Skip to main content
SpringerLink
Log in

A Central Limit Theorem for Diffusion in Sparse Random Graphs

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every vertex has two states: it is either active or inactive. We assume that to each vertex is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of vertices with threshold zero which consists of initially activated vertices, whereas every other vertex is inactive. Subsequently, in each round, if an inactive vertex with threshold \(\theta \) has at least \(\theta \) of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more vertices become activated. The main result of this paper provides a central limit theorem for the final size of activated vertices. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated vertices has asymptotically Gaussian fluctuations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Adler, J., Lev, U.: Bootstrap percolation: visualizations and applications. Braz. J. Phys. 33, 641–644 (2003)

    Article  ADS  Google Scholar 

  2. Amini, H.: Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electron. J. Comb. 17, R25 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Amini, H.: Bootstrap percolation in living neural networks. J. Stat. Phys. 141(3), 459–475 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Amini, H., Cont, R., Minca, A.: Resilience to contagion in financial networks. Math. Financ. 26(2), 329–365 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Amini, H., Fountoulakis, N.: Bootstrap percolation in power-law random graphs. J. Stat. Phys. 155(1), 72–92 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Amini, H., Fountoulakis, N., Panagiotou, K.: Bootstrap percolation in inhomogeneous random graphs. arXiv preprint arXiv:1402.2815, (2014)

  7. Amini, H., Minca, A.: Epidemic spreading and equilibrium social distancing in heterogeneous networks. Dyn. Games Appl. 12(1), 258–287 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Amini, H., Minca, A., Sulem, A.: A dynamic contagion risk model with recovery features. Math. Oper. Res. 47(2), 1412–1442 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  9. Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30(1–2), 257–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bayraktar, E., Chakraborty, S., Zhang, X.: K-core in percolated dense graph sequences. arXiv preprint arXiv:2012.09730, (2020)

  11. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (2013)

    MATH  Google Scholar 

  12. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424(4–5), 175–308 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a Bethe lattice. J. Phys. C 12(1), L31 (1979)

    Article  ADS  Google Scholar 

  14. Cooper, C.: The cores of random hypergraphs with a given degree sequence. Random Struct. Algorithms 25(4), 353–375 (2004)

    Article  MathSciNet  Google Scholar 

  15. Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. Rev. Mod. Phys. 80(4), 1275 (2008)

    Article  ADS  Google Scholar 

  16. Durrett, R.: Random Graph Dynamics. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  17. Fernholz, D., Ramachandran, V.: Cores and Connectivity in Sparse Random Graphs. The University of Texas at Austin, technical report TR-04-13 (2004)

  18. Fountoulakis, N., Kang, M., Koch, C., Makai, T., et al.: A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs. Ann. Appl. Probab. 28(2), 990–1051 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, vol. 288. Springer, New York (2013)

    MATH  Google Scholar 

  20. Janson, S., Luczak, M.J.: A simple solution to the k-core problem. Random Struct. Algorithms 30(1–2), 50–62 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Janson, S., Luczak, M.J.: Asymptotic normality of the k-core in random graphs. Ann. Appl. Probab. 18(3), 1085–1137 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)

    MATH  Google Scholar 

  24. Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)

  25. Kleinberg, J.: Cascading behavior in networks: algorithmic and economic issues. Algorithmic Game Theory 24, 613–632 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lelarge, M.: Diffusion and cascading behavior in random networks. Games Econ. Behav. 75(2), 752–775 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Leskovec, J., Adamic, L.A., Huberman, B.A.: The dynamics of viral marketing. ACM Trans. Web 1(1), 5 (2007)

    Article  Google Scholar 

  28. Liu, Y.-Y., Barabási, A.-L.: Control principles of complex systems. Rev. Mod. Phys. 88(3), 035006 (2016)

    Article  ADS  Google Scholar 

  29. Liu, Y.-Y., Csóka, E., Zhou, H., Pósfai, M.: Core percolation on complex networks. Phys. Rev. Lett. 109(20), 205703 (2012)

    Article  ADS  Google Scholar 

  30. Łuczak, T.: Size and connectivity of the k-core of a random graph. Discret. Math. 91(1), 61–68 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Molloy, M.: Cores in random hypergraphs and boolean formulas. Random Struct. Algorithms 27(1), 124–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6(2–3), 161–179 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Morris, S.: Contagion. Rev. Econ. Stud. 67(1), 57–78 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Newman, M., Barabási, A.-L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  35. Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E 66(1), 016128 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  36. Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  37. Pittel, B., Spencer, J., Wormald, N.: Sudden emergence of a giantk-core in a random graph. J. Comb. Theory Ser. B 67(1), 111–151 (1996)

    Article  MATH  Google Scholar 

  38. Riordan, O.: The k-core and branching processes. Comb. Probab. Comput. 17(1), 111–136 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tlusty, T., Eckmann, J.-P.: Remarks on bootstrap percolation in metric networks. J. Phys. A 42(20), 205004 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. van der Hofstad, R.: Random Graphs and Complex Networks, vol. 43. Cambridge university Press, Cambridge (2016)

    Book  Google Scholar 

  41. Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. 99(9), 5766–5771 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referee for a very detailed report which significantly improved the quality of this article.

Funding

Erhan Bayraktar is partially supported by the National Science Foundation under Grant DMS- 2106556 and by the Susan M. Smith chair. Suman Chakraborty is partially supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-Grant NETWORKS- 024.002.003.

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611, USA

    Hamed Amini

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Erhan Bayraktar

  3. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Suman Chakraborty

Authors
  1. Hamed Amini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erhan Bayraktar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Suman Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hamed Amini.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Communicated by Eric A. Carlen.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amini, H., Bayraktar, E. & Chakraborty, S. A Central Limit Theorem for Diffusion in Sparse Random Graphs. J Stat Phys 190, 57 (2023). https://doi.org/10.1007/s10955-023-03068-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-023-03068-9

Keywords

Search

Navigation