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.
We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and the Miller-Abrahams random resistor network with lower-bounded conductances. The latter is relevant for the analysis of conductivity by Mott variable range hopping in strongly disordered systems. By using the method of randomized algorithms developed by Duminil-Copin et al. we prove that in the subcritical phase the probability that the origin is connected to some point at distance n decays exponentially in n, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to (λ − λ c ), λ being the density of the Poisson point process and λ c being the critical density.
We consider the Miller-Abrahams (MA) random resistor network built on a homogeneous Poisson point process (PPP) on R d , d ≥ 2. Points of the PPP are marked by i.i.d. random variables and the MA random resistor network is obtained by plugging an electrical filament between any pair of distinct points in the PPP. The conductivity of the filament between two points decays exponentially in their distance and depends on their marks in a suitable form prescribed by electron transport in amorphous materials. The graph obtained by keeping filaments with conductivity lower bounded by a threshold ϑ exhibits a phase transition at some ϑ crit . Under the assumption that the marks are nonnegative (or nonpositive) and bounded, we show that in the supercritical phase the maximal number of vertex-disjoint left-right crossings in a box of size n is lower bounded by Cn d−1 apart an event of exponentially small probability. This result is one of the main ingredients entering in the proof of Mott's law in [4].
Consider a graph where each of the n nodes is either in state $$\mathcal {R}$$ R or $$\mathcal {B}$$ 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 a$$\mathcal {R}$$ R majority, when the dynamics is subject to two forms of bias toward state $$\mathcal {B}$$ 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 $$\mathcal {B}$$ B ; in the second form of bias, the agent tries to corrupt nodes directly by making them update to $$\mathcal {B}$$ B . Our main result shows a sharp phase transition in both forms of bias. By considering initial configurations in which every node has probability $$q \in (\frac{1}{2},1]$$ q ∈ ( 1 2 , 1 ] of being in state $$\mathcal {R}$$ R , we prove that for every $$k\ge 3$$ k ≥ 3 there exists a critical value $$p_{k,q}^\star $$ p k , q ⋆ such that, with high probability, the external agent is able to subvert the initial majority either in $$n^{\omega (1)}$$ n ω ( 1 ) rounds, if $$p<p_{k,q}^\star $$ p < p k , q ⋆ , or in O(1) rounds, if $$p>p_{k,q}^\star $$ p > p k , q ⋆ . When $$k<3$$ k < 3 , instead, no phase transition phenomenon is observed and the disruption happens in O(1) rounds for $$p>0$$ p > 0 .
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