53
8

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

Abstract

Consider a graph where each of the nn nodes is in one of two possible states. Herein, we analyze the synchronous kk-Majority dynamics, where nodes sample kk 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 33-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 σ\sigma with some probability pp, regardless of the state of the sampled node, and its true state with probability 1p1-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 ω(logn)\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 pp and on the initial unbalance toward state σ\sigma. More in detail, we prove that for every k3k\geq 3 there exists a pkp_k^\star such that if p>pkp>p_k^\star the process reaches in O(1)O(1) rounds a σ\sigma-almost-consensus, i.e., a configuration where a fraction 1o(1)1-o(1) of the volume is in state σ\sigma. On the other hand, if p<pkp<p_k^\star, we look at random initial configurations in which every node is in state σ\sigma with probability 1q1-q independently of the others. We prove that there exists a constant qp,kq_{p,k}^\star such that if q<qp,kq < q_{p,k}^\star then a σ\sigma-almost-consensus is still reached in O(1)O(1) rounds, while, if q>qp,kq > q_{p,k}^\star, the process spends nω(1)n^{\omega(1)} rounds in a metastable phase where the fraction of volume in state σ\sigma is around a constant value depending only on pp and kk.

View on arXiv
Comments on this paper