# 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 & 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. doi: 10.2307/2235155. Google Scholar [2] N. Ailon, M. Charikar and A. Newman, Aggregating inconsistent information: Ranking and clustering,, Journal of the ACM, 55 (2008). doi: 10.1145/1411509.1411513. Google Scholar [3] J. Bar-Ilan, Comparing rankings of search results on the web,, Information Processing and Management, 41 (2005), 1511. doi: 10.1016/j.ipm.2005.03.008. Google Scholar [4] N. Betzler, R. Bredereck and R. Niedermeier, Partial kernelization for rank aggregation: Theory and experiments,, in Parameterized and Exact Computation, (6478), 26. doi: 10.1007/978-3-642-17493-3_5. Google Scholar [5] T. Biedl, F. J. Brandenburg and X. Deng, Crossings and permutations,, Graph Drawing, (3843), 1. doi: 10.1007/11618058_1. Google Scholar [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. doi: 10.1016/j.physrep.2014.07.001. Google Scholar [7] R. A. Brealey, S. C. Myers and F. Allen, Principios de Finanzas Corporativas (in Spanish),, $8^{th}$ edition, (2006). Google Scholar [8] E. F. Brigham and P. R. Daves, International Financial Management,, South-Western, (2002). Google Scholar [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, (2009), 436. doi: 10.1145/1571941.1572017. Google Scholar [10] W. W. Cohen, R. E. Schapire and Y. Singer, Learning to order things,, Journal of Artificial Intelligence Research, 10 (1999), 243. doi: 10.1613/jair.587. Google Scholar [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. Google Scholar [12] W. D. Cook, M. Kress and L. M. Seiford, An axiomatic approach to distance on partial orderings,, RAIRO, 20 (1986), 115. Google Scholar [13] W. D. Cook, Distance-based and ad hoc consensus models in ordinal preference ranking,, European Journal of Operational Research, 172 (2006), 369. doi: 10.1016/j.ejor.2005.03.048. Google Scholar [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). doi: 10.1063/1.4826446. Google Scholar [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, (2001), 613. doi: 10.1145/371920.372165. Google Scholar [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. doi: 10.1002/mcda.313. Google Scholar [17] R. Fagin, R. Kumar, M. Mahdian, D. Sivakumar and E. Vee, Comparing partial rankings,, SIAM J. Discrete Math., 20 (2006), 628. doi: 10.1137/05063088X. Google Scholar [18] E. Fama and K. French, Common risk factors in the returns of bonds and stocks,, Journal of Financial Economics, 33 (1993), 3. doi: 10.1016/0304-405X(93)90023-5. Google Scholar [19] M. Golumbic, D. Rotem and J. Urrutia, Comparability graphs and intersection graphs,, Discrete Mathematics, 43 (1983), 37. doi: 10.1016/0012-365X(83)90019-5. Google Scholar [20] J. Gómez-Gardeñes, I. Reinares, A. Arenas and L. M. Floria, Evolution of cooperation in multiplex networks,, Sci. Rep., 2 (2012), 1. doi: 10.1038/srep00620. Google Scholar [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). doi: 10.1103/PhysRevLett.110.028701. Google Scholar [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). doi: 10.1145/1391729.1391730. Google Scholar [23] , Invertia, The Economical Website of Telefónica (Spain)., Available from: , (). Google Scholar [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, (6506), 3. doi: 10.1007/978-3-642-17517-6_3. Google Scholar [25] J. G. Kemeny and J. L. Snell, Mathematical Models in the Social Sciences,, $2^{nd}$ edition, (1978). Google Scholar [26] M. G. Kendall, A new measure of rank correlation,, Biometrika, 30 (1938), 81. doi: 10.1093/biomet/30.1-2.81. Google Scholar [27] M. G. Kendall and B. Babington Smith, The problem of $m$ rankings,, Ann. Math. Statist., 10 (1939), 275. doi: 10.1214/aoms/1177732186. Google Scholar [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. doi: 10.1137/S0097539703437855. Google Scholar [29] R. Kumar and S. Vassilvitskii, Generalized distances between rankings,, in Proc. 19th International Conference on World Wide Web, (2010), 571. doi: 10.1145/1772690.1772749. Google Scholar [30] A. N. Langville and C. D. Meyer, Who's #1?: The Science of Rating and Ranking,, $1^{st}$ edition, (2012). Google Scholar [31] , Madrid Stock Market, Official Annual Reports about Madrid and Spanish Stock Markets (in Spanish)., Available from: , (). Google Scholar [32] E. E. Peters, Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility,, Wiley, (1991). Google Scholar [33] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs,, Canadian Journal of Mathematics, 23 (1971), 160. doi: 10.4153/CJM-1971-016-5. Google Scholar [34] R. Savit, When random is not random: An introduction to chaos in market prices,, The Journal of Futures Markets, 8 (1988), 271. doi: 10.1002/fut.3990080303. Google Scholar [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. doi: 10.1111/0020-8833.00113. Google Scholar [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). doi: 10.1145/2000824.2000829. Google Scholar [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. doi: 10.1073/pnas.1004008107. Google Scholar

show all references

##### 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. doi: 10.2307/2235155. Google Scholar [2] N. Ailon, M. Charikar and A. Newman, Aggregating inconsistent information: Ranking and clustering,, Journal of the ACM, 55 (2008). doi: 10.1145/1411509.1411513. Google Scholar [3] J. Bar-Ilan, Comparing rankings of search results on the web,, Information Processing and Management, 41 (2005), 1511. doi: 10.1016/j.ipm.2005.03.008. Google Scholar [4] N. Betzler, R. Bredereck and R. Niedermeier, Partial kernelization for rank aggregation: Theory and experiments,, in Parameterized and Exact Computation, (6478), 26. doi: 10.1007/978-3-642-17493-3_5. Google Scholar [5] T. Biedl, F. J. Brandenburg and X. Deng, Crossings and permutations,, Graph Drawing, (3843), 1. doi: 10.1007/11618058_1. Google Scholar [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. doi: 10.1016/j.physrep.2014.07.001. Google Scholar [7] R. A. Brealey, S. C. Myers and F. Allen, Principios de Finanzas Corporativas (in Spanish),, $8^{th}$ edition, (2006). Google Scholar [8] E. F. Brigham and P. R. Daves, International Financial Management,, South-Western, (2002). Google Scholar [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, (2009), 436. doi: 10.1145/1571941.1572017. Google Scholar [10] W. W. Cohen, R. E. Schapire and Y. Singer, Learning to order things,, Journal of Artificial Intelligence Research, 10 (1999), 243. doi: 10.1613/jair.587. Google Scholar [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. Google Scholar [12] W. D. Cook, M. Kress and L. M. Seiford, An axiomatic approach to distance on partial orderings,, RAIRO, 20 (1986), 115. Google Scholar [13] W. D. Cook, Distance-based and ad hoc consensus models in ordinal preference ranking,, European Journal of Operational Research, 172 (2006), 369. doi: 10.1016/j.ejor.2005.03.048. Google Scholar [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). doi: 10.1063/1.4826446. Google Scholar [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, (2001), 613. doi: 10.1145/371920.372165. Google Scholar [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. doi: 10.1002/mcda.313. Google Scholar [17] R. Fagin, R. Kumar, M. Mahdian, D. Sivakumar and E. Vee, Comparing partial rankings,, SIAM J. Discrete Math., 20 (2006), 628. doi: 10.1137/05063088X. Google Scholar [18] E. Fama and K. French, Common risk factors in the returns of bonds and stocks,, Journal of Financial Economics, 33 (1993), 3. doi: 10.1016/0304-405X(93)90023-5. Google Scholar [19] M. Golumbic, D. Rotem and J. Urrutia, Comparability graphs and intersection graphs,, Discrete Mathematics, 43 (1983), 37. doi: 10.1016/0012-365X(83)90019-5. Google Scholar [20] J. Gómez-Gardeñes, I. Reinares, A. Arenas and L. M. Floria, Evolution of cooperation in multiplex networks,, Sci. Rep., 2 (2012), 1. doi: 10.1038/srep00620. Google Scholar [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). doi: 10.1103/PhysRevLett.110.028701. Google Scholar [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). doi: 10.1145/1391729.1391730. Google Scholar [23] , Invertia, The Economical Website of Telefónica (Spain)., Available from: , (). Google Scholar [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, (6506), 3. doi: 10.1007/978-3-642-17517-6_3. Google Scholar [25] J. G. Kemeny and J. L. Snell, Mathematical Models in the Social Sciences,, $2^{nd}$ edition, (1978). Google Scholar [26] M. G. Kendall, A new measure of rank correlation,, Biometrika, 30 (1938), 81. doi: 10.1093/biomet/30.1-2.81. Google Scholar [27] M. G. Kendall and B. Babington Smith, The problem of $m$ rankings,, Ann. Math. Statist., 10 (1939), 275. doi: 10.1214/aoms/1177732186. Google Scholar [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. doi: 10.1137/S0097539703437855. Google Scholar [29] R. Kumar and S. Vassilvitskii, Generalized distances between rankings,, in Proc. 19th International Conference on World Wide Web, (2010), 571. doi: 10.1145/1772690.1772749. Google Scholar [30] A. N. Langville and C. D. Meyer, Who's #1?: The Science of Rating and Ranking,, $1^{st}$ edition, (2012). Google Scholar [31] , Madrid Stock Market, Official Annual Reports about Madrid and Spanish Stock Markets (in Spanish)., Available from: , (). Google Scholar [32] E. E. Peters, Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility,, Wiley, (1991). Google Scholar [33] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs,, Canadian Journal of Mathematics, 23 (1971), 160. doi: 10.4153/CJM-1971-016-5. Google Scholar [34] R. Savit, When random is not random: An introduction to chaos in market prices,, The Journal of Futures Markets, 8 (1988), 271. doi: 10.1002/fut.3990080303. Google Scholar [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. doi: 10.1111/0020-8833.00113. Google Scholar [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). doi: 10.1145/2000824.2000829. Google Scholar [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. doi: 10.1073/pnas.1004008107. Google Scholar
 [1] Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 [2] Gabrielle Demange. Collective attention and ranking methods. Journal of Dynamics & Games, 2014, 1 (1) : 17-43. doi: 10.3934/jdg.2014.1.17 [3] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006 [4] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 [5] Jian Yang, Youhua (Frank) Chen. On information quality ranking and its managerial implications. Journal of Industrial & Management Optimization, 2010, 6 (4) : 729-750. doi: 10.3934/jimo.2010.6.729 [6] Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019036 [7] Lixin Wu, Fan Zhang. LIBOR market model with stochastic volatility. Journal of Industrial & Management Optimization, 2006, 2 (2) : 199-227. doi: 10.3934/jimo.2006.2.199 [8] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 [9] Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635 [10] Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339 [11] Jean-René Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399 [12] V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73 [13] Hans-Otto Walther. Contracting return maps for monotone delayed feedback. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 259-274. doi: 10.3934/dcds.2001.7.259 [14] Tao Gu, Peiyu Ren, Maozhu Jin, Hua Wang. Tourism destination competitiveness evaluation in Sichuan province using TOPSIS model based on information entropy weights. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 771-782. doi: 10.3934/dcdss.2019051 [15] Litao Guo, Bernard L. S. Lin. Vulnerability of super connected split graphs and bisplit graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1179-1185. doi: 10.3934/dcdss.2019081 [16] Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207 [17] Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial & Management Optimization, 2018, 14 (1) : 199-229. doi: 10.3934/jimo.2017043 [18] Vinicius Albani, Uri M. Ascher, Xu Yang, Jorge P. Zubelli. Data driven recovery of local volatility surfaces. Inverse Problems & Imaging, 2017, 11 (5) : 799-823. doi: 10.3934/ipi.2017038 [19] Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435 [20] Victor Isakov. Recovery of time dependent volatility coefficient by linearization. Evolution Equations & Control Theory, 2014, 3 (1) : 119-134. doi: 10.3934/eect.2014.3.119

2018 Impact Factor: 0.871