Abstract
We study a pluralityconsensus 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 3majority 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 timegap between the pluralityconsensus 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.
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Notes
 1.
We say that a family of events \(\{\mathcal {E}_n\}_n\) holds w.h.p. if a positive constant c exists such that \(\mathbf {P}\left( \mathcal {E}_n \right) \geqslant 1  n^{c}\) for sufficiently large n.
 2.
In the (simple) consensus problem the goal is to reach any stable monochromatic configuration (any color is accepted) starting from any initial configuration.
 3.
We are using the fact that \(\mathbf {P}\left( A\cap B \right) \geqslant 1\mathbf {P}\left( A^{C} \right) \mathbf {P}\left( B^{C} \right) \).
 4.
Notice that the inequality holds in particular for negative a as well.
 5.
See e.g., Chapter 17 in [15] for a summary of martingales and related results.
 6.
A number of pretty similar “folklore” results can be found in specialized mathematical forums, for example http://cstheory.stackexchange.com/questions/14471/reversechernoffbound.
References
 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)
 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)
 3.
Babaee, A., Draief, M.: Distributed multivalued consensus. In: Proceedings of Computer and Information Sciences III, pp. 271–279. Springer (2013)
 4.
Becchetti, L., Clementi, A., Natale, E., Pasquale, F., Silvestri, R.: Plurality consensus in the gossip model. In: Proceedings of the 26th Annual ACMSIAM Symposium on Discrete Algorithms (SODA’15), pp. 371–390. SIAM (2015)
 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)
 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)
 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)
 8.
Draief, M., Vojnovic, M.: Convergence speed of binary interval consensus. SIAM J. Control Optim. 50(3), 1087–1109 (2012)
 9.
Dubhashi, D., Ranjan, D.: Balls and bins: a study in negative dependence. Random Struct. Algorithms 13(2), 99–124 (1998)
 10.
Easley, D., Kleinberg, J.: Networks, Crowds, and Markets. Cambridge University Press, Cambridge (2010)
 11.
Greenberg, S., Mohri, M.: Tight lower bound on the probability of a binomial exceeding its expectation. Stat. Probab. Lett. 86, 91–98 (2014)
 12.
Hassin, Y., Peleg, D.: Distributed probabilistic polling and applications to proportionate agreement. Inf. Comput. 171, 248–268 (2001)
 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)
 14.
Land, M.W.S., Belew, R.K.: No twostate ca for density classification exists. Phys. Rev. Lett. 74(25), 5148–5150 (1995)
 15.
Levin, D., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. AMS, Norwalk (2008)
 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)
 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)
 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)
 19.
Mousavi, N.: How tight is chernoff bound? https://ece.uwaterloo.ca/~nmousavi/Papers/ChernoffTightness.pdf
 20.
Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231–257 (2002)
 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)
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Partially supported by Italian MIURPRIN 2010–2011 Project ARS TechnoMedia, the EU FET Project MULTIPLEX 317532, and by the National Science Foundation under Grants No. CCF 1540685 and CCF 1655215. A preliminary version of this paper appeared in the Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA’14).
Useful bounds
Useful bounds
Lemma 11
(Chernoff bounds) Let \(X = \sum _{i=1}^n X_i\) where \(X_i\)’s are independent Bernoulli random variables and let \(\mu = \mathbf {E}_{} \left[ X \right] \). Then,

1.
For any \(0 < \delta \leqslant 4\), \(\mathbf {P}\left( X > (1 + \delta )\mu \right) < e^{\frac{\delta ^2\mu }{4}}\);

2.
For any \(\delta \geqslant 4\), \(\mathbf {P}\left( X > (1 + \delta )\mu \right) < e^{\delta \mu }\);

3.
For any \(\lambda > 0\), \(\mathbf {P}_{} \left( X \geqslant \mu + \lambda \right) \leqslant e^{2 \lambda ^2 / n}\).
Lemma 12
(Jensen inequality) Let \(\phi \,:\, \mathbb {R} \rightarrow \mathbb {R}\) be a convex function and \(x_1, \ldots , x_k \in \mathbb {R}\) be k real numbers, then
In Sect. 4.4, we use the following “reverse”Chernoff bound [19, Theorem 2]^{Footnote 6}
Theorem 5
(Reverse Chernoff bound) Let X be the sum of m independent Bernoulli variables with probability \(p\leqslant {1} / {4}\) and let \(\mu = pm\). Then, for any \(t \in ( 0, m\mu )\):
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Becchetti, L., Clementi, A., Natale, E. et al. Simple dynamics for plurality consensus. Distrib. Comput. 30, 293–306 (2017). https://doi.org/10.1007/s0044601602894
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Keywords
 Plurality consensus
 Distributed randomized algorithms
 Markov chains
 Dynamics