Skip to main content

Continuous Phase Transitions in Supercritical Explosive Percolation

  • Chapter
  • First Online:
Explosive Percolation in Random Networks

Part of the book series: Springer Theses ((Springer Theses))

  • 669 Accesses

Abstract

Percolation in networks, a phase transition from small, scattered components to large-scale connectivity, is heavily studied and widely applied in technological and social systems [14], biological networks [5, 6], epidemiology [710] and even dynamical models of economic systems [11, 12].

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Strogatz, S.H.: Exploring complex networks. Nature 410, 268276 (2001)

    Article  Google Scholar 

  2. Newman, M.E.J., Watts, D.J., Strogatz, S.H.: Random graph models of social networks. Proc. Natl. Acad. Sci. 99, 2566–2572 (2002)

    Article  MATH  Google Scholar 

  3. Song, C., Havlin, S., Makse, H.A.: Origins of fractality in the growth of complex networks. Nat. Phys. 2, 275–281 (2006)

    Article  Google Scholar 

  4. Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E., Havlin, S.: Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010)

    Article  Google Scholar 

  5. Kim, J., Krapivsky, P.L., Kahng, B., Redner, S.: Infinite-order percolation and giant fluctuations in a protein interaction network. Phys. Rev. E 66, 055101 (2002)

    Article  Google Scholar 

  6. Rozenfeld, H.D., Gallos, L.K., Makse, H.A.: Explosive percolation in the human protein homology network. Eur. Phys. J. B 75, 305–310 (2010)

    Article  MATH  Google Scholar 

  7. Moore, C., Newman, M.E.J.: Epidemics and percolation in small-world networks. Phys. Rev. E 61, 5678–5682 (2000)

    Article  Google Scholar 

  8. Serrano, M.A., Bogun̋á, M.: Percolation and epidemic thresholds in clustered networks. Phys. Rev. Lett. 97, 088701 (2006)

    Google Scholar 

  9. Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008)

    Article  Google Scholar 

  10. Parshani, R., Carmi, S., Havlin, S.: Epidemic threshold for the susceptible-infectious-susceptible model on random networks. Phys. Rev. Lett. 104, 258701 (2010)

    Article  Google Scholar 

  11. Ausloos, M., Lambiotte, R.: Clusters or networks of economies? A macroeconomy study through gross domestic product. Phys. A 382, 16–21 (2007)

    Article  Google Scholar 

  12. Roca, C. P., Draief, M., Helbing, D.: Percolate or die: Multi-percolation decides the struggle between competing innovations. arXiv:1101.0775v1

  13. Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17 (1960)

    MATH  Google Scholar 

  14. Achlioptas, D.D., ’Souza, R.M., Spencer. J.: Explosive percolation in random networks. Science 323, 1453–1455 (2009)

    Google Scholar 

  15. Cho, Y.S., Kim, J.S., Park, J., Kahng, B., Kim, D.: Percolation transitions in scale-free networks under the Achlioptas process. Phys. Rev. Lett. 103, 135702 (2009)

    Article  Google Scholar 

  16. Radicchi, F., Fortunato, S.: Explosive percolation in scale-free networks. Phys. Rev. Lett. 103, 168701 (2009)

    Article  Google Scholar 

  17. Radicchi, F., Fortunato, S.: Explosive percolation: a numerical analysis. Phys. Rev. E 81, 036110 (2010)

    Article  Google Scholar 

  18. Ziff, R.M.: Explosive growth in biased dynamic percolation on two-dimensional regular lattice networks. Phys. Rev. Lett. 103, 045701 (2009)

    Article  Google Scholar 

  19. Ziff, R.M.: Scaling behavior of explosive percolation on the square lattice. Phys. Rev. E 82, 051105 (2010)

    Article  Google Scholar 

  20. da Costa, R.A., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Explosive percolation transition is actually continuous. Phys. Rev. Lett. 105, 255701 (2010)

    Article  Google Scholar 

  21. Nagler, J., Levina, A., Timme, M.: Impact of single links in competitive percolation. Nat. Phys. 7, 265–270 (2011)

    Article  Google Scholar 

  22. Riordan, O., Warnke, L.: Explosive percolation is continuous. Science 333, 322–324 (2011)

    Article  Google Scholar 

  23. Grassberger, P., Christensen, C., Bizhani, G., Son, S.-W., Paczuski, M.: Explosive percolation is continuous, but with unusual finite size behavior. Phys. Rev. Lett. 106, 225701 (2011)

    Article  Google Scholar 

  24. Lee, H.K., Kim, B.J., Park, H.: Continuity of the explosive percolation transition. Phys. Rev. E 84, 020101(R) (2011)

    Google Scholar 

  25. Cho, Y.S., Kahng, B., Kim, D.: Cluster aggregation model for discontinuous percolation transitions. Phys. Rev. E 81, 030103(R) (2010)

    Article  Google Scholar 

  26. Araújo, N.A.M., Herrmann, H.J.: Explosive percolation via control of the largest cluster. Phys. Rev. Lett. 105, 035701 (2010)

    Article  Google Scholar 

  27. Moreira, A.A., Oliveira, E.A., Reis, S.D.S., Herrmann, H.J., Andrade, J.S.: Hamiltonian approach for explosive percolation. Phys. Rev. E 81, 040101(R) (2010)

    Article  Google Scholar 

  28. Schrenk, K.J., Araújo, N.A.M., Herrmann, H.J.: Gaussian model of explosive percolation in three and higher dimensions. Phys. Rev. E 84, 041136 (2011)

    Article  Google Scholar 

  29. Choi, W., Yook, S.-H., Kim, Y.: Explosive site percolation with a product rule. Phys. Rev. E 84, 020102(R) (2011)

    Article  Google Scholar 

  30. Cho, Y.S., Kahng, B.: Discontinuous percolation transitions in real physical systems. Phys. Rev. E 84, 050102(R) (2011)

    Article  Google Scholar 

  31. Cho, Y. S., Kim, Y. W., Kahng, B.: Discontinuous percolation in diffusion-limited cluster aggregation. J. Stat. Mech. P10004 (2012)

    Google Scholar 

  32. Panagiotou, K., Spöhel, R., Steger, A., Thomas, H.: Explosive percolation in Erdös-Rényi-like random graph processes. Electron. Notes Discrete Math. 38, 699–704 (2011)

    Article  Google Scholar 

  33. Boettcher, S., Singh, V., Ziff, R.M.: Ordinary percolation with discontinuous transitions. Nat. Commun. 3, 787 (2012)

    Article  Google Scholar 

  34. Bizhani, G., Paczuski, M., Grassberger, P.: Discontinuous percolation transitions in epidemic processes, surface depinning in random media, and Hamiltonian random graphs. Phys. Rev. E 86, 011128 (2012)

    Article  Google Scholar 

  35. Cao, L., Schwarz, J.M.: Correlated percolation and tricriticality. Phys. Rev. E 86, 061131 (2012)

    Article  Google Scholar 

  36. Cho, Y.S., Kahng, B.: Suppression effect on explosive percolation. Phys. Rev. Lett. 107, 275703 (2011)

    Article  Google Scholar 

  37. Schrenk, K.J., Felder, A., Deflorin, S., Araujo, N.A.M., D’Souza, R.M., Herrmann, H.J.: Bohman-Frieze-Wormald model on the lattice, yielding a discontinuous percolation transition. Phys. Rev. E 85, 031103 (2012)

    Article  Google Scholar 

  38. Bohman, T., Frieze, A., Wormald, N.C.: Avoidance of a giant component in half the edge set of a random graph. Random Struct. Algorithms 25, 432–449 (2004)

    Article  MathSciNet  Google Scholar 

  39. Chen, W., D’Souza, R.M.: Explosive percolation with multiple giant components. Phys. Rev. Lett. 106, 115701 (2011)

    Article  Google Scholar 

  40. Chen, W., Zheng, Z., D’Souza, R.M.: Deriving an underlying mechanism for discontinuous percolation. Europhys. Lett. 100, 66006 (2012)

    Article  Google Scholar 

  41. Bengtsson, M., Kock, S.: Cooperation and competition in relationships between competitors in business networks. J. Bus. Ind. Mark. 14, 178–194 (1999)

    Article  Google Scholar 

  42. Frank, S.A.: Repression of competition and the evolution of cooperation. Evol. Int. J. Org. Evol. 57, 693–705 (2003)

    Google Scholar 

  43. Hirshleifer, J.: Competition, cooperation, and conflict in economics and biology. Am. Econ. Rev. 68 (1978)

    Google Scholar 

  44. Spencer, J.: The giant component: the golden anniversary. Not. AMS 57, 720–724 (2010)

    MATH  Google Scholar 

  45. Ben-Naim, E., Krapivsky, P.L.: Percolation with multiple giant clusters. J. Phys. A 38, L417–L423 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  46. Asztalos, A., Toroczkai, Z.: Network discovery by generalized random walks. Europhys. Lett. 92, 50008 (2010)

    Article  Google Scholar 

  47. Anderson, R.M., May, R.M.: Infectious Diseases in Humans. Oxford University Press, Oxford (1992)

    Google Scholar 

  48. Manna, S.S., Chatterjee, A.: A new route to explosive percolation. Phys. A 390, 177–182 (2011)

    Article  Google Scholar 

  49. Riordan, O., Warnke, L.: Achlioptas processes are not always self-averaging. Phys. Rev. E 86, 011129 (2012)

    Article  Google Scholar 

  50. Nagler, J., Tiessen, T, Gutch, H.W.: Continuous percolation with discontinuities. Phys. Rev. X 2, 031009 (2012)

    Google Scholar 

  51. Landau, D.P., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  52. Bastas, N., Kosmidis, K., Argyrakis, P.: Explosive site percolation and finite-size hysteresis. Phys. Rev. E 84, 066112 (2011)

    Article  Google Scholar 

  53. Zhang, Y., Wei, W., Guo, B., Zhang, R., Zheng, Z.: Formation mechanism and size features of multiple giant clusters in generic percolation processes. Phys. Rev. E 86, 051103 (2012)

    Article  Google Scholar 

  54. Zhang, R., Wei, W., Guo, B., Zhang, Y., Zheng, Z.: Analysis on the evolution process of BFW-like model with discontinuous percolation of multiple giant components. Phys. A 392, 1232–1245 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei Chen .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Chen, W. (2014). Continuous Phase Transitions in Supercritical Explosive Percolation. In: Explosive Percolation in Random Networks. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43739-1_4

Download citation

Publish with us

Policies and ethics