Majority Model on Random Regular Graphs

Gärtner, Bernd; Zehmakan, Ahad N.

doi:10.1007/978-3-319-77404-6_42

Cited by 41 publications

(38 citation statements)

References 26 publications

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“…Therefore, Theorem 9 implies that in the majority model on G ∆ n , if |B(0)| ≤ ( 1 2 − c √ ∆ )n for some large constant c then the process gets fully red a.a.s. This result is already known by Gärtner and Zehmakan [17], however with a much more involved proof.…”

Section: :11supporting

confidence: 67%

“…The majority model has been studied on different classes of graphs, like lattice [16,36,38,15], infinite lattice [10], random regular graphs [17], and infinite trees [22], when the initial configuration is random, meaning each node is independently blue with probability p b and red otherwise (without loss of generality, we always assume p b ≤ 1/2). We are interested in the behavior of the process when the underlying graph is the Erdős-Rényi random graph G n,p , where the node set is [n] = {1, · · · , n} and each edge is added with probability p independently.…”

Section: Introductionmentioning

confidence: 99%

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Opinion forming in Erdős–Rényi random graph and expanders

Zehmakan

2020

Discrete Applied Mathematics

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Assume for a graph G = (V, E) and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called majority model, on the Erdős-Rényi random graph G n,p and regular expanders. First we consider the behavior of the majority model on G n,p with an initial random configuration, where each node is blue independently with probability p b and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely log n n . Furthermore, we say a graph G is λ-expander if the second-largest absolute eigenvalue of its adjacency matrix is λ. We prove that for a ∆-regular λ-expander graph if λ/∆ is sufficiently small, then the majority model by starting from ( 1 2 − δ)n blue nodes (for an arbitrarily small constant δ > 0) results in fully red configuration in sub-logarithmically many rounds. Roughly speaking, this means the majority model is an "efficient" and "fast" density classifier on regular expanders. As a by-product of our results, we show regular Ramanujan graphs are asymptotically optimally immune, that is for an n-node ∆-regular Ramanujan graph if the initial number of blue nodes is s ≤ βn, the number of blue nodes in the next round is at most cs ∆ for some constants c, β > 0. This settles an open problem by Peleg [33]. Eligible for best student paper.

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Section: :11supporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Opinion forming in Erdős–Rényi random graph and expanders

Zehmakan

2020

Discrete Applied Mathematics

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“…In the deterministic majority, every node updates its state according to the majority state of its neighborhood as a whole, loosing the random interaction, which is a fundamental feature of the dynamics previously discussed. This deterministic protocol has been extensively studied in the literature; we mention, for example, its analysis on expander graphs [MNT14,Zeh20], random regular graphs [GZ18], and Erdős-Rnyi random graphs [BCO + 16, Zeh20].…”

Section: Related Workmentioning

confidence: 99%

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

Cruciani

Mimun

Quattropani

et al. 2021

Proceedings of the 22nd International Conference on Distributed Computing and Networking

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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.

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“…In particular, he proved that on a d ‐regular

λ

‐expander graph G , when the initial configuration satisfies

\sum_{v \in V false(G false)} S_{0} false(v false) \geq \frac{4 λ}{d} n

, majority dynamics will reach the configuration where every vertex has state +1 within

O false({normallog}_{d^{2} false/ λ^{2}} n false)

rounds. Also, Gärtner and Zehmakan 5 showed that if the initial density of the −1s is 1/2 − ε for some ε > 0, then the majority dynamics will eventually reach the configuration where every vertex has state +1.…”

Section: Introductionmentioning

confidence: 99%

Resolution of a conjecture on majority dynamics: Rapid stabilization in dense random graphs

Fountoulakis

Kang

Makai

2020

Random Struct Algorithms

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We study majority dynamics on the binomial random graph G(n, p) with p = ∕n and > n 1∕2 , for some large > 0. In this process, each vertex has a state in {−1, +1} and at each round every vertex adopts the state of the majority of its neighbors, retaining its state in the case of a tie. We show that with high probability the process reaches unanimity in at most four rounds. This confirms a conjecture of Benjamini et al.

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Majority Model on Random Regular Graphs

Cited by 41 publications

References 26 publications

Opinion forming in Erdős–Rényi random graph and expanders

Opinion forming in Erdős–Rényi random graph and expanders

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

Resolution of a conjecture on majority dynamics: Rapid stabilization in dense random graphs

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