  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 $p>0$.
