R. Criado, C. I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang and M. Zanin, The structure and dynamics of multilayer networks, Physics Reports, 544 (2014), 1-122. doi: 10.1016/j.physrep.2014.07.001. [7] R. A. Brealey, S. C. Myers and F. Allen, Principios de Finanzas Corporativas (in Spanish), $8^{th}$ edition, Mc Graw-Hill, 2006. [8] E. F. Brigham and P. R. Daves, International Financial Management, South-Western, 2002. [9] B. Carterette, On rank correlation and the distance between rankings, in Proc. of the 32nd Int. ACM Conf. Research and Development in Information Retrieval, ACM, New York, 2009, 436-443. doi: 10.1145/1571941.1572017. [10] W. W. Cohen, R. E. Schapire and Y. Singer, Learning to order things, Journal of Artificial Intelligence Research, 10 (1999), 243-270. doi: 10.1613/jair.587. [11] V. Conitzer, A. Davenport and Y. Heights, Improved bounds for computing Kemeny rankings, in Proceedings of The Twenty-First National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference, 2006, 620-626. [12] W. D. Cook, M. Kress and L. M. Seiford, An axiomatic approach to distance on partial orderings, RAIRO, Recherche Operationelle/Operations Research, 20 (1986), 115-122. [13] W. D. Cook, Distance-based and ad hoc consensus models in ordinal preference ranking, European Journal of Operational Research, 172 (2006), 369-385. doi: 10.1016/j.ejor.2005.03.048. [14] R. Criado, E. García, F. Pedroche and M. Romance, A new method for comparing rankings through complex networks: Model and analysis of competitiveness of major European soccer leagues, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 043114. doi: 10.1063/1.4826446. [15] C. Dwork, R. Kumary, M. Naorz and D. Sivakumarx, Rank aggregation methods for the web, in Proc. 10th International Conference on World Wide Web, WWW'01, ACM, New York, 2001, 613-622. doi: 10.1145/371920.372165. [16] E. J. Emond and D. W. Mason, A new rank correlation coefficient with application to the consensus ranking problem, J. Multi-Crit. Decis. Anal., 11 (2002), 17-28. doi: 10.1002/mcda.313. [17] R. Fagin, R. Kumar, M. Mahdian, D. Sivakumar and E. Vee, Comparing partial rankings, SIAM J. Discrete Math., 20 (2006), 628-648. doi: 10.1137/05063088X. [18] E. Fama and K. French, Common risk factors in the returns of bonds and stocks, Journal of Financial Economics, 33 (1993), 3-56. doi: 10.1016/0304-405X(93)90023-5. [19] M. Golumbic, D. Rotem and J. Urrutia, Comparability graphs and intersection graphs, Discrete Mathematics, 43 (1983), 37-46. doi: 10.1016/0012-365X(83)90019-5. [20] J. Gómez-Gardeñes, I. Reinares, A. Arenas and L. M. Floria, Evolution of cooperation in multiplex networks, Sci. Rep., 2 (2012), 1-6. doi: 10.1038/srep00620. [21] S. Gomez, A. Díaz-Guilera, J. Gómez-Gardeñes, C.J. Pérez-Vicente, Y. Moreno and A. Arenas, Diffusion dynamics on multiplex networks, Physical Review Letters, 110 (2013), 028701. doi: 10.1103/PhysRevLett.110.028701. [22] I. F. Ilyas, G. Beskales and M. A. Soliman, A survey of top-kquery processing techniques in relational database systems, ACM Computing Surveys, 40 (2008), p11. doi: 10.1145/1391729.1391730. [23] , Invertia, The Economical Website of Telefónica (Spain). Available from: http://www.invertia.com/mercados/bolsa/indices/ibex-35/acciones-ib011ibex35. [24] M. Karpinski and W. Schudy, Faster algorithms for feedback arc set tournament, Kemeny rank aggregation and betweenness tournament, in Algorithms and computation. Part I, Lecture Notes in Computer Science, 6506, Springer, Berlin, 2010, 3-14. doi: 10.1007/978-3-642-17517-6_3. [25] J. G. Kemeny and J. L. Snell, Mathematical Models in the Social Sciences, $2^{nd}$ edition, The MIT Press, Cambridge, 1978. [26] M. G. Kendall, A new measure of rank correlation, Biometrika, 30 (1938), 81-93. doi: 10.1093/biomet/30.1-2.81. [27] M. G. Kendall and B. Babington Smith, The problem of $m$ rankings, Ann. Math. Statist., 10 (1939), 275-287. doi: 10.1214/aoms/1177732186. [28] D. Kratsch, R. M. Mcconnell, K. Mehlhorn and J. P. Spinrad, Certifying Algorithms for recognizing interval graphs and permutation graphs, SIAM J. Comput., 36 (2006), 326-353. doi: 10.1137/S0097539703437855. [29] R. Kumar and S. Vassilvitskii, Generalized distances between rankings, in Proc. 19th International Conference on World Wide Web, WWW'10, ACM, New York, 2010, 571-580. doi: 10.1145/1772690.1772749. [30] A. N. Langville and C. D. Meyer, Who's #1?: The Science of Rating and Ranking, $1^{st}$ edition, Princeton University Press, New Jersey, 2012. [31] , Madrid Stock Market, Official Annual Reports about Madrid and Spanish Stock Markets (in Spanish). Available from: http://www.bolsamadrid.es/esp/BMadrid/Protector/Protector3.aspx. [32] E. E. Peters, Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility, Wiley, Somerset, NJ, 1991. [33] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canadian Journal of Mathematics, 23 (1971), 160-175. doi: 10.4153/CJM-1971-016-5. [34] R. Savit, When random is not random: An introduction to chaos in market prices, The Journal of Futures Markets, 8 (1988), 271-290. doi: 10.1002/fut.3990080303. [35] C. S. Signorino and J. M. Ritter, Tau-b or not tau-b: Measuring the similarity of foreign policy positions, International Studies Quarterly, 43 (1999), 115-144. doi: 10.1111/0020-8833.00113. [36] K. Stefanidis, G. Koutrika and E. Pitoura, A survey on representation, composition and application of preferences in database systems, ACM Transactions on Database Systems, 36 (2011), p19. doi: 10.1145/2000824.2000829. [37] M. Szell, R. Lambiotte and S. Thurner, Multirelational organization of large-scale social networks in an online world, Proceedings of the National Academy of Sciences, 107 (2010), 13636-13641. doi: 10.1073/pnas.1004008107.
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