A model to classify users of social networks based on PageRank

In this paper, we present a model to classify users of Social Networks. In particular, we focus on Social Network Sites. The model is based on the PageRank algorithm. We use the personalization vector to bias the PageRank to some users. We give an explicit expression of the personalization vector that allows the introduction of some typical features of the users of SNSs. We describe the model as a seven-step process. We illustrate the applicability of the model with two examples. One example is based on real links of a Facebook network. We also indicate how to take into account real actions of Facebook users to implement the model.This work is supported by Spanish DGI grant MTM2010-18674.Pedroche Sánchez, F. (2012). A model to classify users of social networks based on PageRank. International Journal of Bifurcation and Chaos. 22(7):1-14. https://doi.org/10.1142/S0218127412501623S114227Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469(3), 93-153. doi:10.1016/j.physrep.2008.09.002BOCCALETTI, S., LATORA, V., MORENO, Y., CHAVEZ, M., & HWANG, D. (2006). Complex networks: Structure and dynamics. Physics Reports, 424(4-5), 175-308. doi:10.1016/j.physrep.2005.10.009Boldi, P., Santini, M., & Vigna, S. (2009). PageRank. ACM Transactions on Information Systems, 27(4), 1-23. doi:10.1145/1629096.1629097Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661-703. doi:10.1137/070710111Criado, R., Flores, J., González-Vasco, M. I., & Pello, J. (2007). Choosing a leader on a complex network. Journal of Computational and Applied Mathematics, 204(1), 10-17. doi:10.1016/j.cam.2006.04.024C. De Kerchove and P. Van Dooren, Lectures Notes in Control and Information Sciences 389 (2009) pp. 3–16.Dorogovtsev, S. (2010). Lectures on Complex Networks. doi:10.1093/acprof:oso/9780199548927.001.0001Easley, D., & Kleinberg, J. (2010). Networks, Crowds, and Markets. doi:10.1017/cbo9780511761942Estrada, E., & Higham, D. J. (2010). Network Properties Revealed through Matrix Functions. SIAM Review, 52(4), 696-714. doi:10.1137/090761070Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3-5), 75-174. doi:10.1016/j.physrep.2009.11.002Granovetter, M. S. (1973). The Strength of Weak Ties. American Journal of Sociology, 78(6), 1360-1380. doi:10.1086/225469Haveliwala, T. H. (2003). Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE Transactions on Knowledge and Data Engineering, 15(4), 784-796. doi:10.1109/tkde.2003.1208999Langville, A. N., & Meyer, C. D. (2006). Google’s PageRank and Beyond. doi:10.1515/9781400830329Lazer, D., Pentland, A., Adamic, L., Aral, S., Barabasi, A.-L., Brewer, D., … Van Alstyne, M. (2009). SOCIAL SCIENCE: Computational Social Science. Science, 323(5915), 721-723. doi:10.1126/science.1167742Lewis, K., Kaufman, J., Gonzalez, M., Wimmer, A., & Christakis, N. (2008). Tastes, ties, and time: A new social network dataset using Facebook.com. Social Networks, 30(4), 330-342. doi:10.1016/j.socnet.2008.07.002Nan Lin, Dayton, P. W., & Greenwald, P. (1978). Analyzing the Instrumental Use of Relations in the Context of Social Structure. Sociological Methods & Research, 7(2), 149-166. doi:10.1177/004912417800700203Mayer, A., & Puller, S. L. (2008). The old boy (and girl) network: Social network formation on university campuses. Journal of Public Economics, 92(1-2), 329-347. doi:10.1016/j.jpubeco.2007.09.001Newman, M. (2010). Networks. doi:10.1093/acprof:oso/9780199206650.001.0001Pedroche Sánchez, F. (2010). Competitivity groups on social network sites. Mathematical and Computer Modelling, 52(7-8), 1052-1057. doi:10.1016/j.mcm.2010.02.031Sabater, J., & Sierra, C. (2005). Review on Computational Trust and Reputation Models. Artificial Intelligence Review, 24(1), 33-60. doi:10.1007/s10462-004-0041-5Serra-Capizzano, S. (2005). Jordan Canonical Form of the Google Matrix: A Potential Contribution to the PageRank Computation. SIAM Journal on Matrix Analysis and Applications, 27(2), 305-312. doi:10.1137/s0895479804441407Vasalou, A., Joinson, A. N., & Courvoisier, D. (2010). Cultural differences, experience with social networks and the nature of «true commitment» in Facebook. International Journal of Human-Computer Studies, 68(10), 719-728. doi:10.1016/j.ijhcs.2010.06.00

Corrected Evolutive Kendall's tau Coefficients for Incomplete Rankings with Ties: Application to Case of Spotify Lists

[EN] Mathematical analysis of rankings is essential for a wide range of scientific, public, and industrial applications (e.g., group decision-making, organizational methods, R&D sponsorship, recommender systems, voter systems, sports competitions, grant proposals rankings, web searchers, Internet streaming-on-demand media providers, etc.). Recently, some methods for incomplete aggregate rankings (rankings in which not all the elements are ranked) with ties, based on the classic Kendall's tau coefficient, have been presented. We are interested in ordinal rankings (that is, we can order the elements to be the first, the second, etc.) allowing ties between the elements (e.g., two elements may be in the first position). We extend a previous coefficient for comparing a series of complete rankings with ties to two new coefficients for comparing a series of incomplete rankings with ties. We make use of the newest definitions of Kendall's tau extensions. We also offer a theoretical result to interpret these coefficients in terms of the type of interactions that the elements of two consecutive rankings may show (e.g., they preserve their positions, cross their positions, and they are tied in one ranking but untied in the other ranking, etc.). We give some small examples to illustrate all the newly presented parameters and coefficients. We also apply our coefficients to compare some series of Spotify charts, both Top 200 and Viral 50, showing the applicability and utility of the proposed measures.This research was funded by the Spanish Government, Ministerio de Economia y Competividad, grant number MTM2016-75963-P.Pedroche Sánchez, F.; Conejero, JA. (2020). Corrected Evolutive Kendall's tau Coefficients for Incomplete Rankings with Ties: Application to Case of Spotify Lists. Mathematics. 8(10):1-30. https://doi.org/10.3390/math8101828S130810Diaconis, P., & Graham, R. L. (1977). Spearman’s Footrule as a Measure of Disarray. Journal of the Royal Statistical Society: Series B (Methodological), 39(2), 262-268. doi:10.1111/j.2517-6161.1977.tb01624.xMoreno-Centeno, E., & Escobedo, A. R. (2015). Axiomatic aggregation of incomplete rankings. IIE Transactions, 48(6), 475-488. doi:10.1080/0740817x.2015.1109737Criado, R., García, E., Pedroche, F., & Romance, M. (2013). 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(4), 043114. doi:10.1063/1.4826446Fortune 500https://fortune.com/fortune500/Academic Ranking of World Universities ARWU 2020http://www.shanghairanking.com/ARWU2020.htmlCWTS Leiden Ranking 2020https://www.leidenranking.com/ranking/2020/listThe Hot 100https://www.billboard.com/charts/hot-100Fagin, R., Kumar, R., Mahdian, M., Sivakumar, D., & Vee, E. (2006). Comparing Partial Rankings. SIAM Journal on Discrete Mathematics, 20(3), 628-648. doi:10.1137/05063088xCook, W. D., Kress, M., & Seiford, L. M. (1986). An axiomatic approach to distance on partial orderings. RAIRO - Operations Research, 20(2), 115-122. doi:10.1051/ro/1986200201151Yoo, Y., Escobedo, A. R., & Skolfield, J. K. (2020). A new correlation coefficient for comparing and aggregating non-strict and incomplete rankings. European Journal of Operational Research, 285(3), 1025-1041. doi:10.1016/j.ejor.2020.02.027Pedroche, F., Criado, R., García, E., Romance, M., & Sánchez, V. E. (2015). Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Networks and Heterogeneous Media, 10(1), 101-125. doi:10.3934/nhm.2015.10.101Criado, R., García, E., Pedroche, F., & Romance, M. (2016). On graphs associated to sets of rankings. Journal of Computational and Applied Mathematics, 291, 497-508. doi:10.1016/j.cam.2015.03.009KENDALL, M. G. (1938). A NEW MEASURE OF RANK CORRELATION. Biometrika, 30(1-2), 81-93. doi:10.1093/biomet/30.1-2.81Kendall, M. G., & Smith, B. B. (1939). The Problem of $m$ Rankings. The Annals of Mathematical Statistics, 10(3), 275-287. doi:10.1214/aoms/1177732186Bogart, K. P. (1973). Preference structures I: Distances between transitive preference relations†. The Journal of Mathematical Sociology, 3(1), 49-67. doi:10.1080/0022250x.1973.9989823Bogart, K. P. (1975). Preference Structures. II: Distances Between Asymmetric Relations. SIAM Journal on Applied Mathematics, 29(2), 254-262. doi:10.1137/0129023Cicirello, V. (2020). Kendall tau sequence distance: Extending Kendall tau from ranks to sequences. EAI Endorsed Transactions on Industrial Networks and Intelligent Systems, 7(23), 163925. doi:10.4108/eai.13-7-2018.163925Armstrong, R. A. (2019). Should Pearson’s correlation coefficient be avoided? Ophthalmic and Physiological Optics, 39(5), 316-327. doi:10.1111/opo.12636Redman, W. (2019). An O(n) method of calculating Kendall correlations of spike trains. PLOS ONE, 14(2), e0212190. doi:10.1371/journal.pone.0212190Pihur, V., Datta, S., & Datta, S. (2009). RankAggreg, an R package for weighted rank aggregation. BMC Bioinformatics, 10(1). doi:10.1186/1471-2105-10-62Pnueli, A., Lempel, A., & Even, S. (1971). Transitive Orientation of Graphs and Identification of Permutation Graphs. Canadian Journal of Mathematics, 23(1), 160-175. doi:10.4153/cjm-1971-016-5Gervacio, S. V., Rapanut, T. A., & Ramos, P. C. F. (2013). Characterization and Construction of Permutation Graphs. Open Journal of Discrete Mathematics, 03(01), 33-38. doi:10.4236/ojdm.2013.31007Golumbic, M. C., Rotem, D., & Urrutia, J. (1983). Comparability graphs and intersection graphs. Discrete Mathematics, 43(1), 37-46. doi:10.1016/0012-365x(83)90019-5Emond, E. J., & Mason, D. W. (2002). A new rank correlation coefficient with application to the consensus ranking problem. Journal of Multi-Criteria Decision Analysis, 11(1), 17-28. doi:10.1002/mcda.313Spotify Reports Second Quarter 2020 Earningshttps://newsroom.spotify.com/2020-07-29/spotify-reports-second-quarter-2020-earningsCompany infohttps://newsroom.spotify.com/company-info/Bussines Wirehttps://www.businesswire.com/news/home/20200429005216/en/Swanson, K. (2013). A Case Study on Spotify: Exploring Perceptions of the Music Streaming Service. Journal of the Music and Entertainment Industry Educators Association, 13(1), 207-230. doi:10.25101/13.10Microsoft Retires Groove Music Service, Partners with Spotifyhttps://www.theverge.com/2017/10/2/16401898/microsoft-groove-music-pass-discontinued-spotify-partnerSpotify Launches on PlayStation Music Todayhttps://blog.playstation.com/2015/03/30/spotify-launches-on-playstation-music-today/You Can Now Share Music from Spotify to Facebook Storieshttps://techcrunch.com/2019/08/30/you-can-now-share-music-from-spotify-to-facebook-storiesMähler, R., & Vonderau, P. (2017). Studying Ad Targeting with Digital Methods: The Case of Spotify. Culture Unbound, 9(2), 212-221. doi:10.3384/cu.2000.1525.1792212Analyzing Spotify Data. Exploring the Possibilities of User Data from a Scientific and Business Perspective. (Supervised by Sandjai Bhulai). Report from Vrije Universiteit Amsterdamhttps://www.math.vu.nl/~sbhulai/papers/paper-vandenhoven.pdfGreenberg, D. M., Kosinski, M., Stillwell, D. J., Monteiro, B. L., Levitin, D. J., & Rentfrow, P. J. (2016). The Song Is You. Social Psychological and Personality Science, 7(6), 597-605. doi:10.1177/1948550616641473Spotify Charts Regionalhttps://spotifycharts.com/regionalSpotify Charts Launch Globally, Showcase 50 Most Listened to and Most Viral Tracks Weeklyhttps://www.engadget.com/2013-05-21-spotify-charts-launch.htmlSpotify says its Viral-50 chart reaches the parts other charts don’thttps://musically.com/2014/07/15/spotify-says-its-viral-50-chart-reaches-the-parts-other-charts-dont/Spotify Reveals New Viral 50 Charthttps://www.musicweek.com/news/read/spotify-launches-the-viral-50-chart/059027Reports Results for Fiscal Second Quarter Ended 31 March 2020https://www.wmg.com/news/warner-music-group-corp-reports-results-fiscal-second-quarter-ended-march-31-2020-34751COVID-19’s Effect on the Global Music Business, Part 1: Genrehttps://blog.chartmetric.com/covid-19-effect-on-the-global-music-business-part-1-genre/Top 200https://spotifycharts.com/regional/global/weeklySpotify Chartshttps://spotifycharts.com/viral

Modelling Social Network Sites with PageRank and Social Competences

[EN] In this communication a recent method to classify the users of an SNS into Competitivity groups is recalled. This method is based on the PageRank algorithm. Competitivity groups are sets of nodes that compete among each other to gain PageRank via the personalization vector. Specific features of the SNSs (such as number of friends or activity of the users) can be considered as Social Competences. By means of these Social Competences a node can modify its ranking inside a Competitivity group.This work is supported by Spanish DGI grant MTM2010-18674.Pedroche Sánchez, F. (2011). Modelling Social Network Sites with PageRank and Social Competences. International Journal of Complex Systems in Science. 1(1):65-68. http://hdl.handle.net/10251/46059S65681

Leadership groups on Social Network Sites based on Personalized PageRank

n this paper we present a new framework to identify leaders in a Social Network Site using the Personalized PageRank vector. The methodology is based on the concept of Leadership group recently introduced by one of the authors. We show how to analyze the structure of the Leadership group as a function of a single parameter. Zachary¿s network and a Facebook university network are used to illustrate the applicability of the model.We thank an unknown referee who made some suggestive comments that improved the readability of the paper. This work is supported by Spanish DGI grant MTM2010-18674.Pedroche Sánchez, F.; Moreno, F.; González, A.; Valencia, A. (2013). Leadership groups on Social Network Sites based on Personalized PageRank. Mathematical and Computer Modelling. 57(7-8):1891-1896. https://doi.org/10.1016/j.mcm.2011.12.026S18911896577-

Parametric controllability of the personalized PageRank: Classic model vs biplex approach

[EN] Measures of centrality in networks defined by means of matrix algebra, like PageRank-type centralities, have been used for over 70 years. Recently, new extensions of PageRank have been formulated and may include a personalization (or teleportation) vector. It is accepted that one of the key issues for any centrality measure formulation is to what extent someone can control its variability. In this paper, we compare the limits of variability of two centrality measures for complex networks that we call classic PageRank (PR) and biplex approach PageRank (BPR). Both centrality measures depend on the so-called damping parameter alpha that controls the quantity of teleportation. Our first result is that the intersection of the intervals of variation of both centrality measures is always a nonempty set. Our second result is that when alpha is lower that 0.48 (and, therefore, the ranking is highly affected by teleportation effects) then the upper limits of PR are more controllable than the upper limits of BPR; on the contrary, when alpha is greater than 0.5 (and we recall that the usual PageRank algorithm uses the value 0.85), then the upper limits of PR are less controllable than the upper limits of BPR, provided certain mild assumptions on the local structure of the graph. Regarding the lower limits of variability, we give a result for small values of alpha. We illustrate the results with some analytical networks and also with a real Facebook network.This work has been partially supported by the Spanish Ministry of Science, Innovation and Universities under Project Nos. PGC2018-101625-B-I00, MTM2016-76808-P, and MTM2017-84194-P (AEI/FEDER, UE).Flores, J.; García, E.; Pedroche Sánchez, F.; Romance, M. (2020). 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Sharp estimates for the personalized Multiplex PageRank

[EN] PageRank can be understood as the stationary distribution of a Markov chain that occurs in a two-layer network with the same set of nodes in both layers: the physical layer and the teleportation layer. In this paper we present some bounds for the extension of this two-layer approach to Multiplex networks, establishing sharp estimates for this Multiplex PageRank and locating the possible values of the personalized PageRank for each node of a network. Several examples are shown to compare the values obtained for both algorithms, the classic and the two-layer PageRank. (C) 2017 Elsevier B.V. All rights reserved.This work has been partially supported by the projects MTM2014-59906-P, MTM2014-52470-P (Spanish Ministry) and the grant Grupo de excelencia investigadora URJC-Banco de Santander GARECOM 30VCPIGI11. The authors would like to thank an anonymous referee for his/her valuable comments and remarks that have improved the readability of the manuscript.Pedroche Sánchez, F.; García, E.; Romance, M.; Criado Herrero, R. (2018). Sharp estimates for the personalized Multiplex PageRank. Journal of Computational and Applied Mathematics. 330:1030-1040. https://doi.org/10.1016/j.cam.2017.02.013S1030104033

On the intersection of the classes of doubly diagonally dominant matrices and S-strictly diagonally dominant matrices

We denote by H0 the subclass of H-matrices consisting of all the matrices that lay simultaneously on the classes of doubly diagonally dominant (DDD) matrices (A = [aij ] ∈ Cn×n : |aii||ajj | ≥ k =i |aik| k =j |ajk|, i = j) and S-strictly diagonally dominant (S-SDD) matrices. Notice that strictly doubly diagonally dominant matrices (also called Ostrowsky matrices) are a subclass of H0. Strictly diagonally dominant matrices (SDD) are also a subclass of H0. In this paper we analyze some properties of the class H0 = DDD ∩ S-SDD

On the spectrum of two-layer approach and Multiplex PageRank

[EN] In this paper, we present some results about the spectrum of the matrix associated with the computation of the Multiplex PageRank defined by the authors in a previous paper. These results can be considered as a natural extension of the known results about the spectrum of the Google matrix. In particular, we show that the eigenvalues of the transition matrix associated with the multiplex network can be deduced from the eigenvalues of a block matrix containing the stochastic matrices defined for each layer. We also show that, as occurs in the classic PageRank, the spectrum is not affected by the personalization vectors defined on each layer but depends on the parameter a that controls the teleportation. We also give some analytical relations between the eigenvalues and we include some small examples illustrating the main results. (C) 2018 Elsevier B.V. All rights reserved.We thank the two anonymous reviewers for their constructive comments, which helped us to improve the manuscript. This work has been partially supported by the projects MTM2014-59906-P, MTM2014-52470-P (Spanish Ministry and FEDER, EU, Spain), MTM2017-84194-P (AEI/FEDER, EU, Spain) and the grant URJC-Grupo de Excelencia Investigadora GARECOM (2014-2017), Spain.Pedroche Sánchez, F.; García, E.; Romance, M.; Criado Herrero, R. (2018). On the spectrum of two-layer approach and Multiplex PageRank. Journal of Computational and Applied Mathematics. 344:161-172. https://doi.org/10.1016/j.cam.2018.05.033S16117234

Combining the Two-Layers PageRank Approach with the APA Centrality in Networks with Data

[EN] Identifying the influential nodes in complex networks is a fundamental and practical topic at the moment. In this paper, a new centrality measure for complex networks is proposed based on two contrasting models that have their common origin in the well-known PageRank centrality. On the one hand, the essence of the model proposed is taken from the Adapted PageRank Algorithm (APA) centrality, whose main characteristic is that constitutes a measure to establish a ranking of nodes considering the importance of some dataset associated to the network. On the other hand, a technique known as two-layers PageRank approach is applied to this model. This technique focuses on the idea that the PageRank centrality can be understood as a two-layer network, the topological and teleportation layers, respectively. The main point of the proposed centrality is that it combines the APA centrality with the idea of two-layers; however, the difference now is that the teleportation layer is replaced by a layer that collects the data present in the network. This combination gives rise to a new algorithm for ranking the nodes according to their importance. Subsequently, the coherence of the new measure is demonstrated by calculating the correlation and the quantitative differences of both centralities (APA and the new centrality). A detailed study of the differences of both centralities, taking different types of networks, is performed. A real urban network with data randomly generated is evaluated as well as the well-known Zachary's karate club network. Some numerical results are carried out by varying the values of the alpha parameter-known as dumping factor in PageRank model-that varies the importance given to the two layers (topology and data) within the computation of the new centrality. The proposed algorithm takes the best characteristics of the models on which it is based: on the one hand, it is a measure of centrality, in complex networks with data, whose calculation is stable numerically and, on the other hand, it is able to separate the topological properties of the network and the influence of the data.Partially supported by the Spanish Government, Ministerio de Economia y Competividad, grant number TIN2017-84821-P.Agryzkov, T.; Pedroche Sánchez, F.; Tortosa, L.; Vicent, JF. (2018). Combining the Two-Layers PageRank Approach with the APA Centrality in Networks with Data. ISPRS International Journal of Geo-Information. 7(12):1-22. https://doi.org/10.3390/ijgi7120480S122712Crucitti, P., Latora, V., & Porta, S. (2006). Centrality measures in spatial networks of urban streets. Physical Review E, 73(3). doi:10.1103/physreve.73.036125Bonacich, P. (1991). Simultaneous group and individual centralities. Social Networks, 13(2), 155-168. doi:10.1016/0378-8733(91)90018-oStephenson, K., & Zelen, M. (1989). Rethinking centrality: Methods and examples. Social Networks, 11(1), 1-37. doi:10.1016/0378-8733(89)90016-6Szell, M., Lambiotte, R., & Thurner, S. (2010). Multirelational organization of large-scale social networks in an online world. Proceedings of the National Academy of Sciences, 107(31), 13636-13641. doi:10.1073/pnas.1004008107Calabrese, F., Colonna, M., Lovisolo, P., Parata, D., & Ratti, C. (2011). Real-Time Urban Monitoring Using Cell Phones: A Case Study in Rome. IEEE Transactions on Intelligent Transportation Systems, 12(1), 141-151. doi:10.1109/tits.2010.2074196Chen, D., Lü, L., Shang, M.-S., Zhang, Y.-C., & Zhou, T. (2012). Identifying influential nodes in complex networks. Physica A: Statistical Mechanics and its Applications, 391(4), 1777-1787. doi:10.1016/j.physa.2011.09.017Zhou, Y.-B., Lü, L., & Li, M. (2012). Quantifying the influence of scientists and their publications: distinguishing between prestige and popularity. New Journal of Physics, 14(3), 033033. doi:10.1088/1367-2630/14/3/033033Porta, S., Crucitti, P., & Latora, V. (2006). The network analysis of urban streets: A dual approach. Physica A: Statistical Mechanics and its Applications, 369(2), 853-866. doi:10.1016/j.physa.2005.12.063Jiang, B. (2009). Ranking spaces for predicting human movement in an urban environment. International Journal of Geographical Information Science, 23(7), 823-837. doi:10.1080/13658810802022822Bonacich, P. (1987). Power and Centrality: A Family of Measures. American Journal of Sociology, 92(5), 1170-1182. doi:10.1086/228631Boldi, P., & Vigna, S. (2014). Axioms for Centrality. Internet Mathematics, 10(3-4), 222-262. doi:10.1080/15427951.2013.865686Freeman, L. C. (1977). A Set of Measures of Centrality Based on Betweenness. Sociometry, 40(1), 35. doi:10.2307/3033543Brandes, U. (2001). A faster algorithm for betweenness centrality*. The Journal of Mathematical Sociology, 25(2), 163-177. doi:10.1080/0022250x.2001.9990249Haveliwala, T. H. (2003). Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE Transactions on Knowledge and Data Engineering, 15(4), 784-796. doi:10.1109/tkde.2003.1208999Berkhin, P. (2005). A Survey on PageRank Computing. Internet Mathematics, 2(1), 73-120. doi:10.1080/15427951.2005.10129098García, E., Pedroche, F., & Romance, M. (2013). On the localization of the personalized PageRank of complex networks. Linear Algebra and its Applications, 439(3), 640-652. doi:10.1016/j.laa.2012.10.051Langville, A., & Meyer, C. (2004). Deeper Inside PageRank. Internet Mathematics, 1(3), 335-380. doi:10.1080/15427951.2004.10129091Bianchini, M., Gori, M., & Scarselli, F. (2005). Inside PageRank. ACM Transactions on Internet Technology, 5(1), 92-128. doi:10.1145/1052934.1052938Migallón, H., Migallón, V., Palomino, J. A., & Penadés, J. (2018). A heuristic relaxed extrapolated algorithm for accelerating PageRank. Advances in Engineering Software, 120, 88-95. doi:10.1016/j.advengsoft.2016.01.024Agryzkov, T., Oliver, J. L., Tortosa, L., & Vicent, J. F. (2012). An algorithm for ranking the nodes of an urban network based on the concept of PageRank vector. Applied Mathematics and Computation, 219(4), 2186-2193. doi:10.1016/j.amc.2012.08.064Agryzkov, T., Tortosa, L., & Vicent, J. F. (2016). New highlights and a new centrality measure based on the Adapted PageRank Algorithm for urban networks. Applied Mathematics and Computation, 291, 14-29. doi:10.1016/j.amc.2016.06.036Agryzkov, T., Tortosa, L., Vicent, J. F., & Wilson, R. (2017). A centrality measure for urban networks based on the eigenvector centrality concept. Environment and Planning B: Urban Analytics and City Science, 46(4), 668-689. doi:10.1177/2399808317724444Conti, M., & Kumar, M. (2010). Opportunities in Opportunistic Computing. Computer, 43(1), 42-50. doi:10.1109/mc.2010.19Zhang, Y., Song, L., Jiang, C., Tran, N. H., Dawy, Z., & Han, Z. (2017). A Social-Aware Framework for Efficient Information Dissemination in Wireless Ad Hoc Networks. IEEE Communications Magazine, 55(1), 174-179. doi:10.1109/mcom.2017.1600029cmBoccaletti, S., Bianconi, G., Criado, R., del Genio, C. I., Gómez-Gardeñes, J., Romance, M., … Zanin, M. (2014). The structure and dynamics of multilayer networks. Physics Reports, 544(1), 1-122. doi:10.1016/j.physrep.2014.07.001Kivela, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., & Porter, M. A. (2014). Multilayer networks. Journal of Complex Networks, 2(3), 203-271. doi:10.1093/comnet/cnu016Pedroche, F., Romance, M., & Criado, R. (2016). A biplex approach to PageRank centrality: From classic to multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(6), 065301. doi:10.1063/1.4952955Zachary, W. W. (1977). 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Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index)

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in [Networks and Heterogeneous Media] following peer review. The definitive publisher-authenticated version [Pedroche, F… (et al.) (2015). Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Springfield, MO: American Institute of Mathematical Science. Networks and Heterogeneous Media.Volume 10, Number 1, March 2015, pp. 101-125. eISSN 1556-181X] is available online at: https://aimsciences.org/journals/pdfs.jsp?paperID=10842&mode=fullIn 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 coe cient for two partial rankings to the concept of evolutive Kendall's coe cient 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 de ne 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 de ned by 25 companies that have contributed to the IBEX-35 return and volatility values over the period 2003 to 2013.This work was partially supported by Spanish MICINN Funds and FEDER Funds MTM2009-13848, MTM2010-16153 and MTM2010-18674, and Junta de Andalucia Funds FQM-264. The authors would like to thank the referees for their valuable comments and remarks.Pedroche Sánchez, F.; Criado, R.; García, EH.; Romance, M.; Sánchez, VE. (2015). Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Networks and Heterogeneous Media. 10(1):101-125. https://doi.org/10.3934/nhm.2015.10.101S10112510