The Dynamics and Stability of Probabilistic Population Processes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10616)

Abstract

We study here the dynamics and stability of Probabilistic Population Processes, via the differential equations approach. We provide a quite general model following the work of Kurtz [15] for approximating discrete processes with continuous differential equations. We show that it includes the model of Angluin et al. [1], in the case of very large populations. We require that the long-term behavior of the family of increasingly large discrete processes is a good approximation to the long-term behavior of the continuous process, i.e., we exclude population protocols that are extremely unstable such as parity-dependent decision processes. For the general model, we give a sufficient condition for stability that can be checked in polynomial time. We also study two interesting sub cases: (a) Protocols whose specifications (in our terms) are configuration independent. We show that they are always stable and that their eventual subpopulation percentages are actually a Markov Chain stationary distribution. (b) Protocols that have dynamics resembling virus spread. We show that their dynamics are actually similar to the well-known Replicator Dynamics of Evolutionary Games. We also provide a sufficient condition for stability in this case.

References

  1. 1.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. In: 23rd Annual ACM Symposium on Principles of Distributed Computing (PODC), New York, NY, USA, pp. 290–299 (2004)Google Scholar
  2. 2.
    Angluin, D., Aspnes, J., Eisenstat, D.: Fast computation by population protocols with a leader. In: Dolev, S. (ed.) DISC 2006. LNCS, vol. 4167, pp. 61–75. Springer, Heidelberg (2006). doi:10.1007/11864219_5 CrossRefGoogle Scholar
  3. 3.
    Belleville, A., Doty, D., Soloveichik, D.: Hardness of computing and approximating predicates and functions with leaderless population protocols. In: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), Leibniz International Proceedings in Informatics (LIPIcs), vol. 80, pp. 141:1–141:14, Dagstuhl, Germany (2017). Schloss Dagstuhl-Leibniz-Zentrum fuer InformatikGoogle Scholar
  4. 4.
    Chatzigiannakis, I., Michail, O., Nikolaou, S., Pavlogiannis, A., Spirakis, P.G.: Passively mobile communicating machines that use restricted space. Theor. Comput. Sci. 412(46), 6469–6483 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chatzigiannakis, I., Mylonas, G., Vitaletti, A.: Urban pervasive applications: challenges, scenarios and case studies. Comput. Sci. Rev. 5(1), 103–118 (2011)CrossRefGoogle Scholar
  6. 6.
    Chatzigiannakis, I., Spirakis, P.G.: The dynamics of probabilistic population protocols. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 498–499. Springer, Heidelberg (2008). doi:10.1007/978-3-540-87779-0_35 CrossRefGoogle Scholar
  7. 7.
    Czyzowicz, J., Ga̧sieniec, L., Kosowski, A., Kranakis, E., Spirakis, P.G., Uznański, P.: On convergence and threshold properties of discrete Lotka-Volterra population protocols. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 393–405. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47672-7_32 CrossRefGoogle Scholar
  8. 8.
    Diamadi, Z., Fischer, M.J.: A simple game for the study of trust in distributed systems. Wuhan Univ. J. Nat. Sci. 6(1–2), 72–82 (2001)CrossRefGoogle Scholar
  9. 9.
    Galstyan, A., Lerman, K.: Analysis of a stochastic model of adaptive task allocation in robots. In: Brueckner, S.A., Di Marzo Serugendo, G., Karageorgos, A., Nagpal, R. (eds.) ESOA 2004. LNCS, vol. 3464, pp. 167–179. Springer, Heidelberg (2005). doi:10.1007/11494676_11 CrossRefGoogle Scholar
  10. 10.
    Guerraoui, R., Ruppert, E.: Names trump malice: tiny mobile agents can tolerate Byzantine failures. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 484–495. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02930-1_40 CrossRefGoogle Scholar
  11. 11.
    Hartman, P.: A lemma in the theory of structural stability of differential equations. Am. Math. Soc. 11(4), 610–620 (1960)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hirsch, M., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, London (1974)MATHGoogle Scholar
  13. 13.
    Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  14. 14.
    Kleinrock, L.: Queueing Systems, Theory, vol. I. Wiley, Hoboken (1975)MATHGoogle Scholar
  15. 15.
    Kurtz, T.G.: Approximation of Population Processes (1981)Google Scholar
  16. 16.
    Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Mediated population protocols. Theor. Comput. Sci. 412(22), 2434–2450 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mitzenmacher, M.: Analyses of load stealing models based on families of differential equations. Theory Comput. Syst. 34(1), 77–98 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)CrossRefMATHGoogle Scholar
  19. 19.
    Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge (1997)MATHGoogle Scholar
  20. 20.
    Wormald, N.C.: Differential equations for random processes and random graphs. Ann. Appl. Probab. 5, 1217–1235 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer, Control and Management EngineeringSapienza University of RomeRomeItaly
  2. 2.Computer Science DepartmentUniversity of LiverpoolLiverpoolUK
  3. 3.Computer Engineering and Informatics DepartmentPatras UniversityPatrasGreece

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