The Dynamics and Stability of Probabilistic Population Processes
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  for approximating discrete processes with continuous differential equations. We show that it includes the model of Angluin et al. , 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.
- 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
- 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
- 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
- 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
- 15.Kurtz, T.G.: Approximation of Population Processes (1981)Google Scholar