Consider a graph where each of the n nodes is in one of two possible states . Herein, we analyze the synchronous k-Majority dynamics, where nodes sample k neighbors uniformly at random with replacement and adopt the majority state among the nodes in the sample ( potential ties are broken uniformly at random). This class of dynamics generalizes other well-known dynamics, e. g., Voter and 3-Majority, which have been studied in the literature as distributed algorithms for consensus . We consider a biased communication model: whenever nodes sample a neighbor they see state \sigma with some probability p, regardless of the state of the sampled node, and its true state with probability 1-p. Differently from previous works where specific graph topologies are considered, our analysis only requires the graphs to be sufficiently dense, i.e., to have minimum degree \omega(\log n), without any further topological assumption. In this setting we prove two phase transition phenomena, both occurring with high probability, depending on the bias p and on the initial unbalance toward state \sigma. More in detail,  we prove that for every k \geq 3 \geq3 there exists a p_k^\star such that if p>p _k^\star the process reaches in O(1) rounds a \sigma-almost-consensus, i.e., a configuration where a fraction 1-o(1) of the volume is in state \sigma. On the other hand, if p < p_k^\star, we look at random initial configurations in which every node is in state \sigma with probability 1-q independently of the others. We prove that there exists a constant q_{p,k^\star such that if q < q\SUBSCRIPT}%DIFDELCMD < {%%% p , k ^\star then a \sigma-almost-consensus is still reached in O(1) rounds, while, if q > q _{p,k^\star, the process spends n^{\omega(1)} rounds in a metastable phase where the fraction of volume in state \sigma is around a constant value depending only on p and k}  .
Consider a graph where each of the n nodes is either in state \mathcal{R . 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 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 and on the same critical value. By considering initial configurations in which each node has probability q \in (\frac{1 of being in state \mathcal{R we prove that for every k \geq3 there exists a critical value p_{k,q if p>p _{k,q < ^\star such that if q < q\SUBSCRIPT}%DIFDELCMD < {%%% p \SUBSCRIPT k , q ^\star, the process spends n^{\omega(1)} rounds in a metastable phase where the fraction of volume in state \sigma is around a constant value depending only on p and k} ^*, the external agent needs at least a superpolynomial number of rounds to subvert the initial majority .
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