Limit Laws for Consensus Protocols on the Complete Graph

We study a distributed consensus problem on a complete communication network of $n$ vertices, each holding one of two opinions. The vertices communicate in rounds, possibly in the presence of adversarial noise, and exchange information until they all agree on a single opinion. We consider a general class of protocols, where the vertices randomly sample neighbors and update their own opinion according to an update function $f$ depending on the sampled opinions. A prominent example is the $k$-maj protocol, where every vertex adopts the majority opinion of $k$ randomly sampled neighbors. We consider the runtime $R_n$ that is the number of rounds until all vertices agree on the same opinion, which we call the dominating opinion $D_n$. In our main result we describe the limiting distributions of these two key quantities for a large class of update functions $f$, for arbitrary initial configurations and under the presence of an adversary who may alter the opinions of up to $o(\sqrt{n})$ vertices in each round. We show that there are $f$-specific constants $\gamma, m > 0$ such that $R_n$ centers around $\mu_n = \frac{1}{2}\log_\gamma n + \log_m\ln n$, and we describe the asymptotic distribution of $R_n - \mu_n$. In particular, we show that it does not converge, and that it becomes asymptotically periodic both in the $\log n$ as well as the $\log\log n$ scale. Applied to $k$-maj, our results show, among other things, that $\gamma_{k\text{-maj}} = \binom{k-1}{\lfloor k/2 \rfloor}2^{1-k}k \sim ({2k}/{\pi})^{1/2}$.