On the Localization of the Personalized PageRank of Complex Networks

In this paper new results on personalized PageRank are shown. We consider directed graphs that may contain dangling nodes. The main result presented gives an analytical characterization of all the possible values of the personalized PageRank for any node.We use this result to give a theoretical justification of a recent model that uses the personalized PageRank to classify users of Social Networks Sites. We introduce new concepts concerning competitivity and leadership in complex networks. We also present some theoretical techniques to locate leaders and competitors which are valid for any personalization vector and by using only information related to the adjacency matrix of the graph and the distribution of its dangling nodes

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

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-

Confirmación de la ausencia del alga marina asiática Codium fragile subsp. fragile (Codiaceae, Chlorophyta) en el Pacífico de México, mediante datos moleculares

Background. Codium fragile is a green alga that inhabits the Pacific coast of Baja California, Mexico, whose first records date back to 1909. This species has several subspecies, one of them Codium fragile subsp. fragile, originally from Japan, has proven to be an invasive organism in different parts of the world. Objectives. Confirm the presence or absence of this invasive strain on the coast of the Mexican Pacific, comparing with individuals from Japan, the United States and Mexico using molecular tools. Methods. We analyzed 20 individuals of the different species recorded for the Mexican Pacific and ten outside the region, including two as outgroup. The genomic DNA was extracted using the Sanger method, regions of the psb and 23S markers were amplified, the sequences obtained were edited and aligned in MEGA and MESQUITE, subsequently phylogenetic analyses of maximum likelihood were carried out in PAUP and MEGA and Bayesian inference in MrBayes. Genetic distances were obtained in MEGA and PAUP. Results. It is shown that, genetically, the individuals from Mexico integrate a clade different from the Asian entity, with inter-species genetic distances that were located at 6% for the 23S marker, while for the subclades of C. fragile the distance between them was 0.4%. For psb, the distance was 25% between species and 2.2% between these two subspecies. Conclusions. Individuals from California and Mexico nested in the native clade C. fragile, while others also from California are recognized with that of Japan as belonging to the invasive clade (C. fragile subsp. fragile). At the moment, the absence of this invasive strain in the Pacific of Mexico is confirmed.Antecedentes. Codium fragile es un alga verde que habita en las costas del Pacífico de Baja California, México, cuyos primeros registros datan de 1909. Esta especie que posee varias subespecies, una de ellas Codium fragile subsp. fragile, oriunda de Japón, ha demostrado ser un organismo invasor en diferentes partes del mundo. Objetivos. Confirmar la presencia o ausencia de esta cepa invasora en las costas del Pacífico mexicano, comparando individuos de Japón, Estados Unidos y México mediante el uso de herramientas moleculares. Métodos. Se analizaron 20 individuos de las especies registradas para el Pacífico mexicano y diez fuera de la región, incluyendo además dos como grupo externo. El DNA genómico se extrajo mediante el método Sanger, se amplificaron regiones de los marcadores psb y 23S, las secuencias obtenidas se editaron y alinearon en MEGA y MESQUITE, posteriormente se realizaron análisis filogenéticos de máxima verosimilitud en PAUP y MEGA y de inferencia bayesiana en MrBayes. Las distancias genéticas se obtuvieron en MEGA y PAUP. Resultados. Se demuestra que, genéticamente, los individuos de México integran un clado diferente a la entidad asiática, con distancias genéticas inter-especie que se ubicaron en 6% para el marcador 23S, mientras que para los subclados de C. fragile la distancia entre ellos fue de 0.4%. Para psb, la distancia fue del 25% entre especies y de 2.2% entre estas dos subespecies. Conclusiones. Individuos de California y México se anidaron en el clado nativo C. fragile, mientras que otros procedentes también de California se reconocen con el de Japón como pertenecientes al clado invasor (C. fragile subsp. fragile); así por el momento, se confirma la ausencia de esta cepa invasora en el Pacífico de México

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

A biplex approach to PageRank centrality: from classic to multiplex networks

In this paper, we present a new view of the PageRank algorithm inspired by multiplex networks. This new approach allows to introduce a new centrality measure for classic complex networks and a new proposal to extend the usual PageRank algorithm to multiplex networks. We give some analytical relations between these new approaches and the classic PageRank centrality measure, and we illustrate the new parameters presented by computing them on real underground networks. © 2016 Author(s).This work has been partially supported by the project MTM2014-59906 (Spanish Ministry) and the Grant URJC-Grupo de Excelencia Investigadora GARECOM (2014-2016).Pedroche Sánchez, F.; Romance, M.; Criado Herrero, R. (2016). A biplex approach to PageRank centrality: from classic to multiplex networks. Chaos. 26(6):065301-1-065301-9. https://doi.org/10.1063/1.4952955S065301-1065301-926