Simple dynamics for plurality consensus

Abstract

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.

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

$$\begin{aligned} \phi \left( \frac{1}{k} \sum _{i=1}^k x_i \right) \leqslant \frac{1}{k} \sum _{i=1}^k \phi (x_i). \end{aligned}$$

In Sect. 4.4, we use the following “reverse”-Chernoff bound [19, Theorem 2]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 $t \in ( 0, m-\mu )$:

$$\begin{aligned} \mathbf {P}\left( X - \mu > t \right) \geqslant \frac{1}{4}e^{-\frac{2t^2}{\mu }}. \end{aligned}$$