Majority Consensus on Random Graphs of a Given Degree Sequence

We study the local majority protocol on graphs of a given degree sequence for certain classes of degree sequences. We show that subject to a sufficiently large bias determined by the parameters of the graph, in time $A \log_d \log_d n$ for $A$ constant, the local majority protocol achieves correct consensus with probability $1-n^{-\Omega((\log n)^{\varepsilon})}$, where $\varepsilon$ is a constant. $d$ is a paramter of the graph; the smallest integer which is the degree of $\Theta(n)$ vertices in the graph. We 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 $1-e^{-\Omega(n^{1-o(1)})}$. We further show that for the special case of random $d$-regular graphs with $d$ constant, we can get a stronger error probability at the expense of time. Specifically, with probability $1-O\brac{c^{-n^{\varepsilon}}}$, the local majority has converged to the correct consensus 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 complete graph.