Journal of Statistical Physics

, Volume 168, Issue 6, pp 1180–1190 | Cite as

Information Transmission and Criticality in the Contact Process



In the present paper, we study the relation between criticality and information transmission in the one-dimensional contact process with infection parameter \(\lambda .\) We introduce a notion of sensitivity of the process to its initial condition and prove that it increases not only for values of \(\lambda < \lambda _c, \) the value of the critical parameter, but keeps increasing even after \( \lambda _c , \) before finally starting to decrease for values of \(\lambda \) sufficiently above \(\lambda _c.\) This provides a counterexample to the common belief that associates maximal information transmission to criticality.


Contact process Criticality Information transmission Duality and coupling 

Mathematics Subject Classification

60K35 82B27 


  1. 1.
    Bak, P.P.: How Nature Works. Springer, New York (1996)CrossRefMATHGoogle Scholar
  2. 2.
    Barnett, L.L., Lizier, J.T.J.T., Harré, M.M., Seth, A.K.A.K., Bossomaier, T.T.: Information flow in a kinetic Ising model peaks in the disordered phase. Phys. Rev. Lett. 111, 177203 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    Beggs, J.M.J.M., Plenz, D.D.: Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167–11177 (2003)Google Scholar
  4. 4.
    Bertein, F.F., Galves, A.A.: Une classe de systèmes de particules stable par association. Z. Wahr. Verw. Gebiete 41, 73–85 (1977)CrossRefMATHGoogle Scholar
  5. 5.
    Bezuidenhout, C.C., Grimmett, G.R.G.R.: The critical contact process dies out. Ann. Probab. 18, 1462–1482 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cavagna, A.A., Cimarelli, A.A., Giardina, I.I., Parisi, G.G., Santagati, R.R., Stefanini, F.F., Viale, M.M.: Scale-free correlations in starling flocks. Proc. Natl. Acad. Sci. USA 107, 11865–11870 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Erten, E.Y., Lizier, J.T., Piraveenan, M., Prokopenko, M.: Criticality and Information dynamics in epidemiological models. Entropy 19(5), 194 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    Galves, A.A., Garcia, N.L.N.L., Prieur, C.C.: Perfect simulation of a coupling achieving the \({\bar{d}}\)-distance between ordered pairs of binary chains of infinite order. J. Stat. Phys. 141, 669–682 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Harris, T.E.T.E.: Contact interactions on a lattice. Ann. Probab. 2, 969–988 (1974)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Harris, T.E.T.E.: On a class of set-valued Markov processes. Ann. Probab. 4, 175–194 (1976)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Harris, T.E.T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6, 355–378 (1978)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kinouchi, O.O., Copelli, M.M.: Optimal dynamical range of excitable networks at criticality. Nat. Phys. 2, 348–351 (2006)CrossRefGoogle Scholar
  13. 13.
    Liggett, T.M.T.M.: Interacting Particle Systems. Springer, Berlin (1985)CrossRefMATHGoogle Scholar
  14. 14.
    Schreiber, T.T.: Measuring information transfer. Phys. Rev. Lett. 85, 461–464 (2000)ADSCrossRefGoogle Scholar
  15. 15.
    Vanni, F.F., Luković, M.M., Grigolini, P.P.: Criticality and transmission of information in a swarm of cooperative units. Phys. Rev. Lett. 107, 078103 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Vijayaraghavan, V.S., James, R.G., Crutchfield, J.P.: Anatomy of a spin: the information-theoretic structure of classical spin systems. Entropy 19(5), 214 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Gran Sasso Science InstituteL’AquilaItaly
  2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil
  3. 3.AGM, CNRS-UMR 8088Université de Cergy-PontoiseCergy-PontoiseFrance

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