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

Abstract

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 σ 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-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 with high probability, depending on the bias p and on the initial unbalance toward state σ. More in detail, we prove that for every k ≥ 3 there exists a p k such that if p > p k the process reaches in O(1) rounds a σ-almost-consensus, i.e., a configuration where a fraction 1 − γ of the volume is in state σ, for any arbitrarily-small positive constant γ. On the other hand, if p < p k , we look at random initial configurations in which every node is in state σ with probability 1−q independently of the others. We prove that there exists a constant q p,k such that if q < q p,k then a σ-almost-consensus is still reached in O(1) rounds, while, if q > q p,k , the process spends n ω(1) rounds in a metastable phase where the fraction of volume in state σ 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.

Phase Transition of the 3-Majority Dynamics with Uniform Communication Noise

“…Their result holds for any sufficiently dense graph. We remark that our work differs from [16] in that there is no preferred opinion, and the noise affecting communications may result in any possible opinion.…”

“…As for the non-uniform communication noise case, in [16] it is considered the h-Majority dynamics with a binary opinion set {alpha, beta}, with a probability p that any received message is flipped towards a fixed preferred opinion, say beta, while with probability 1 − p the former message keeps intact. They suppose there is an initial majority agreeing on alpha, and they analyze the time of disruption, that is the time the initial majority is subverted.…”

“…Indeed, they involve sending complicated codes through communication links, and it is reasonable to assume that biological type entities communicate between each other in a simpler way. For this reason, in recent years many works have been focusing on the study of the (majority) consensus problem where the communication between entities is unreliable and subjected to uniform noise [16,17,23,24].…”

Biased Majority Opinion Dynamics: Exploiting Graph k-domination

“…In [Mukhopadhyay et al, 2020], each agent updates each of its opinions at points of different independent Poisson point processes, which introduces a bias towards the opinion with the lowest firing rate frequency. The works closest to ours are [Anagnostopoulos et al, 2020;Cruciani et al, 2021].…”

“…One might wonder if the converse occurs, namely, whether the biased majority dynamics always affords (expected) polynomial convergence to the absorbing state when the network is not dense. While this is indeed the case for cycles, trees, and disconnected cliques of size O(log n), understanding the behavior of the dynamics remains open for bounded degree topologies, inducing challenging open problems formulated in [Anagnostopoulos et al, 2020;Cruciani et al, 2021].…”