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

Fountoulakis, Nikolaos; Kang, Mihyun; Makai, Tamás

doi:10.1002/rsa.20970

Cited by 15 publications

(25 citation statements)

References 10 publications

(26 reference statements)

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“…Results on the evolution of majority dynamics on the random graph G(n, p) were obtained recently by Benjamini et al [2]. The last author in collaboration with Kang, and Makai [10] proved the following theorem confirming the rapid stabilisation of the majority dynamics process on a suitably dense G(n, p), confirming a conjecture stated in [2]. Let M 0 be the most popular vertex state seen across the initial configuration.…”

Section: Now We Can Boundmentioning

confidence: 53%

“…Theorem 5.1 [10]. For all ε ∈ [0, 1) there exist Λ, n 0 such that for all n > n 0 , if p ≥ Λn − 1 2 , then G(n, p) is such that with probability at least 1 − ε, across the product space of G(n, p) and S 1/2 , the vertices in V n following the majority dynamics rule, unanimously have state M 0 after four rounds.…”

Section: Now We Can Boundmentioning

confidence: 99%

“…Some results on the majority regime. We will consider a selection of results from [10] (Lemmas 3.5, 3.6 therein), which will be useful in the minority regime analysis. The first result concerns the expectation and variance of n 1 (v; 1), given that E + c has occurred.…”

Section: Now We Can Boundmentioning

confidence: 99%

“…In this scenario, we observe that the interacting node system reduces to the so-called majority or minority dynamics on G(n, p). We sample a selection of results from [10], which concern the rapid stabilisation of agent strategies when playing a majority game on a dense random graph. Using these results as a basis, we proceed to prove our result concerning the formation of consensus strategies in the minority game analogue of the random graph majority game.…”

Section: Introductionmentioning

confidence: 99%

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Best response dynamics on random graphs

Chellig¹,

Durbac²,

Fountoulakis³

2020

Preprint

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We consider evolutionary games on a population whose underlying topology of interactions is determined by a binomial random graph G(n, p). Our focus is on 2player symmetric games with 2 strategies played between the incident members of such a population. Players update their strategies synchronously. At each round, each player selects the strategy that is the best response to the current set of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for p in a range that depends on a certain characteristic of the payoff matrix. In the presence of a bias among the pure Nash equilibria of the game, we determine a sharp threshold on p above which the largest connected component reaches unanimity with high probability. For p below this critical value, where this does not happen, we identify those substructures inside the largest component that remain discordant throughout the evolution of the system.

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Section: Now We Can Boundmentioning

confidence: 53%

Section: Now We Can Boundmentioning

confidence: 99%

Section: Now We Can Boundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Best response dynamics on random graphs

Chellig¹,

Durbac²,

Fountoulakis³

2020

Preprint

Self Cite

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“…For a finite graph, beginning with any initial configuration C 0 , the process of majority dynamics will become recurrent at some point, and interestingly, it is shown in [6] that the period is at most 2 when t is sufficiently large. Given a random initial configuration C 0 with each vertex independently taking state +1 with probability p + and −1 with probability p − = 1 − p + , majority dynamics has been investigated for several classes of graphs including lattice [7,8], infinite lattice [9], infinite trees [10], random regular graphs [11], and Erdős-Rényi random graphs [2,[12][13][14][15]. For example, it is shown in [14] that majority dynamics undergoes a phase transition at the threshold of connectivity p n = n −1 ln n for Erdős-Rényi random graph G(n, p n ), where p n is the edge probability [16].…”

Section: (): V-volmentioning

confidence: 99%

A Note on the Majority Dynamics in Inhomogeneous Random Graphs

Shang

2021

Results Math

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In this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

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Majority dynamics on sparse random graphs

Chakraborti

Kim

Lee

et al. 2023

Random Struct Algorithms

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Majority dynamics on a graph G is a deterministic process such that every vertex updates its ±1-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erdős-Rényi random graph G(n, p), the random initial ±1-assignment converges to a 99%-agreement with high probability whenever p = 𝜔(1∕n). This conjecture was first confirmed for p ≥ 𝜆n −1∕2 for a large constant 𝜆 by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p < 𝜆n −1∕2 . We break this Ω(n −1∕2 )-barrier by proving the conjecture for sparser random graphs G(n, p), where 𝜆 ′ n −3∕5 log n ≤ p ≤ 𝜆n −1∕2 with a large constant 𝜆 ′ > 0.

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

Cited by 15 publications

References 10 publications

Best response dynamics on random graphs

Best response dynamics on random graphs

A Note on the Majority Dynamics in Inhomogeneous Random Graphs

Majority dynamics on sparse random graphs

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