A new bound in Majority Dynamics on Random Graphs

We study the evolution of majority dynamics on Erd\H{o}s-R\'enyi $G(n,p)$ random graphs. In this process, each vertex of a graph is assigned one of two initial states. Subsequently, on every day, each vertex simultaneously updates its state to the most common state in its neighbourhood. If the difference in the numbers of vertices in each state on day $0$ is larger than $\max \left\{\frac{1}{\sqrt{p}} \exp\left[A\sqrt{\log \left(\frac{1}{p}\right)}\right] , Bp^{-3/2} n^{-1/2} \right\}$ for constants $A$ and $B$, we demonstrate that the state with the initial majority wins with overwhelmingly high probability. This extends work by Linh Tran and Van Vu (2023), who previously considered this phenomenon. We also study majority dynamics with a random initial assignment of vertex states. When each vertex is assigned to a state with equal probability, we show that unanimity occurs with high probability for every $p \geq \lambda n^{-2/3}$, for some constant $\lambda$. This improves work by Fountoulakis, Kang and Makai (2020). Furthermore, we also consider a random initial assignment of vertex states where a vertex is slightly more likely to be in the first state than the second state. Previous work by Zehmakan (2018) and Tran and Vu (2023) provided conditions on how big this bias needs to be for the first colour to achieve unanimity with high probability. We strengthen these results by providing a weaker sufficient condition.