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Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling

  • Dane TaylorEmail author
  • Mason A. Porter
  • Peter J. Mucha
Chapter
Part of the Computational Social Sciences book series (CSS)

Abstract

We describe centralities in temporal networks using a supracentrality framework to study centrality trajectories, which characterize how the importances of nodes change in time. We study supracentrality generalizations of eigenvector-based centralities, a family of centrality measures for time-independent networks that includes PageRank, hub and authority scores, and eigenvector centrality. We start with a sequence of adjacency matrices, each of which represents a time layer of a network at a different point or interval of time. Coupling centrality matrices across time layers with weighted interlayer edges yields a supracentrality matrix \(\mathbb {C}(\omega )\), where ω controls the extent to which centrality trajectories change over time. We can flexibly tune the weight and topology of the interlayer coupling to cater to different scientific applications. The entries of the dominant eigenvector of \(\mathbb {C}(\omega )\) represent joint centralities, which simultaneously quantify the importance of every node in every time layer. Inspired by probability theory, we also compute marginal and conditional centralities. We illustrate how to adjust the coupling between time layers to tune the extent to which nodes’ centrality trajectories are influenced by the oldest and newest time layers. We support our findings by analysis in the limits of small and large ω.

Keywords

Temporal networks Centrality PageRank Multilayer networks Multiplex networks 

Notes

Acknowledgements

We thank Petter Holme and Jari Saramäki for the invitation to write this chapter. We thank Deryl DeFord, Tina Eliassi-Rad, Des Higham, Christine Klymko, Marianne McKenzie, Scott Pauls, and Michael Schaub for fruitful conversations. DT was supported by the Simons Foundation under Award #578333. PJM was supported by the James S. McDonnell Foundation 21st Century Science Initiative—Complex Systems Scholar Award #220020315.

References

  1. 1.
    Newman, M.E.J.: Networks, 2nd edn. Oxford University Press, Oxford (2018)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bonacich, P.: J. Math. Sociol. 2(1), 113 (1972)CrossRefGoogle Scholar
  3. 3.
    Faust, K.: Soc. Networks 19(2), 157 (1997)CrossRefGoogle Scholar
  4. 4.
    Borgatti, S.P., Jones, C., Everett, M.G.: Connections 21(2), 27 (1998)Google Scholar
  5. 5.
    Kempe, D., Kleinberg, J., Tardos, É.: In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146. ACM, New York, NY (2003)Google Scholar
  6. 6.
    Brin, S., Page, L.: In: Proceedings of the Seventh International World Wide Web Conference, pp. 107–117 (1998)Google Scholar
  7. 7.
    Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank citation ranking: bringing order to the Web. Technical Report 1999-66. Stanford InfoLab (1999)Google Scholar
  8. 8.
    Kleinberg, J.: J. ACM 46(5), 604 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Callaghan, T., Porter, M.A., Mucha, P.J.: Am. Math. Mon. 114(9), 761 (2007)CrossRefGoogle Scholar
  10. 10.
    Saavedra, S., Powers, S., McCotter, T., Porter, M.A., Mucha, P.J.: Physica A 389(5), 1131 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Chartier, T.P., Kreutzer, E., Langville, A.N., Pedings, K.E.: SIAM J. Sci. Comput. 33(3), 1077 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Holme, P.: Adv. Complex Syst. 6(02), 163 (2003)CrossRefGoogle Scholar
  13. 13.
    Guimerà, R., Mossa, S., Turtschi, A., Amaral, L.A.N.: Proc. Natl. Acad. Sci. U. S. A. 102(22), 7794 (2005)Google Scholar
  14. 14.
    Leicht, E.A., Clarkson, G., Shedden, K., Newman, M.E.J: Eur. Phys. J. B 59(1), 75 (2007)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Fowler, J.H., Johnson, T.R., Spriggs II, J.F., Jeon, S., Wahlbeck, P.J.: Policy Anal. 15(3), 324 (2007)CrossRefGoogle Scholar
  16. 16.
    Fowler, J.H., Jeon, S.: Soc. Networks 30(1), 16 (2008)CrossRefGoogle Scholar
  17. 17.
    Bergstrom, C.T., West, J.D., Wiseman, M.A.: J. Neurosci. 28(45), 11433 (2008)CrossRefGoogle Scholar
  18. 18.
    Jeong, H., Mason, S.P., Barabási, A.L., Oltvai, Z.N.: Nature 411(6833), 41 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Holme, P., Saramäki, J.: Phys. Rep. 519(3), 97 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    Holme, P., Saramäki, J. (eds.): Temporal Networks. Springer-Verlag, Berlin (2013)Google Scholar
  21. 21.
    Holme, P.: Eur. Phys. J. B 88(9), 234 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    Liao, H., Mariani, M.S., Medo, M., Zhang, Y.C., Zhou, M.Y.: Phys. Rep. 689, 1 (2017)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Tang, M.Y., Musolesi, M., Mascolo, C., Latora, V., Nicosia, V.: In: Proceedings of the 3rd Workshop on Social Network Systems—SNS ’10, pp. 1–6 (2010)Google Scholar
  24. 24.
    Kim, H., Tang, J., Anderson, R., Mascolo, C.: Comput. Netw. 56(3), 983 (2012)CrossRefGoogle Scholar
  25. 25.
    Williams, M.J., Musolesi, M.: R. Soc. Open Sci. 3(6) (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Alsayed, A., Higham, D.J.: Chaos, Solitons Fractals 72, 35 (2015)Google Scholar
  27. 27.
    Fenu, C., Higham, D.J.: SIAM J. Matrix Anal. Appl. 38(2), 343 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pan, R., Saramäki, J.: Phys. Rev. E 84(1), 016105 (2011)ADSCrossRefGoogle Scholar
  29. 29.
    Lerman, K., Ghosh, R., Kang, J.H.: In: Proceedings of the Eighth Workshop on Mining and Learning with Graphs, pp. 70–77. ACM, New York, NY (2010)Google Scholar
  30. 30.
    Grindrod, P., Higham, D.J.: Proc. R. Soc. A 470(2165), 20130835 (2014)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Motegi, S., Masuda, N.: Sci. Rep. 2, 904 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Grindrod, P., Parsons, M.C., Higham, D.J., Estrada, E.: Phys. Rev. E 83(4), 046120 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    Estrada, E: Phys. Rev. E 88(4), 042811 (2013)Google Scholar
  34. 34.
    Grindrod, P, Higham, D.J.: SIAM Rev. 55(1), 118 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Chen, I., Benzi, M., Chang, H.H., Hertzberg, V.S.: J. Complex Networks 5(2), 274 (2016)Google Scholar
  36. 36.
    Arrigo, F., Higham, D.J.: Appl. Network Sci. 2(1), 17 (2017)CrossRefGoogle Scholar
  37. 37.
    Huang, D.W., Yu, Z.G.: Sci. Rep. 7, 41454 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    Takaguchi, T., Yano, Y., Yoshida, Y.: Eur. Phys. J. B 89(2), 1 (2016)CrossRefGoogle Scholar
  39. 39.
    Walker, D., Xie, H., Yan, K.K., Maslov, S.: J. Stat. Mech. 2007(06), P06010 (2007)CrossRefGoogle Scholar
  40. 40.
    Mariani, M.S., Medo, M., Zhang, Y.C.: arXiv preprint, arXiv: 1608.08414 (2016)Google Scholar
  41. 41.
    You, K., Tempo, R., Qiu, L.: IEEE Trans. Autom. Control 62(5), 2080 (2017)CrossRefGoogle Scholar
  42. 42.
    Rossi, R.A., Gleich, D.F.: Algorithms and Models for the Web Graph, pp. 126–137. Springer, Berlin (2012)Google Scholar
  43. 43.
    Mariani, M.S., Medo, M., Zhang, Y.C.: Sci. Rep. 5, 16181 (2015)ADSCrossRefGoogle Scholar
  44. 44.
    Praprotnik, S., Batagelj, V.: Ars Mat. Contemp. 11(1), 11 (2015)CrossRefGoogle Scholar
  45. 45.
    Huang, Q., Zhao, C., Zhang, X., Wang, X., Yi, D.: Europhys. Lett. 118(3), 36001 (2017)ADSCrossRefGoogle Scholar
  46. 46.
    Flores, J., Romance, M.: J. Comput. Appl. Math. 330, 1041 (2018)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Kossinets, G., Kleinberg, J., Watts, D.: In: Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 435–443. ACM, New York, NY (2008)Google Scholar
  48. 48.
    Kostakos, V.: Physica A 388(6), 1007 (2009)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Taylor, D., Myers, S.A., Clauset, A., Porter, M.A., Mucha, P.J.: Multiscale Model. Simul. 15(1), 537 (2017)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Taylor, D., Porter, M.A., Mucha, P.J.: arXiv preprint arXiv:1904.02059 (2019)Google Scholar
  51. 51.
    Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: J. Complex Networks 2(3), 203 (2014)CrossRefGoogle Scholar
  52. 52.
    Porter, M.A.: Not. Am. Math. Soc. 65(11), 1419 (2018)Google Scholar
  53. 53.
    Taylor, D., Caceres, R.S., Mucha, P.J.: Phys. Rev. X 7(3), 031056 (2017)Google Scholar
  54. 54.
    Mucha, P.J., Porter, M.A.: Chaos 20(4), 041108 (2010)ADSCrossRefGoogle Scholar
  55. 55.
    Bassett, D.S., Porter, M.A., Wymbs, N.F., Grafton, S.T., Carlson, J.M., Mucha, P.J.: Chaos 23(1), 013142 (2013)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Weir, W.H., Emmons, S., Gibson, R., Taylor, D., Mucha, P.J.: Algorithms 10(3), 93 (2017)CrossRefGoogle Scholar
  57. 57.
    Pamfil, A.R., Howison, S.D., Lambiotte, R., Porter, M.A.: SIAM J. Math. Data Sci. (in press) arXiv:1804.01964Google Scholar
  58. 58.
    Magnani, M., Rossi, L.: In: 2011 International Conference on Advances in Social Networks Analysis and Mining (ASONAM), pp. 5–12. IEEE, Piscataway, NJ (2011)Google Scholar
  59. 59.
    De Domenico, M., Solé-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M.A., Gómez, S., Arenas, A.: Phys. Rev. X 3(4), 041022 (2013)Google Scholar
  60. 60.
    Battiston, F., Nicosia, V., Latora, V.: Phys. Rev. E 89(3), 032804 (2014)ADSCrossRefGoogle Scholar
  61. 61.
    Tavassoli, S., Zweig, K.A.: In: 2016 Third European Network Intelligence Conference (ENIC), pp. 25–32. IEEE, Wrocław, Poland (2016)Google Scholar
  62. 62.
    Magnani, M., Micenkova, B., Rossi, L.: arXiv preprint arXiv:1303.4986 (2013)Google Scholar
  63. 63.
    Solé-Ribalta, A., De Domenico, M., Gómez, S., Arenas, A.: In: Proceedings of the 2014 ACM Conference on Web Science, pp. 149–155. ACM, New York, NY (2014)Google Scholar
  64. 64.
    Chakraborty, T., Narayanam, R.: In: 2016 IEEE 32nd International Conference on Data Engineering (ICDE), pp. 397–408. IEEE, Piscataway, NJ (2016)Google Scholar
  65. 65.
    Solé-Ribalta, A., De Domenico, M., Gómez, S., Arenas, A.: Physica D 323, 73 (2016)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    Spatocco, C., Stilo, G., Domeniconi, C.: arXiv preprint arXiv:1801.08026 (2018)Google Scholar
  67. 67.
    Rahmede, C., Iacovacci, J., Arenas, A., Bianconi, G.: J. Complex Networks 6(5), 733 (2017)CrossRefGoogle Scholar
  68. 68.
    Tudisco, F., Arrigo, F., Gautier, A.: J. SIAM Appl. Math. 78(2), 853 (2018)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Solá, L., Romance, M., Criado, R., Flores, J., del Amo, A.G., Boccaletti, S.: Chaos 23(3), 033131 (2013)ADSCrossRefGoogle Scholar
  70. 70.
    DeFord, D.R., Pauls, S.D.: J. Complex Networks 6(3), 353 (2017)CrossRefGoogle Scholar
  71. 71.
    DeFord, D.R.: In: International Workshop on Complex Networks and their Applications, pp. 1111–1123. Springer, Berlin (2017)Google Scholar
  72. 72.
    Ng, M.K.P., Li, X., Ye, Y.: In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1217–1225. ACM, New York, NY (2011)Google Scholar
  73. 73.
    Halu, A., Mondragón, R.J., Panzarasa, P., Bianconi, G.: PLoS ONE 8(10), e78293 (2013)ADSCrossRefGoogle Scholar
  74. 74.
    Ding, C., Li, K.: Neurocomputing 312, 263 (2018)CrossRefGoogle Scholar
  75. 75.
    Gleich, D.F.: SIAM Rev. 57(3), 321 (2015)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Taylor, D.: Data release: Ph.D. exchange in the Mathematical Genealogy Project. Available at https://sites.google.com/site/danetaylorresearch/data
  77. 77.
    Langville, A.N., Meyer, C.D.: Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, Princeton, NJ (2006)zbMATHCrossRefGoogle Scholar
  78. 78.
    Masuda, N., Porter, M.A., Lambiotte, R.: Phys. Rep. 716–717, 1 (2017)ADSCrossRefGoogle Scholar
  79. 79.
    Burris, V.: Am. Sociol. Rev. 69(2), 239 (2004)CrossRefGoogle Scholar
  80. 80.
    Myers, S.A., Mucha, P.J., Porter, M.A.: Chaos 21(4), 041104 (2011)ADSCrossRefGoogle Scholar
  81. 81.
    Clauset, A., Arbesman, S., Larremore, D.B.: Sci. Adv. 1(1), e1400005 (2015)ADSCrossRefGoogle Scholar
  82. 82.
    The Mathematics Genealogy Project. Available at http://www.genealogy.ams.org; data provided 19 October 2009
  83. 83.
    Ahmad, W., Porter, M.A., Beguerisse-Díaz, M.: arXiv preprint arXiv:1805.00193 (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dane Taylor
    • 1
    Email author
  • Mason A. Porter
    • 2
  • Peter J. Mucha
    • 3
  1. 1.University at Buffalo, State University of New YorkBuffaloUSA
  2. 2.University of CaliforniaLos AngelesUSA
  3. 3.University of North CarolinaChapel HillUSA

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