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

Abstract

[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