Consensus on the Initial Global Majority by Local Majority Polling for a Class of Sparse Graphs

We study the local majority protocol on simple graphs of a given degree sequence, for a certain class of degree sequences. We show that for almost all such graphs, subject to a sufficiently large bias, within time $A \log_d \log_d n$ the local majority protocol achieves consensus on the initial global majority with probability $1-n^{-\Omega((\log n)^{\varepsilon})}$, where $\varepsilon>0$ is a constant. $A$ is bounded by a universal constant and $d$ is a parameter of the graph; the smallest integer which is the degree of $\Theta(n)$ vertices in the graph. We further show that under the assumption that a vertex $v$ does not change its colour if it and all of its neighbours are the same colour, \emph{any} local protocol $\mathcal{P}$ takes time at least $(1-o(1))\log_d\log_d n$, with probability $1-e^{-\Omega(n^{1-o(1)})}$ on such graphs. We further show that for almost all $d$-regular simple graphs with $d$ constant, we can get a stronger probability to convergence of initial majority at the expense of time. Specifically, with probability $1-O\brac{c^{-n^{\varepsilon}}}$, the local majority protocol achieves consensus on the initial majority by time $O(\log n)$. Finally, we show how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erd\H{o}s--Renyi random graphs in the connected regime.