Phase Transition of the k-Majority Dynamics in Biased Communication Models

Consider a graph where each of the $n$ nodes is either in state $\mathcal{R}$ or $\mathcal{B}$. Herein, we analyze the 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 the nodes in the sample (breaking ties uniformly at random). Differently from previous work, we study the robustness of the $k$-Majority in maintaining a majority, that we consider $\mathcal{R}$ w.l.o.g., when the dynamics is subject to two forms of adversarial noise, or bias, toward state $\mathcal{B}$. We consider an external agent that wants to subvert the initial majority and, in each round, either tries to alter the communication between each pair of nodes transmitting state $\mathcal{B}$ (first form of bias), or tries to corrupt each node directly making it update to $\mathcal{B}$ (second form of bias), with a probability of success $p$. Our results show a phase transition in both forms of bias and on the same critical value. By considering initial configurations in which each 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: if $p>p_{k,q}^*$, the external agent is able to subvert the initial majority within a constant number of rounds; if $p<p_{k,q}^*$, the external agent needs at least a superpolynomial number of rounds to subvert the initial majority.