Search Engine for Halal Linked Open Data Using Entity Ranking Approach

Halal concept is an essential aspect of Muslim daily life. Currently, many organizations around the world provide halal certification services as known as halal certification bodies. In the majority, these organizations provide halal product information on their website. However, information is presented in different formats, such as pdf, table, and text. As a result, the user is difficult to search for information on these websites. Therefore, we develop search engine on halal linked open data to facilitate users for searching halal products. We use an entity ranking approach to retrieve relevant items based on user queries that consist of an independent-ranking and dependent-ranking method. Independent ranking employs a link-count approach to indicate the information richness of the entity. Dependent ranking employs term frequency-inverse entity frequency  (TF-IEF) to measure the similarity of an entity based on terms. We use Apache Lucene to perform indexing and search process. Also, we use the Neo4j graph database to save entity ranking computation results. The results show that the system delivers excellent results. The Mean Average Precision (MAP) for top-5 results is 91,2%

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

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). Parametric controllability of the personalized PageRank: Classic model vs biplex approach. Chaos An Interdisciplinary Journal of Nonlinear Science. 30(2):1-15. https://doi.org/10.1063/1.5128567S115302Agryzkov, T., Curado, M., Pedroche, F., Tortosa, L., & Vicent, J. (2019). Extending the Adapted PageRank Algorithm Centrality to Multiplex Networks with Data Using the PageRank Two-Layer Approach. Symmetry, 11(2), 284. doi:10.3390/sym11020284Agryzkov, T., Pedroche, F., Tortosa, L., & Vicent, J. (2018). Combining the Two-Layers PageRank Approach with the APA Centrality in Networks with Data. ISPRS International Journal of Geo-Information, 7(12), 480. doi:10.3390/ijgi7120480Allcott, H., Gentzkow, M., & Yu, C. (2019). Trends in the diffusion of misinformation on social media. Research & Politics, 6(2), 205316801984855. doi:10.1177/2053168019848554Aleja, D., Criado, R., García del Amo, A. J., Pérez, Á., & Romance, M. (2019). Non-backtracking PageRank: From the classic model to hashimoto matrices. Chaos, Solitons & Fractals, 126, 283-291. doi:10.1016/j.chaos.2019.06.017Barabási, A.-L., & Albert, R. (1999). Emergence of Scaling in Random Networks. Science, 286(5439), 509-512. doi:10.1126/science.286.5439.509Bavelas, A. (1948). A Mathematical Model for Group Structures. Human Organization, 7(3), 16-30. doi:10.17730/humo.7.3.f4033344851gl053Benson, A. R. (2019). Three Hypergraph Eigenvector Centralities. SIAM Journal on Mathematics of Data Science, 1(2), 293-312. doi:10.1137/18m1203031Boccaletti, 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.2014.07.001Boldi, P., & Vigna, S. (2014). Axioms for Centrality. Internet Mathematics, 10(3-4), 222-262. doi:10.1080/15427951.2013.865686Boldi, P., Santini, M., & Vigna, S. (2009). PageRank. ACM Transactions on Information Systems, 27(4), 1-23. doi:10.1145/1629096.1629097Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. The Journal of Mathematical Sociology, 2(1), 113-120. doi:10.1080/0022250x.1972.9989806Borgatti, S. P., & Everett, M. G. (2006). A Graph-theoretic perspective on centrality. Social Networks, 28(4), 466-484. doi:10.1016/j.socnet.2005.11.005Buzzanca, M., Carchiolo, V., Longheu, A., Malgeri, M., & Mangioni, G. (2018). Black hole metric: Overcoming the pagerank normalization problem. Information Sciences, 438, 58-72. doi:10.1016/j.ins.2018.01.033De Domenico, M., Solé-Ribalta, A., Omodei, E., Gómez, S., & Arenas, A. (2015). Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications, 6(1). doi:10.1038/ncomms7868DeFord, D. R., & Pauls, S. D. (2017). A new framework for dynamical models on multiplex networks. Journal of Complex Networks, 6(3), 353-381. doi:10.1093/comnet/cnx041Del Corso, G. M., & Romani, F. (2016). A multi-class approach for ranking graph nodes: Models and experiments with incomplete data. Information Sciences, 329, 619-637. doi:10.1016/j.ins.2015.09.046Estrada, E., & Silver, G. (2017). Accounting for the role of long walks on networks via a new matrix function. Journal of Mathematical Analysis and Applications, 449(2), 1581-1600. doi:10.1016/j.jmaa.2016.12.062Festinger, L. (1949). The Analysis of Sociograms using Matrix Algebra. Human Relations, 2(2), 153-158. doi:10.1177/001872674900200205Votruba, J. (1975). On the determination of χl,η+−0 AND η000 from bubble chamber measurements. Czechoslovak Journal of Physics, 25(6), 619-625. doi:10.1007/bf01591018Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215-239. doi:10.1016/0378-8733(78)90021-7Ermann, L., Frahm, K. M., & Shepelyansky, D. L. (2015). Google matrix analysis of directed networks. Reviews of Modern Physics, 87(4), 1261-1310. doi:10.1103/revmodphys.87.1261Frahm, K. M., & Shepelyansky, D. L. (2019). Ising-PageRank model of opinion formation on social networks. Physica A: Statistical Mechanics and its Applications, 526, 121069. doi:10.1016/j.physa.2019.121069García, 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.2012.10.051Gu, C., Jiang, X., Shao, C., & Chen, Z. (2018). A GMRES-Power algorithm for computing PageRank problems. Journal of Computational and Applied Mathematics, 343, 113-123. doi:10.1016/j.cam.2018.03.017Halu, A., Mondragón, R. J., Panzarasa, P., & Bianconi, G. (2013). Multiplex PageRank. PLoS ONE, 8(10), e78293. doi:10.1371/journal.pone.0078293Horn, R. A., & Johnson, C. R. (1991). Topics in Matrix Analysis. doi:10.1017/cbo9780511840371Iacovacci, J., & Bianconi, G. (2016). Extracting information from multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(6), 065306. doi:10.1063/1.4953161Iacovacci, J., Rahmede, C., Arenas, A., & Bianconi, G. (2016). Functional Multiplex PageRank. EPL (Europhysics Letters), 116(2), 28004. doi:10.1209/0295-5075/116/28004Iván, G., & Grolmusz, V. (2010). When the Web meets the cell: using personalized PageRank for analyzing protein interaction networks. Bioinformatics, 27(3), 405-407. doi:10.1093/bioinformatics/btq680Kalecky, K., & Cho, Y.-R. (2018). PrimAlign: PageRank-inspired Markovian alignment for large biological networks. Bioinformatics, 34(13), i537-i546. doi:10.1093/bioinformatics/bty288Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39-43. doi:10.1007/bf02289026Langville, A., & Meyer, C. (2004). Deeper Inside PageRank. Internet Mathematics, 1(3), 335-380. doi:10.1080/15427951.2004.10129091Liu, Y.-Y., Slotine, J.-J., & Barabási, A.-L. (2011). Controllability of complex networks. Nature, 473(7346), 167-173. doi:10.1038/nature10011Lv, L., Zhang, K., Zhang, T., Bardou, D., Zhang, J., & Cai, Y. (2019). PageRank centrality for temporal networks. Physics Letters A, 383(12), 1215-1222. doi:10.1016/j.physleta.2019.01.041Massucci, F. A., & Docampo, D. (2019). Measuring the academic reputation through citation networks via PageRank. Journal of Informetrics, 13(1), 185-201. doi:10.1016/j.joi.2018.12.001Masuda, N., Porter, M. A., & Lambiotte, R. (2017). Random walks and diffusion on networks. Physics Reports, 716-717, 1-58. doi:10.1016/j.physrep.2017.07.007Migallón, H., Migallón, V., & Penadés, J. (2018). Parallel two-stage algorithms for solving the PageRank problem. Advances in Engineering Software, 125, 188-199. doi:10.1016/j.advengsoft.2018.03.002Newman, M. (2010). Networks. doi:10.1093/acprof:oso/9780199206650.001.0001Nicosia, V., Criado, R., Romance, M., Russo, G., & Latora, V. (2012). Controlling centrality in complex networks. Scientific Reports, 2(1). doi:10.1038/srep00218Pedroche, F., García, E., Romance, M., & Criado, R. (2018). Sharp estimates for the personalized Multiplex PageRank. Journal of Computational and Applied Mathematics, 330, 1030-1040. doi:10.1016/j.cam.2017.02.013Pedroche, F., Tortosa, L., & Vicent, J. F. (2019). An Eigenvector Centrality for Multiplex Networks with Data. Symmetry, 11(6), 763. doi:10.3390/sym11060763Pedroche, 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:10.1063/1.4952955Sciarra, C., Chiarotti, G., Laio, F., & Ridolfi, L. (2018). A change of perspective in network centrality. Scientific Reports, 8(1). doi:10.1038/s41598-018-33336-8Scholz, M., Pfeiffer, J., & Rothlauf, F. (2017). Using PageRank for non-personalized default rankings in dynamic markets. European Journal of Operational Research, 260(1), 388-401. doi:10.1016/j.ejor.2016.12.022Shen, Y., Gu, C., & Zhao, P. (2019). Structural Vulnerability Assessment of Multi-energy System Using a PageRank Algorithm. Energy Procedia, 158, 6466-6471. doi:10.1016/j.egypro.2019.01.132Shen, Z.-L., Huang, T.-Z., Carpentieri, B., Wen, C., Gu, X.-M., & Tan, X.-Y. (2019). Off-diagonal low-rank preconditioner for difficult PageRank problems. Journal of Computational and Applied Mathematics, 346, 456-470. doi:10.1016/j.cam.2018.07.015Shepelyansky, D. L., & Zhirov, O. V. (2010). Towards Google matrix of brain. Physics Letters A, 374(31-32), 3206-3209. doi:10.1016/j.physleta.2010.06.007Solá, L., Romance, M., Criado, R., Flores, J., García del Amo, A., & Boccaletti, S. (2013). Eigenvector centrality of nodes in multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(3), 033131. doi:10.1063/1.4818544Tian, Z., Liu, Y., Zhang, Y., Liu, Z., & Tian, M. (2019). The general inner-outer iteration method based on regular splittings for the PageRank problem. Applied Mathematics and Computation, 356, 479-501. doi:10.1016/j.amc.2019.02.066Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393(6684), 440-442. doi:10.1038/30918Yun, T.-S., Jeong, D., & Park, S. (2019). «Too central to fail» systemic risk measure using PageRank algorithm. Journal of Economic Behavior & Organization, 162, 251-272. doi:10.1016/j.jebo.2018.12.02

Eye-Tracking-Based Classification of Information Search Behavior Using Machine Learning: Evidence from Experiments in Physical Shops and Virtual Reality Shopping Environments

Classifying information search behavior helps tailor recommender systems to individual customers’ shopping motives. But how can we identify these motives without requiring users to exert too much effort? Our research goal is to demonstrate that eye tracking can be used at the point of sale to do so. We focus on two frequently investigated shopping motives: goal-directed and exploratory search. To train and test a prediction model, we conducted two eye-tracking experiments in front of supermarket shelves. The first experiment was carried out in immersive virtual reality; the second, in physical reality—in other words, as a field study in a real supermarket. We conducted a virtual reality study, because recently launched virtual shopping environments suggest that there is great interest in using this technology as a retail channel. Our empirical results show that support vector machines allow the correct classification of search motives with 80% accuracy in virtual reality and 85% accuracy in physical reality. Our findings also imply that eye movements allow shopping motives to be identified relatively early in the search process: our models achieve 70% prediction accuracy after only 15 seconds in virtual reality and 75% in physical reality. Applying an ensemble method increases the prediction accuracy substantially, to about 90%. Consequently, the approach that we propose could be used for the satisfiable classification of consumers in practice. Furthermore, both environments’ best predictor variables overlap substantially. This finding provides evidence that in virtual reality, information search behavior might be similar to the one used in physical reality. Finally, we also discuss managerial implications for retailers and companies that are planning to use our technology to personalize a consumer assistance system

On PageRank versatility for multiplex networks: properties and some useful bounds

[EN] In this paper, some results concerning the PageRank versatility measure for multiplex networks are given. This measure extends to the multiplex setting the well-known classic PageRank. Particularly, we focus on some spectral properties of the Laplacian matrix of the multiplex and on obtaining boundaries for the ranking value of a given node when some personalization vector is added, as in the classic setting.This work has been partially supported by the projects MTM2017-84194-P (Ministerio de Ciencia y Tecnología, AEI/FEDER, UE) and PGC2018-101625B-100 (Ministerio de Ciencia y Tecnología, AEI/FEDER, UE).Pedroche Sánchez, F.; Criado, R.; Flores, J.; García, E.; Romance, M. (2020). On PageRank versatility for multiplex networks: properties and some useful bounds. Mathematical Methods in the Applied Sciences. 43(14):8158-8176. https://doi.org/10.1002/mma.6274815881764314BOCCALETTI, 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.2005.10.009Criado, R., Flores, J., García del Amo, A., & Romance, M. (2011). Analytical relationships between metric and centrality measures of a network and its dual. Journal of Computational and Applied Mathematics, 235(7), 1775-1780. doi:10.1016/j.cam.2010.04.011De Domenico, M., Solé-Ribalta, A., Omodei, E., Gómez, S., & Arenas, A. (2015). Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications, 6(1). doi:10.1038/ncomms7868Freeman, L. C. (1977). A Set of Measures of Centrality Based on Betweenness. Sociometry, 40(1), 35. doi:10.2307/3033543Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215-239. doi:10.1016/0378-8733(78)90021-7Nicosia, V., Criado, R., Romance, M., Russo, G., & Latora, V. (2012). Controlling centrality in complex networks. Scientific Reports, 2(1). doi:10.1038/srep00218Solá, L., Romance, M., Criado, R., Flores, J., García del Amo, A., & Boccaletti, S. (2013). Eigenvector centrality of nodes in multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(3), 033131. doi:10.1063/1.4818544Boccaletti, 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.2014.07.001Criado, 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.2015.03.009Guimera, R., Mossa, S., Turtschi, A., & Amaral, L. A. N. (2005). The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proceedings of the National Academy of Sciences, 102(22), 7794-7799. doi:10.1073/pnas.0407994102Jeong, H., Mason, S. P., Barabási, A.-L., & Oltvai, Z. N. (2001). Lethality and centrality in protein networks. Nature, 411(6833), 41-42. doi:10.1038/35075138PageL BrinS MotwaniR WinogradT.The PageRank citation ranking: Bringing order to the Web   Tech. Rep. 66;1998.  Stanford University.Agryzkov, 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.2016.06.036Boldi, P., Santini, M., & Vigna, S. (2009). PageRank. ACM Transactions on Information Systems, 27(4), 1-23. doi:10.1145/1629096.1629097Criado, R., Moral, S., Pérez, Á., & Romance, M. (2018). On the edges’ PageRank and line graphs. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(7), 075503. doi:10.1063/1.5020127García, 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.2012.10.051Halu, A., Mondragón, R. J., Panzarasa, P., & Bianconi, G. (2013). Multiplex PageRank. PLoS ONE, 8(10), e78293. doi:10.1371/journal.pone.0078293Pedroche Sánchez, F. (2010). Competitivity groups on social network sites. Mathematical and Computer Modelling, 52(7-8), 1052-1057. doi:10.1016/j.mcm.2010.02.031Scholz, M., Pfeiffer, J., & Rothlauf, F. (2017). Using PageRank for non-personalized default rankings in dynamic markets. European Journal of Operational Research, 260(1), 388-401. doi:10.1016/j.ejor.2016.12.022Shen, Z.-L., Huang, T.-Z., Carpentieri, B., Gu, X.-M., & Wen, C. (2017). An efficient elimination strategy for solving PageRank problems. Applied Mathematics and Computation, 298, 111-122. doi:10.1016/j.amc.2016.10.031Tan, X. (2017). A new extrapolation method for PageRank computations. Journal of Computational and Applied Mathematics, 313, 383-392. doi:10.1016/j.cam.2016.08.034Wen, C., Huang, T.-Z., & Shen, Z.-L. (2017). A note on the two-step matrix splitting iteration for computing PageRank. Journal of Computational and Applied Mathematics, 315, 87-97. doi:10.1016/j.cam.2016.10.020Pedroche, 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:10.1063/1.4952955Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., & Onnela, J.-P. (2010). Community Structure in Time-Dependent, Multiscale, and Multiplex Networks. Science, 328(5980), 876-878. doi:10.1126/science.1184819Szell, 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.1004008107Radicchi, F., & Arenas, A. (2013). Abrupt transition in the structural formation of interconnected networks. Nature Physics, 9(11), 717-720. doi:10.1038/nphys2761Battiston, F., Nicosia, V., & Latora, V. (2017). The new challenges of multiplex networks: Measures and models. The European Physical Journal Special Topics, 226(3), 401-416. doi:10.1140/epjst/e2016-60274-8Romance, M., Solá, L., Flores, J., García, E., García del Amo, A., & Criado, R. (2015). A Perron–Frobenius theory for block matrices associated to a multiplex network. Chaos, Solitons & Fractals, 72, 77-89. doi:10.1016/j.chaos.2014.12.020Wang, D., & Zou, X. (2018). A new centrality measure of nodes in multilayer networks under the framework of tensor computation. Applied Mathematical Modelling, 54, 46-63. doi:10.1016/j.apm.2017.07.012PadgettJF.Marriage and Elite Structure in Renaissance Florence 1282‐1500. Paper delivered to the Social Science History Association;1994.Sciarra, C., Chiarotti, G., Laio, F., & Ridolfi, L. (2018). A change of perspective in network centrality. Scientific Reports, 8(1). doi:10.1038/s41598-018-33336-8Horn, R. A., & Johnson, C. R. (1991). Topics in Matrix Analysis. doi:10.1017/cbo9780511840371OMODEI, E., DE DOMENICO, M., & ARENAS, A. (2016). Evaluating the impact of interdisciplinary research: A multilayer network approach. Network Science, 5(2), 235-246. doi:10.1017/nws.2016.15Horn, R. A., & Johnson, C. R. (1985). Matrix Analysis. doi:10.1017/cbo9780511810817Merris, R. (1998). Laplacian graph eigenvectors. Linear Algebra and its Applications, 278(1-3), 221-236. doi:10.1016/s0024-3795(97)10080-

On PageRank versatility for multiplex networks: properties and some useful bounds

[EN] In this paper, some results concerning the PageRank versatility measure for multiplex networks are given. This measure extends to the multiplex setting the well-known classic PageRank. Particularly, we focus on some spectral properties of the Laplacian matrix of the multiplex and on obtaining boundaries for the ranking value of a given node when some personalization vector is added, as in the classic setting.This work has been partially supported by the projects MTM2017-84194-P (Ministerio de Ciencia y Tecnología, AEI/FEDER, UE) and PGC2018-101625B-100 (Ministerio de Ciencia y Tecnología, AEI/FEDER, UE).Pedroche Sánchez, F.; Criado, R.; Flores, J.; García, E.; Romance, M. (2020). On PageRank versatility for multiplex networks: properties and some useful bounds. Mathematical Methods in the Applied Sciences. 43(14):8158-8176. https://doi.org/10.1002/mma.6274S815881764314BOCCALETTI, 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.2005.10.009Criado, R., Flores, J., García del Amo, A., & Romance, M. (2011). Analytical relationships between metric and centrality measures of a network and its dual. Journal of Computational and Applied Mathematics, 235(7), 1775-1780. doi:10.1016/j.cam.2010.04.011De Domenico, M., Solé-Ribalta, A., Omodei, E., Gómez, S., & Arenas, A. (2015). Ranking in interconnected multilayer networks reveals versatile nodes. Nature Communications, 6(1). doi:10.1038/ncomms7868Freeman, L. C. (1977). A Set of Measures of Centrality Based on Betweenness. Sociometry, 40(1), 35. doi:10.2307/3033543Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215-239. doi:10.1016/0378-8733(78)90021-7Nicosia, V., Criado, R., Romance, M., Russo, G., & Latora, V. (2012). Controlling centrality in complex networks. Scientific Reports, 2(1). doi:10.1038/srep00218Solá, L., Romance, M., Criado, R., Flores, J., García del Amo, A., & Boccaletti, S. (2013). Eigenvector centrality of nodes in multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(3), 033131. doi:10.1063/1.4818544Boccaletti, 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.2014.07.001Criado, 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.2015.03.009Guimera, R., Mossa, S., Turtschi, A., & Amaral, L. A. N. (2005). The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proceedings of the National Academy of Sciences, 102(22), 7794-7799. doi:10.1073/pnas.0407994102Jeong, H., Mason, S. P., Barabási, A.-L., & Oltvai, Z. N. (2001). Lethality and centrality in protein networks. Nature, 411(6833), 41-42. doi:10.1038/35075138PageL BrinS MotwaniR WinogradT.The PageRank citation ranking: Bringing order to the Web   Tech. Rep. 66;1998.  Stanford University.Agryzkov, 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.2016.06.036Boldi, P., Santini, M., & Vigna, S. (2009). PageRank. ACM Transactions on Information Systems, 27(4), 1-23. doi:10.1145/1629096.1629097Criado, R., Moral, S., Pérez, Á., & Romance, M. (2018). On the edges’ PageRank and line graphs. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(7), 075503. doi:10.1063/1.5020127García, 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.2012.10.051Halu, A., Mondragón, R. J., Panzarasa, P., & Bianconi, G. (2013). Multiplex PageRank. PLoS ONE, 8(10), e78293. doi:10.1371/journal.pone.0078293Pedroche Sánchez, F. (2010). Competitivity groups on social network sites. Mathematical and Computer Modelling, 52(7-8), 1052-1057. doi:10.1016/j.mcm.2010.02.031Scholz, M., Pfeiffer, J., & Rothlauf, F. (2017). Using PageRank for non-personalized default rankings in dynamic markets. European Journal of Operational Research, 260(1), 388-401. doi:10.1016/j.ejor.2016.12.022Shen, Z.-L., Huang, T.-Z., Carpentieri, B., Gu, X.-M., & Wen, C. (2017). An efficient elimination strategy for solving PageRank problems. Applied Mathematics and Computation, 298, 111-122. doi:10.1016/j.amc.2016.10.031Tan, X. (2017). A new extrapolation method for PageRank computations. Journal of Computational and Applied Mathematics, 313, 383-392. doi:10.1016/j.cam.2016.08.034Wen, C., Huang, T.-Z., & Shen, Z.-L. (2017). A note on the two-step matrix splitting iteration for computing PageRank. Journal of Computational and Applied Mathematics, 315, 87-97. doi:10.1016/j.cam.2016.10.020Pedroche, 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:10.1063/1.4952955Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., & Onnela, J.-P. (2010). Community Structure in Time-Dependent, Multiscale, and Multiplex Networks. Science, 328(5980), 876-878. doi:10.1126/science.1184819Szell, 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.1004008107Radicchi, F., & Arenas, A. (2013). Abrupt transition in the structural formation of interconnected networks. Nature Physics, 9(11), 717-720. doi:10.1038/nphys2761Battiston, F., Nicosia, V., & Latora, V. (2017). The new challenges of multiplex networks: Measures and models. The European Physical Journal Special Topics, 226(3), 401-416. doi:10.1140/epjst/e2016-60274-8Romance, M., Solá, L., Flores, J., García, E., García del Amo, A., & Criado, R. (2015). A Perron–Frobenius theory for block matrices associated to a multiplex network. Chaos, Solitons & Fractals, 72, 77-89. doi:10.1016/j.chaos.2014.12.020Wang, D., & Zou, X. (2018). A new centrality measure of nodes in multilayer networks under the framework of tensor computation. Applied Mathematical Modelling, 54, 46-63. doi:10.1016/j.apm.2017.07.012PadgettJF.Marriage and Elite Structure in Renaissance Florence 1282‐1500. Paper delivered to the Social Science History Association;1994.Sciarra, C., Chiarotti, G., Laio, F., & Ridolfi, L. (2018). A change of perspective in network centrality. Scientific Reports, 8(1). doi:10.1038/s41598-018-33336-8Horn, R. A., & Johnson, C. R. (1991). Topics in Matrix Analysis. doi:10.1017/cbo9780511840371OMODEI, E., DE DOMENICO, M., & ARENAS, A. (2016). Evaluating the impact of interdisciplinary research: A multilayer network approach. Network Science, 5(2), 235-246. doi:10.1017/nws.2016.15Horn, R. A., & Johnson, C. R. (1985). Matrix Analysis. doi:10.1017/cbo9780511810817Merris, R. (1998). Laplacian graph eigenvectors. Linear Algebra and its Applications, 278(1-3), 221-236. doi:10.1016/s0024-3795(97)10080-