Phase transition of the k-majority dynamics in biased communication models

Abstract

Consider a graph where each of the n nodes is either in state mathcal {R} or mathcal {B}. Herein, we analyze the synchronousk-Majoritydynamics, 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 maintaining amathcal {R}majority, when the dynamics is subject to two forms of 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 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 kge 3 there exists a critical value p_{k,q}^star 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}^star , or in O(1) rounds, if p>p_{k,q}^star . When k<3, instead, no phase transition phenomenon is observed and the disruption happens in O(1) rounds for p>0.