Majority Dynamics: The Power of One

Consider $n=\ell+m$ individuals, where $\ell\le m$, with $\ell$ individuals holding an opinion $A$ and $m$ holding an opinion $B$. Suppose that the individuals communicate via an undirected network $G$, and in each time step, each individual updates her opinion according to a majority rule (that is, according to the opinion of the majority of the individuals she can communicate with in the network). This simple and well studied process is known as "majority dynamics in social networks". Here we consider the case where $G$ is a random network, sampled from the binomial model $\mathbb{G}(n,p)$, where $(\log n)^{-1/16}\le p\le 1-(\log n)^{-1/16}$. We show that for $n=\ell+m$ with $\Delta=m-\ell\le(\log n)^{1/4}$, the above process terminates whp after three steps when a consensus is reached. Furthermore, we calculate the (asymptotically) correct probability for opinion $B$ to "win" and show it is \[\Phi\bigg(\frac{p\Delta\sqrt{2}}{\sqrt{\pi p(1-p)}}\bigg) + O(n^{-c}),\] where $\Phi$ is the Gaussian CDF. This answers two conjectures of Tran and Vu and also a question raised by Berkowitz and Devlin. The proof technique involves iterated degree revelation and analysis of the resulting degree-constrained random graph models via graph enumeration techniques of McKay and Wormald as well as Canfield, Greenhill, and McKay.
Comment: 30 pages