Distributed Computing

, Volume 30, Issue 4, pp 293–306 | Cite as

Simple dynamics for plurality consensus

  • Luca Becchetti
  • Andrea Clementi
  • Emanuele Natale
  • Francesco Pasquale
  • Riccardo Silvestri
  • Luca Trevisan


We study a plurality-consensus process in which each of n anonymous agents of a communication network initially supports a color chosen from the set [k]. Then, in every round, each agent can revise his color according to the colors currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large bias s towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by s additional nodes. The goal is having the process to converge to the stable configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the 3-majority dynamics) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time \(\mathcal {O}( \min \{ k, (n/\log n)^{1/3} \} \, \log n )\) with high probability, provided that \(s \geqslant c \sqrt{ \min \{ 2k, (n/\log n)^{1/3} \}\, n \log n}\). We then prove that our upper bound above is tight as long as \(k \leqslant (n/\log n)^{1/4}\). This fact implies an exponential time-gap between the plurality-consensus process and the median process (see Doerr et al. in Proceedings of the 23rd annual ACM symposium on parallelism in algorithms and architectures (SPAA’11), pp 149–158. ACM, 2011). A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: in particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only.


Plurality consensus Distributed randomized algorithms Markov chains Dynamics 


  1. 1.
    Abdullah, M.A., Draief, M.: Global majority consensus by local majority polling on graphs of a given degree sequence. Discrete Appl. Math. 180, 1–10 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angluin, D., Aspnes, J., Eisenstat, D.: A simple population protocol for fast robust approximate majority. Distrib. Comput. 21(2), 87–102 (2008). (Preliminary version in DISC’07)CrossRefzbMATHGoogle Scholar
  3. 3.
    Babaee, A., Draief, M.: Distributed multivalued consensus. In: Proceedings of Computer and Information Sciences III, pp. 271–279. Springer (2013)Google Scholar
  4. 4.
    Becchetti, L., Clementi, A., Natale, E., Pasquale, F., Silvestri, R.: Plurality consensus in the gossip model. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15), pp. 371–390. SIAM (2015)Google Scholar
  5. 5.
    Bénézit, F., Thiran, P., Vetterli, M.: Interval consensus: from quantized gossip to voting. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’09), pp. 3661–3664. IEEE (2009)Google Scholar
  6. 6.
    Clementi, A., Di Ianni, M., Gambosi, G., Natale, E., Silvestri, R.: Distributed community detection in dynamic graphs. Theor. Comput. Sci. 584, 19–41 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Doerr, B., Goldberg, L.A., Minder, L., Sauerwald, T., Scheideler, C.: Stabilizing consensus with the power of two choices. In: Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA’11), pp. 149–158. ACM (2011)Google Scholar
  8. 8.
    Draief, M., Vojnovic, M.: Convergence speed of binary interval consensus. SIAM J. Control Optim. 50(3), 1087–1109 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dubhashi, D., Ranjan, D.: Balls and bins: a study in negative dependence. Random Struct. Algorithms 13(2), 99–124 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Easley, D., Kleinberg, J.: Networks, Crowds, and Markets. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Greenberg, S., Mohri, M.: Tight lower bound on the probability of a binomial exceeding its expectation. Stat. Probab. Lett. 86, 91–98 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hassin, Y., Peleg, D.: Distributed probabilistic polling and applications to proportionate agreement. Inf. Comput. 171, 248–268 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kearns, M., Tan, J.: Biased voting and the democratic primary problem. In: Proceedings of the 4th Workshop on Internet and Network Economics (WINE), pp. 639–652 (2008)Google Scholar
  14. 14.
    Land, M.W.S., Belew, R.K.: No two-state ca for density classification exists. Phys. Rev. Lett. 74(25), 5148–5150 (1995)CrossRefGoogle Scholar
  15. 15.
    Levin, D., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. AMS, Norwalk (2008)CrossRefGoogle Scholar
  16. 16.
    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: Internship Colloquium on Automata, Languages, and Programming (ICALP’14), pp. 871–882 (2014)Google Scholar
  17. 17.
    Mossel, E., Neeman, J., Tamuz, O.: Majority dynamics and aggregation of information in social networks. Auton. Agent. Multi Agent Syst. 28(3), 408–429 (2014)CrossRefGoogle Scholar
  18. 18.
    Mossel, E., Schoenebeck, G.: Reaching consensus on social networks. In: Proceedings of the 2nd Innovations in Computer Science (ICS’10), pp. 214–229 (2010)Google Scholar
  19. 19.
    Mousavi, N.: How tight is chernoff bound?
  20. 20.
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Perron, E., Vasudevan, D., Vojnovic, M.: Using three states for binary consensus on complete graphs. In: Proceedings of the 28th IEEE INFOCOM, pp. 2527–2535 (2009)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Sapienza Università di RomaRomeItaly
  2. 2.Max Planck Institute for InformaticsSaarbrückenGermany
  3. 3.Università Tor Vergata di RomaRomeItaly
  4. 4.RomeItaly
  5. 5.UC BerkeleyBerkeleyUSA

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