Targeting Influential Nodes for Recovery in Bootstrap Percolation on Hyperbolic Networks

  • Christine MarshallEmail author
  • Colm O’Riordan
  • James Cruickshank
Part of the Lecture Notes in Social Networks book series (LNSN)


The influence of our peers is a powerful reinforcement for our social behaviour, evidenced in voter behaviour and trend adoption. Bootstrap percolation is a simple method for modelling this process. In this work we look at bootstrap percolation on hyperbolic random geometric graphs, which have been used to model the Internet graph, and introduce a form of bootstrap percolation with recovery, showing that random targeting of nodes for recovery will delay adoption, but this effect is enhanced when nodes of high degree are selectively targeted.


Bootstrap percolation Bootstrap percolation with recovery Hyperbolic random geometric graphs 


  1. 1.
    Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance of complex networks. Nature 406(6794), 378–382 (2000)CrossRefGoogle Scholar
  2. 2.
    Amini, H., Fountoulakis, N.: Bootstrap percolation in power-law random graphs. J. Stat. Phys. 155(1), 72–92 (2014)CrossRefGoogle Scholar
  3. 3.
    Balister, P., Bollobàs, B., Johnson, J.R., Walters, M.: Random majority percolation. Random Struct. Algoritm. 36(3), 315–340 (2010)Google Scholar
  4. 4.
    Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Struct. Algoritm. 30(12), 257–286 (2007)CrossRefGoogle Scholar
  5. 5.
    Barabási, A.L.: Network Science. Cambridge University Press, Cambridge (2016)Google Scholar
  6. 6.
    Baxter, G.J., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.: Bootstrap percolation on complex networks. Phys. Rev. E 82(1), 011103 (2010)CrossRefGoogle Scholar
  7. 7.
    Bénézit, F., Dimakis, A.G., Thiran, P., Vetterli, M.: Order-optimal consensus through randomized path averaging. IEEE Trans. Inf. Theory 56(10), 5150–5167 (2010)CrossRefGoogle Scholar
  8. 8.
    Bringmann, K., Keusch, R., Lengler, J.: Geometric inhomogeneous random graphs. Preprint (2015). arXiv:1511.00576Google Scholar
  9. 9.
    Bullmore, E., Bassett, D.: Brain graphs: graphical models of the human brain connectome. Annu. Rev. Clin. Psychol. 7, 113–140 (2011)CrossRefGoogle Scholar
  10. 10.
    Candellero, E., Fountoulakis, N.: Clustering and the hyperbolic geometry of complex networks. In: Bonato, A., Graham, F., Pralat, P. (eds.) Algorithms and Models for the Web Graph. WAW 2014. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8882, pp. 1–12. Springer, Cham (2014)Google Scholar
  11. 11.
    Candellero, E., Fountoulakis, N.: Bootstrap percolation and the geometry of complex networks. Stoch. Process. Appl. 126, 234–264 (2015)CrossRefGoogle Scholar
  12. 12.
    Centola, D.: The spread of behavior in an online social network experiment. Science 329(5996), 1194–1197 (2010)CrossRefGoogle Scholar
  13. 13.
    Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a bethe lattice. J. Phys. C Solid State Phys. 12(1), L31 (1979)CrossRefGoogle Scholar
  14. 14.
    Coker, T., Gunderson, K.: A sharp threshold for a modified bootstrap percolation with recovery. J. Stat. Phys. 157(3), 531–570 (2014)CrossRefGoogle Scholar
  15. 15.
    Domingos, P., Richardson, M.: Mining the network value of customers. In: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’01, pp. 57–66. ACM, New York (2001)Google Scholar
  16. 16.
    Gleeson, J.P.: Cascades on correlated and modular random networks. Phys. Rev. E 77(4), 046117 (2008)CrossRefGoogle Scholar
  17. 17.
    Gomez Rodriguez, M., Leskovec, J., Krause, A.: Inferring networks of diffusion and influence. In: Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1019–1028. ACM, New York (2010)Google Scholar
  18. 18.
    Jackson, M.O., López-Pintado, D.: Diffusion and contagion in networks with heterogeneous agents and homophily. Netw. Sci. 1(01), 49–67 (2013)CrossRefGoogle Scholar
  19. 19.
    Janson, S., Łuczak, T., Turova, T., Vallier, T.: Bootstrap percolation on the random graph g n,p. Ann. Appl. Probab. 22(5), 1989–2047 (2012)CrossRefGoogle Scholar
  20. 20.
    Kempe, D., Kleinberg, J.M., Tardos, É.: Influential nodes in a diffusion model for social networks. In: ICALP, vol. 5, pp. 1127–1138. Springer, Berlin (2005)CrossRefGoogle Scholar
  21. 21.
    Kempe, D., Kleinberg, J.M., Tardos, É.: Maximizing the spread of influence through a social network. Theory Comput. 11(4), 105–147 (2015)CrossRefGoogle Scholar
  22. 22.
    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)CrossRefGoogle Scholar
  23. 23.
    Leskovec, J., Backstrom, L., Kleinberg, J.: Meme-tracking and the dynamics of the news cycle. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 497–506. ACM, New York (2009)Google Scholar
  24. 24.
    Liben-Nowell, D., Kleinberg, J.: Tracing information flow on a global scale using internet chain-letter data. Proc. Natl. Acad. Sci. 105(12), 4633–4638 (2008)CrossRefGoogle Scholar
  25. 25.
    Myers, S.A., Zhu, C., Leskovec, J.: Information diffusion and external influence in networks. In: Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 33–41. ACM, New York (2012)Google Scholar
  26. 26.
    Papadopoulos, F., Psomas, C., Krioukov, D.: Network mapping by replaying hyperbolic growth. IEEE/ACM Trans. Networking 23(1), 198–211 (2015)CrossRefGoogle Scholar
  27. 27.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)CrossRefGoogle Scholar
  28. 28.
    Pržulj, N.: Biological network comparison using graphlet degree distribution. Bioinformatics 23(2), e177–e183 (2007)CrossRefGoogle Scholar
  29. 29.
    Rocchini, C.: Order-3 heptakis heptagonal tiling. (2007). Accessed 15 May 2017
  30. 30.
    Sahini, M., Sahimi, M.: Applications of Percolation Theory. CRC Press, Boca Raton (1994)Google Scholar
  31. 31.
    Shrestha, M., Moore, C.: Message-passing approach for threshold models of behavior in networks. Phys. Rev. E 89(2), 022805 (2014)CrossRefGoogle Scholar
  32. 32.
    Tassier, T.: Simple epidemics and SIS models. In: The Economics of Epidemiology, pp. 9–16. Springer, Berlin (2013)Google Scholar
  33. 33.
    von Looz, M., Staudt, C.L., Meyerhenke, H., Prutkin, R.: Fast generation of dynamic complex networks with underlying hyperbolic geometry. Preprint (2015). arXiv:1501.03545Google Scholar
  34. 34.
    Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. 99(9), 5766–5771 (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Christine Marshall
    • 1
    Email author
  • Colm O’Riordan
    • 1
  • James Cruickshank
    • 2
  1. 1.Discipline of Information TechnologyNational University of IrelandGalwayIreland
  2. 2.School of MathematicsNational University of IrelandGalwayIreland

Personalised recommendations