Phase Transitions of the k-Majority Dynamics in a Biased Communication Model

Consider a graph where each of the n nodes is in one of two possible states, say or . Herein, we analyze the synchronous k-majoritydynamics, where nodes sample k neighbors uniformly at random with replacement and adopt the majority binary 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, w.l.o.g., state with some probability p, regardless of the state of the sampled node, and its true state with probability 1 − p. Such a communication model allows to reason about the robustness of a consensus protocol when communication channels between nodes are noisy. Differently from previous works where specific graph topologies—typically characterized by good expansion properties—are considered, our analysis only requires the graphs to be sufficiently dense, i.e., to have minimum degree ω(log n), without any further topological assumption. In this setting we prove two phase transition phenomena, both occurring asymptotically almost surely, depending on the bias p and on the initial unbalance toward state . More in detail, we prove that for every k ≥ 3 there exists a such that if the process reaches in rounds a -almost-consensus, i.e., a configuration where a fraction 1 − γ of the volume is in state , for any arbitrarily-small positive constant γ. On the other hand, if , we look at random initial configurations in which every node is in state with probability 1 − q independently of the others. We prove that there exists a constant such that if then a -almost-consensus is still reached in rounds, while, if , the process spends nω(1) rounds in a metastable phase where the fraction of volume in state is around a constant value depending only on p and k. Finally we also investigate, in such a biased setting, the differences and similarities between k-majority and other closely-related dynamics, namely voter and deterministic majority.