Global Majority Consensus by Local Majority Polling on Graphs of a Given Degree Sequence

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them, or all less one if the degree of $v$ is even. We study this protocol on graphs of a given degree sequence, in the following setting: initially each vertex of $G$ is red independently with probability $\alpha < \frac{1}{2}$, and is otherwise blue. We show that if $\alpha$ is sufficiently biased, then with high probability consensus is reached on the initial global majority within $O(\log_k \log_k n)$ steps if $5 \leq k \leq d$, and $O(\log_d \log_d n)$ steps if $k > d$. Here, $d\geq 5$ is the effective minimum degree, the smallest integer which occurs $\Theta(n)$ times in the degree sequence. We further show that on such graphs, any local protocol in which a vertex does not change colour if all its neighbours have that same colour, takes time at least $\Omega(\log_d \log_d n)$, with high probability. Additionally, we demonstrate how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erd\H{o}s-R\'enyi random graphs in the connected regime.