On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols

  • Jurek Czyzowicz
  • Leszek Ga̧sieniec
  • Adrian Kosowski
  • Evangelos Kranakis
  • Paul G. Spirakis
  • Przemysław Uznański
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)

Abstract

In this work we focus on a natural class of population protocols whose dynamics are modeled by the discrete version of Lotka-Volterra equations with no linear term. In such protocols, when an agent \(a\) of type (species) \(i\) interacts with an agent \(b\) of type (species) \(j\) with \(a\) as the initiator, then \(b\)’s type becomes \(i\) with probability \(P_{ij}\). In such an interaction, we think of \(a\) as the predator, \(b\) as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random.

We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an \(n\)-agent population converges to some absorbing state in time polynomial in \(n\), w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, there exist protocols of the considered type, such as Rock-Paper-Scissors, which require exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We present a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but “control” convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a “coin-flip consensus” in a distributed system. Some of our techniques may be of independent value.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abdullah, M.A., Draief, M.: Global majority consensus by local majority polling on graphs of a given degree sequence. Discrete Applied Mathematics 180, 1–10 (2015)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distributed Computing 18(4), 235–253 (2006)MATHCrossRefGoogle Scholar
  3. 3.
    Angluin, D., Aspnes, J., Eisenstat, D.: A simple population protocol for fast robust approximate majority. Distributed Computing 21(2), 87–102 (2008)MATHCrossRefGoogle Scholar
  4. 4.
    Angluin, D., Aspnes, J., Eisenstat, D., Ruppert, E.: The computational power of population protocols. Distributed Computing 20(4), 279–304 (2007)MATHCrossRefGoogle Scholar
  5. 5.
    Aspnes, J., Ruppert, E.: An introduction to population protocols. In: Middleware for Network Eccentric and Mobile Applications, pp. 97–120. Springer Verlag (2009)Google Scholar
  6. 6.
    Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Silvestri, R., Trevisan, L.: Simple Dynamics for Majority Consensus. In: Proc. SPAA, pp. 247–256 (2014)Google Scholar
  7. 7.
    Cooper, C., Elsässer, R., Radzik, T.: The power of two choices in distributed voting. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 435–446. Springer, Heidelberg (2014) Google Scholar
  8. 8.
    Cruise, J., Ganesh, A.: Probabilistic consensus via polling and majority rules. Queueing Systems: Theory and Applications 78(2), 99–120 (2014)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Diamadi, Z., Fischer, M.J.: A simple game for the study of trust in distributed systems. Wuhan University Journal of Natural Sciences 6(1–2), 72–82 (2001)CrossRefGoogle Scholar
  10. 10.
    Dobrinevski, A., Frey, E.: Extinction in neutrally stable stochastic Lotka-Volterra models. Phys. Rev. E 85, 051903 (2012)CrossRefGoogle Scholar
  11. 11.
    Hassin, Y., Peleg, D.: Distributed probabilistic polling and applications to proportionate agreement. Information & Computation 171(2), 248–268 (2001)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press (1998)Google Scholar
  13. 13.
    Kerr, B., Riley, M.A., Feldman, M.W., Bohannan, B.J.M.: Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418(6894), 171–174 (2002)CrossRefGoogle Scholar
  14. 14.
    Kirkup, B.C., Riley, M.A.: Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo. Nature 428(6981), 412–414 (2004)CrossRefGoogle Scholar
  15. 15.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. American Mathematical Society (2006)Google Scholar
  16. 16.
    Lotka, A.J.: Contribution to the Theory of Periodic Reactions. J. Phys. Chem. 14(3), 271–274 (1910)CrossRefGoogle Scholar
  17. 17.
    Mertzios, G.B., Nikoletseas, S.E., Raptopoulos, C.L., Spirakis, P.G.: Determining majority in networks with local interactions and very small local memory. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 871–882. Springer, Heidelberg (2014) Google Scholar
  18. 18.
    Michail, O., Chatzigiannakis, I., Spirakis, P.G.: New Models for Population Protocols. Morgan & Claypool Synthesis Lectures on Distributed Computing Theory (2011)Google Scholar
  19. 19.
    Szolnoki, A., Mobilia, M., Jiang, L.-L., Szczesny, B., Rucklidge, A.M., Perc, M.: Cyclic dominance in evolutionary games: a review. J. R. Soc. Interface 11, 20140735 (2014)CrossRefGoogle Scholar
  20. 20.
    Parker, M., Kamenev, A.: Extinction in the Lotka-Volterra model. Phys. Rev. E 80, 021129 (2009)CrossRefGoogle Scholar
  21. 21.
    Reichenbach, T., Mobilia, M.: M, and E. Frey. Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. Phys. Rev. E 74, 051907 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jurek Czyzowicz
    • 1
  • Leszek Ga̧sieniec
    • 2
  • Adrian Kosowski
    • 3
  • Evangelos Kranakis
    • 4
  • Paul G. Spirakis
    • 2
    • 5
  • Przemysław Uznański
    • 6
  1. 1.Department d’InformatiqueUniversité du Québec en OutaouaisGatineauCanada
  2. 2.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  3. 3.Inria Paris and LIAFAUniversité Paris DiderotParisFrance
  4. 4.Carleton University, School of Computer ScienceOttawaCanada
  5. 5.CTIPatrasGreece
  6. 6.Helsinki Institute for Information Technology HIITAalto UniversityEspooFinland

Personalised recommendations