Consider a graph where each of the $n$ nodes is either in state $\mathcal{R}$ or $\mathcal{B}$. Herein, we analyze the \emph{synchronous $k$-Majority dynamics}, where in each discrete-time round nodes simultaneously sample $k$ neighbors uniformly at random with replacement and adopt the majority state among those of the nodes in the sample (breaking ties uniformly at random). Differently from previous work, we study the robustness of the $k$-Majority in \emph{maintaining a $\mathcal{R}$ majority}, when the dynamics is subject to two forms of \emph{bias} toward state $\mathcal{B}$. The bias models an external agent that attempts to subvert the initial majority by altering the communication between nodes, with a probability of success $p$ in each round: in the first form of bias, the agent tries to alter the communication links by transmitting state $\mathcal{B}$; in the second form of bias, the agent tries to corrupt nodes directly by making them update to $\mathcal{B}$. Our main result shows a \emph{sharp phase transition} in both forms of bias. By considering initial configurations in which every node has probability $q \in (\frac{1}{2},1]$ of being in state $\mathcal{R}$, we prove that for every $k\geq3$ there exists a critical value $p_{k,q}^*$ such that, with high probability, the external agent is able to subvert the initial majority either in $n^{\omega(1)}$ rounds, if $p<p_{k,q}^*$, or in $O(1)$ rounds, if $p>p_{k,q}^*$. When $k<3$, instead, no phase transition phenomenon is observed and the disruption happens in $O(1)$ rounds for $p>0$.

翻译：考虑一个图，其中$n$个节点每个都处于状态$\mathcal{R}$或$\mathcal{B}$。本文分析了\emph{同步$k$-多数动力学}，其中在每个离散时间轮次中，节点同时从邻居中均匀随机有放回地采样$k$个节点，并采纳采样节点中多数状态（平局时均匀随机打破）。与以往工作不同，我们研究了当动力学受到两种形式的向状态$\mathcal{B}$的\emph{偏向}时，$k$-多数在\emph{维持$\mathcal{R}$多数}方面的鲁棒性。偏向模型了一个外部代理，该代理试图通过改变节点间的通信来颠覆初始多数，在每一轮中以概率$p$成功：在第一种偏向形式中，代理试图通过传输状态$\mathcal{B}$来改变通信链路；在第二种偏向形式中，代理试图直接通过使节点更新为$\mathcal{B}$来腐化节点。我们的主要结果表明两种偏向形式中都存在\emph{尖锐相变}。考虑每个节点处于状态$\mathcal{R}$的概率为$q \in (\frac{1}{2},1]$的初始配置，我们证明对于每个$k\geq3$，存在一个临界值$p_{k,q}^*$，使得当$p<p_{k,q}^*$时，外部代理在$n^{\omega(1)}$轮内能够颠覆初始多数；当$p>p_{k,q}^*$时，则在$O(1)$轮内颠覆（均以高概率发生）。而当$k<3$时，没有观察到相变现象，且对于$p>0$，破坏在$O(1)$轮内发生。