Skip to main content
SpringerLink
Log in

Dynamical Response of Networks Under External Perturbations: Exact Results

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

Abstract

We give exact statistical distributions for the dynamic response of influence networks subjected to external perturbations. We consider networks whose nodes have two internal states labeled 0 and 1. We let \(N_0\) nodes be frozen in state 0, \(N_1\) in state 1, and the remaining nodes change by adopting the state of a connected node with a fixed probability per time step. The frozen nodes can be interpreted as external perturbations to the subnetwork of free nodes. Analytically extending \(N_0\) and \(N_1\) to be smaller than 1 enables modeling the case of weak coupling. We solve the dynamical equations exactly for fully connected networks, obtaining the equilibrium distribution, transition probabilities between any two states and the characteristic time to equilibration. Our exact results are excellent approximations for other topologies, including random, regular lattice, scale-free and small world networks, when the numbers of fixed nodes are adjusted to take account of the effect of topology on coupling to the environment. This model can describe a variety of complex systems, from magnetic spins to social networks to population genetics, and was recently applied as a framework for early warning signals for real-world self-organized economic market crises.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  3. Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)

  5. Sporns, O.: Networks of the Brain. MIT Press, Cambridge (2011)

  6. Gross, T., Sayama, H. (eds.): Adaptive Networks. Theory, models and applications (Understanding Complex Systems). Springer, Berlin (2009)

  7. Bar-Yam, Y., Epstein, I.: Response of complex networks to stimuli. PNAS 101, 4341–4345 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Albert, R., Jeong, H., Barabasi, A.-L.: Error and attack tolerance of complex networks. Nat. Lond. 406, 378–482 (2000)

    Article  ADS  Google Scholar 

  9. Cohen, R., Erez, K., ben-Avraham, D., Havlin, S.: Resilience of the internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000)

    Article  ADS  Google Scholar 

  10. 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  ADS  Google Scholar 

  11. Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks, vol. 1. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  12. Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001)

    Article  ADS  Google Scholar 

  13. Barahona, M., Pecora, L.M.: Synchronization in small-world systems. Phys. Rev. Lett. 89, 054101 (4 pp.) (2002)

  14. Nishikawa, T., Motter, A.E., Lai, Y.-C., Hoppensteadt, F.C.: Heterogeneity in oscillator networks: are smaller worlds easier to synchronize?. Phys. Rev. Lett. 91, 014101 (4 pp.) (2003)

  15. Moreno, Y., Nekovee, M., Pacheco, A.F.: Dynamics of rumor spreading in complex networks. Phys. Rev. E 69, 066130 (7 pp.) (2004)

  16. Laguna, M.F., Abramson, G., Zanette, D.H.: Vector opinion dynamics in a model for social influence. Physica A 329, 459–472 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Guimera et al. R.: Optimal network topologies for local search with congestion. Phys. Rev. Lett. 89, 248701 (4 pp.) (2002)

  18. Braha, D., BarGYam, Y.: From centrality to temporary fame: dynamic centrality in complex networks. Complexity 12(2), 1–5 (2006)

    Article  Google Scholar 

  19. Hill, S.A., Braha, D.: Dynamic model of time-dependent complex networks. Phys. Rev. E 82(4), 046105 (7 pp.) (2010)

  20. Wang, X., Lai, Y.-C., Lai, C.H.: Oscillations of complex networks. Phys. Rev. E 74, 066104 (5 pp.) (2006)

  21. Lee, E.J., Goh, K.-I., Kahng, B., Kim, D.: Robustness of the avalanche dynamics in data-packet transport on scale-free networks. Phys. Rev. E 71, 056108 (4 pp.) (2005)

  22. Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985)

  23. Mobilia, M.: Does a single zealot affect an infinite group of voters?. Phys. Rev. Lett. 91, 028701 (4 pp.) (2003)

  24. Mobilia, M., Petersen, A., Redner, S.: On the role of zealotry in the voter model. J. Stat. Mech. P08029 (2007). doi:10.1088/1742-5468/2007/08/P08029

  25. Yildiz, E., Ozdaglar, A., Acemoglu, D., Saberi, A., Scaglione, A.: Binary opinion dynamics with stubborn agents. ACM Trans. Econ. Comp. 1(4), 19 (19 pp.) (2013). doi:10.1145/2538508

  26. Harmon, D., de Aguiar, M.A.M., Chinellato, D., Braha, D., Epstein, I.R., Bar-Yam, Y.: Predicting economic market crises using measures of collective panic. Eprint arXiv:1102.2620 [q-fin.ST] (2011)

  27. Chinellato, D.D., de Aguiar, M.A.M., Epstein, I.R., Braha, D., Bar-Yam, Y.: Dynamical response of networks under external perturbations: exact results. E-print arXiv:0705.4607v2 [nlin.SI] (2007)

  28. Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294–307 (1963)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Vilone, D., Castellano, C.: Solution of voter model dynamics on annealed small-world networks. Phys. Rev. E 69, 016109 (8 pp.) (2004)

  30. Sood, V., Redner, S.: Voter model on heterogeneous graphs. Phys. Rev. Lett. 94, 178701 (4 pp.) (2005)

  31. Moran, P.A.P.: Random processes in genetics. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 54, pp. 60–72. Cambridge University Press, Cambridge (1958)

  32. de Aguiar, M.A.M., Bar-Yam, Y.: Moran model as a dynamical process on networks and its implications for neutral speciation. Phys. Rev. E 84, 031901 (10 pp.) (2011)

  33. de Aguiar, M.A.M., Epstein, I.R., Bar-Yam, Y.: Analytically solvable model of probabilistic network dynamics. Phys. Rev. E 72, 067102 (4 pp.) (2005)

  34. Boccara, N.: Models of opinion formation: influence of opinion leaders. Eprint arXiv:0704.1790 [nlin.AO] (2007)

  35. Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’. Netw. Nat. 393, 440–442 (1998)

    Article  Google Scholar 

Download references

Acknowledgments

M.A.M.A. and D.D.C. acknowledge financial support from CNPq and FAPESP. I.R.E. was supported by National Science Foundation Grant CHE-1362477.

Author information

Authors and Affiliations

  1. Instituto de Física ‘Gleb Wataghin’, Universidade Estadual de Campinas, Unicamp, Campinas, SP, 13083-970, Brazil

    David D. Chinellato & Marcus A. M. de Aguiar

  2. New England Complex Systems Institute, Cambridge, MA, 02138, USA

    Irving R. Epstein, Dan Braha, Yaneer Bar-Yam & Marcus A. M. de Aguiar

  3. Department of Chemistry, MS015, Brandeis University, Waltham, MA, 02454, USA

    Irving R. Epstein

  4. University of Massachusetts, Dartmouth, MA, 02747, USA

    Dan Braha

Authors
  1. David D. Chinellato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irving R. Epstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dan Braha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yaneer Bar-Yam
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Marcus A. M. de Aguiar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcus A. M. de Aguiar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chinellato, D.D., Epstein, I.R., Braha, D. et al. Dynamical Response of Networks Under External Perturbations: Exact Results. J Stat Phys 159, 221–230 (2015). https://doi.org/10.1007/s10955-015-1189-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-015-1189-x

Keywords

Search

Navigation