Skip to main content
SpringerLink
Log in

Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling

  • Chapter
  • First Online:
Temporal Network Theory

Part of the book series: Computational Social Sciences ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that PageRank has had intellectual impact well beyond web searches [75].

References

  1. Newman, M.E.J.: Networks, 2nd edn. Oxford University Press, Oxford (2018)

    Book  MATH  Google Scholar 

  2. Bonacich, P.: J. Math. Sociol. 2(1), 113 (1972)

    Article  Google Scholar 

  3. Faust, K.: Soc. Networks 19(2), 157 (1997)

    Article  Google Scholar 

  4. Borgatti, S.P., Jones, C., Everett, M.G.: Connections 21(2), 27 (1998)

    Google Scholar 

  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. Brin, S., Page, L.: In: Proceedings of the Seventh International World Wide Web Conference, pp. 107–117 (1998)

    Google Scholar 

  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. Kleinberg, J.: J. ACM 46(5), 604 (1999)

    Article  MathSciNet  Google Scholar 

  9. Callaghan, T., Porter, M.A., Mucha, P.J.: Am. Math. Mon. 114(9), 761 (2007)

    Article  Google Scholar 

  10. Saavedra, S., Powers, S., McCotter, T., Porter, M.A., Mucha, P.J.: Physica A 389(5), 1131 (2010)

    Article  ADS  Google Scholar 

  11. Chartier, T.P., Kreutzer, E., Langville, A.N., Pedings, K.E.: SIAM J. Sci. Comput. 33(3), 1077 (2011)

    Article  MathSciNet  Google Scholar 

  12. Holme, P.: Adv. Complex Syst. 6(02), 163 (2003)

    Article  Google Scholar 

  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. Leicht, E.A., Clarkson, G., Shedden, K., Newman, M.E.J: Eur. Phys. J. B 59(1), 75 (2007)

    Article  ADS  MATH  Google Scholar 

  15. Fowler, J.H., Johnson, T.R., Spriggs II, J.F., Jeon, S., Wahlbeck, P.J.: Policy Anal. 15(3), 324 (2007)

    Article  Google Scholar 

  16. Fowler, J.H., Jeon, S.: Soc. Networks 30(1), 16 (2008)

    Article  Google Scholar 

  17. Bergstrom, C.T., West, J.D., Wiseman, M.A.: J. Neurosci. 28(45), 11433 (2008)

    Article  Google Scholar 

  18. Jeong, H., Mason, S.P., Barabási, A.L., Oltvai, Z.N.: Nature 411(6833), 41 (2001)

    Article  ADS  Google Scholar 

  19. Holme, P., Saramäki, J.: Phys. Rep. 519(3), 97 (2012)

    Article  ADS  Google Scholar 

  20. Holme, P., Saramäki, J. (eds.): Temporal Networks. Springer-Verlag, Berlin (2013)

    Google Scholar 

  21. Holme, P.: Eur. Phys. J. B 88(9), 234 (2015)

    Article  ADS  Google Scholar 

  22. Liao, H., Mariani, M.S., Medo, M., Zhang, Y.C., Zhou, M.Y.: Phys. Rep. 689, 1 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  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. Kim, H., Tang, J., Anderson, R., Mascolo, C.: Comput. Netw. 56(3), 983 (2012)

    Article  Google Scholar 

  25. Williams, M.J., Musolesi, M.: R. Soc. Open Sci. 3(6) (2016)

    Article  ADS  MathSciNet  Google Scholar 

  26. Alsayed, A., Higham, D.J.: Chaos, Solitons Fractals 72, 35 (2015)

    Google Scholar 

  27. Fenu, C., Higham, D.J.: SIAM J. Matrix Anal. Appl. 38(2), 343 (2017)

    Article  MathSciNet  Google Scholar 

  28. Pan, R., Saramäki, J.: Phys. Rev. E 84(1), 016105 (2011)

    Article  ADS  Google Scholar 

  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. Grindrod, P., Higham, D.J.: Proc. R. Soc. A 470(2165), 20130835 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  31. Motegi, S., Masuda, N.: Sci. Rep. 2, 904 (2012)

    Article  ADS  Google Scholar 

  32. Grindrod, P., Parsons, M.C., Higham, D.J., Estrada, E.: Phys. Rev. E 83(4), 046120 (2011)

    Article  ADS  Google Scholar 

  33. Estrada, E: Phys. Rev. E 88(4), 042811 (2013)

    Google Scholar 

  34. Grindrod, P, Higham, D.J.: SIAM Rev. 55(1), 118 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chen, I., Benzi, M., Chang, H.H., Hertzberg, V.S.: J. Complex Networks 5(2), 274 (2016)

    Google Scholar 

  36. Arrigo, F., Higham, D.J.: Appl. Network Sci. 2(1), 17 (2017)

    Article  Google Scholar 

  37. Huang, D.W., Yu, Z.G.: Sci. Rep. 7, 41454 (2017)

    Article  ADS  Google Scholar 

  38. Takaguchi, T., Yano, Y., Yoshida, Y.: Eur. Phys. J. B 89(2), 1 (2016)

    Article  Google Scholar 

  39. Walker, D., Xie, H., Yan, K.K., Maslov, S.: J. Stat. Mech. 2007(06), P06010 (2007)

    Article  Google Scholar 

  40. Mariani, M.S., Medo, M., Zhang, Y.C.: arXiv preprint, arXiv: 1608.08414 (2016)

    Google Scholar 

  41. You, K., Tempo, R., Qiu, L.: IEEE Trans. Autom. Control 62(5), 2080 (2017)

    Article  Google Scholar 

  42. Rossi, R.A., Gleich, D.F.: Algorithms and Models for the Web Graph, pp. 126–137. Springer, Berlin (2012)

    Google Scholar 

  43. Mariani, M.S., Medo, M., Zhang, Y.C.: Sci. Rep. 5, 16181 (2015)

    Article  ADS  Google Scholar 

  44. Praprotnik, S., Batagelj, V.: Ars Mat. Contemp. 11(1), 11 (2015)

    Article  Google Scholar 

  45. Huang, Q., Zhao, C., Zhang, X., Wang, X., Yi, D.: Europhys. Lett. 118(3), 36001 (2017)

    Article  ADS  Google Scholar 

  46. Flores, J., Romance, M.: J. Comput. Appl. Math. 330, 1041 (2018)

    Article  MathSciNet  Google Scholar 

  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. Kostakos, V.: Physica A 388(6), 1007 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  49. Taylor, D., Myers, S.A., Clauset, A., Porter, M.A., Mucha, P.J.: Multiscale Model. Simul. 15(1), 537 (2017)

    Article  MathSciNet  Google Scholar 

  50. Taylor, D., Porter, M.A., Mucha, P.J.: arXiv preprint arXiv:1904.02059 (2019)

    Google Scholar 

  51. Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: J. Complex Networks 2(3), 203 (2014)

    Article  Google Scholar 

  52. Porter, M.A.: Not. Am. Math. Soc. 65(11), 1419 (2018)

    Google Scholar 

  53. Taylor, D., Caceres, R.S., Mucha, P.J.: Phys. Rev. X 7(3), 031056 (2017)

    Google Scholar 

  54. Mucha, P.J., Porter, M.A.: Chaos 20(4), 041108 (2010)

    Article  ADS  Google Scholar 

  55. Bassett, D.S., Porter, M.A., Wymbs, N.F., Grafton, S.T., Carlson, J.M., Mucha, P.J.: Chaos 23(1), 013142 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  56. Weir, W.H., Emmons, S., Gibson, R., Taylor, D., Mucha, P.J.: Algorithms 10(3), 93 (2017)

    Article  Google Scholar 

  57. Pamfil, A.R., Howison, S.D., Lambiotte, R., Porter, M.A.: SIAM J. Math. Data Sci. (in press) arXiv:1804.01964

    Google Scholar 

  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. 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. Battiston, F., Nicosia, V., Latora, V.: Phys. Rev. E 89(3), 032804 (2014)

    Article  ADS  Google Scholar 

  61. Tavassoli, S., Zweig, K.A.: In: 2016 Third European Network Intelligence Conference (ENIC), pp. 25–32. IEEE, Wrocław, Poland (2016)

    Google Scholar 

  62. Magnani, M., Micenkova, B., Rossi, L.: arXiv preprint arXiv:1303.4986 (2013)

    Google Scholar 

  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. Chakraborty, T., Narayanam, R.: In: 2016 IEEE 32nd International Conference on Data Engineering (ICDE), pp. 397–408. IEEE, Piscataway, NJ (2016)

    Google Scholar 

  65. Solé-Ribalta, A., De Domenico, M., Gómez, S., Arenas, A.: Physica D 323, 73 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  66. Spatocco, C., Stilo, G., Domeniconi, C.: arXiv preprint arXiv:1801.08026 (2018)

    Google Scholar 

  67. Rahmede, C., Iacovacci, J., Arenas, A., Bianconi, G.: J. Complex Networks 6(5), 733 (2017)

    Article  Google Scholar 

  68. Tudisco, F., Arrigo, F., Gautier, A.: J. SIAM Appl. Math. 78(2), 853 (2018)

    Article  MathSciNet  Google Scholar 

  69. Solá, L., Romance, M., Criado, R., Flores, J., del Amo, A.G., Boccaletti, S.: Chaos 23(3), 033131 (2013)

    Article  ADS  Google Scholar 

  70. DeFord, D.R., Pauls, S.D.: J. Complex Networks 6(3), 353 (2017)

    Article  Google Scholar 

  71. DeFord, D.R.: In: International Workshop on Complex Networks and their Applications, pp. 1111–1123. Springer, Berlin (2017)

    Google Scholar 

  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. Halu, A., Mondragón, R.J., Panzarasa, P., Bianconi, G.: PLoS ONE 8(10), e78293 (2013)

    Article  ADS  Google Scholar 

  74. Ding, C., Li, K.: Neurocomputing 312, 263 (2018)

    Article  Google Scholar 

  75. Gleich, D.F.: SIAM Rev. 57(3), 321 (2015)

    Article  MathSciNet  Google Scholar 

  76. Taylor, D.: Data release: Ph.D. exchange in the Mathematical Genealogy Project. Available at https://sites.google.com/site/danetaylorresearch/data

  77. Langville, A.N., Meyer, C.D.: Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, Princeton, NJ (2006)

    Book  MATH  Google Scholar 

  78. Masuda, N., Porter, M.A., Lambiotte, R.: Phys. Rep. 716–717, 1 (2017)

    Article  ADS  Google Scholar 

  79. Burris, V.: Am. Sociol. Rev. 69(2), 239 (2004)

    Article  Google Scholar 

  80. Myers, S.A., Mucha, P.J., Porter, M.A.: Chaos 21(4), 041104 (2011)

    Article  ADS  Google Scholar 

  81. Clauset, A., Arbesman, S., Larremore, D.B.: Sci. Adv. 1(1), e1400005 (2015)

    Article  ADS  Google Scholar 

  82. The Mathematics Genealogy Project. Available at http://www.genealogy.ams.org; data provided 19 October 2009

  83. Ahmad, W., Porter, M.A., Beguerisse-Díaz, M.: arXiv preprint arXiv:1805.00193 (2018)

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. University at Buffalo, State University of New York, Buffalo, NY, USA

    Dane Taylor

  2. University of California, Los Angeles, CA, USA

    Mason A. Porter

  3. University of North Carolina, Chapel Hill, NC, USA

    Peter J. Mucha

Authors
  1. Dane Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mason A. Porter
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter J. Mucha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dane Taylor .

Editor information

Editors and Affiliations

  1. Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Tokyo, Japan

    Petter Holme

  2. Department of Computer Science, Aalto University, Espoo, Finland

    Jari Saramäki

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Taylor, D., Porter, M.A., Mucha, P.J. (2019). Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling. In: Holme, P., Saramäki, J. (eds) Temporal Network Theory. Computational Social Sciences. Springer, Cham. https://doi.org/10.1007/978-3-030-23495-9_17

Download citation

Publish with us

Policies and ethics

Search

Navigation