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

Can the PageRank centrality be manipulated to obtain any desired ranking?

The significance of the PageRank algorithm in shaping the modern Internet cannot be overstated, and its Complex Network theory foundations continue to be a subject of research. In this article we carry out a systematic study of the structural and parametric controllability of PageRank's outcomes, translating a spectral Graph Theory problem into a geometric one, where a natural characterization of its rankings emerges. Furthermore, we show that the change of perspective employed can be applied to the biplex PageRank proposal, performing numerical computations on both real and synthetic network datasets to compare centrality measures used.Comment: 18 pages, 6 figure

Uplifting edges in higher order networks: spectral centralities for non-uniform hypergraphs

Spectral analysis of networks states that many structural properties of graphs, such as centrality of their nodes, are given in terms of their adjacency matrices. The natural extension of such spectral analysis to higher order networks is strongly limited by the fact that a given hypergraph could have several different adjacency hypermatrices, hence the results obtained so far are mainly restricted to the class of uniform hypergraphs, which leaves many real systems unattended. A new method for analysing non-linear eigenvector-like centrality measures of non-uniform hypergraphs is presented in this paper that could be useful for studying properties of $\mathcal{H}$-eigenvectors and $\mathcal{Z}$-eigenvectors in the non-uniform case. In order to do so, a new operation - the $\textit{uplift}$ - is introduced, incorporating auxiliary nodes in the hypergraph to allow for a uniform-like analysis. We later argue why this is a mathematically sound operation, and we furthermore use it to classify a whole family of hypergraphs with unique Perron-like $\mathcal{Z}$-eigenvectors. We supplement the theoretical analysis with several examples and numerical simulations on synthetic and real datasets.Comment: 28 pages, 6 figure

Modeling the Multi-layer Nature of the European Air Transport Network: Resilience and Passengers Re-scheduling under random failures

We study the dynamics of the European Air Transport Network by using a multiplex network formalism. We will consider the set of flights of each airline as an interdependent network and we analyze the resilience of the system against random flight failures in the passenger's rescheduling problem. A comparison between the single-plex approach and the corresponding multiplex one is presented illustrating that the multiplexity strongly affects the robustness of the European Air Network.Comment: 12 pages, 5 figures - Accepted for publication in European Physical Journal Special Topic

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

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