Latent voter model on locally tree-like random graphs

Huo, Ran; Durrett, Rick

doi:10.1016/j.spa.2017.08.004

Cited by 5 publications

(8 citation statements)

References 26 publications

(42 reference statements)

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“…This result can be easily proved using the techniques in [12]. The key idea is that when ε is small most of the steps in the dual are random walk steps, and the random walk is transient, so any coalescence occurs soon after branching, and the dual is essentially a coalescing branching random walk.…”

Section: Results For Galton-watson Treesmentioning

confidence: 96%

The zealot voter model

Huo¹,

Durrett²

2019

Ann. Appl. Probab.

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Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0's are ordinary voters and 1's are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if x has degree d(x) then at rate d(x)p k the individual at x consults k ≥ 1 neighbors. If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate p 0 individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealots survive and when they will survive locally.

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Section: Results For Galton-watson Treesmentioning

confidence: 96%

The zealot voter model

Huo¹,

Durrett²

2019

Ann. Appl. Probab.

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“…• In Section 3.6 we put the pieces together to prove the result. As in Section 3.5 of [20] we do this by showing that if the density U n (t) reaches |U n (t) − 1/2| = 4 , then with very high probability (i.e., for any k with probability ≥ 1 − n −k for large n) it will return to |U n (t) − 1/2| ≤ before we have |U n (t) − 1/2| > 5 . Taking δ = 5…”

Section: Ode Limitmentioning

confidence: 99%

“…This has the consequence that if there are n • u 1's at time t/ n − n b , then at time t/ n the process is close to the voter equilibrium ν u . The argument here is an improvement over the one in Section 3.1 of [20]. We use Azuma's inequality to get error estimates that are stretched exponentially small, i.e., ≤ C exp(cn −α ) with α > 0 rather than polynomial, i.e., ≤ Ct −p .…”

Section: Ode Limitmentioning

confidence: 99%

“…To prove Theorem 1.3, we will follow the approach of Huo and Durrett [20] who proved a similar result for the latent voter model on a random graph generated by the configuration model. Although the random graph has a more complicated geometry than the torus, the proof in that setting is simpler than the one given here, since on the graph random walks mix in time O(log n) rather that in time O(n 2/3 ).…”

Section: Ode Limitmentioning

confidence: 99%

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The q-voter model on the torus

Agarwal

Simper

Durrett

2021

Electron. J. Probab.

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In the q-voter model, the voter at x changes its opinion at rate f qx , where fx is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if q < 1 and clustering if q > 1. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for q close to 1. More precisely, we show that if q < 1, then for any m < ∞ the process on the three-dimensional torus with n points survives for time n m , and after an initial transient phase has a density that it is always close to 1/2. Readers familiar with long time survival results for the contact process and other praticle systems might expect the conjecture to say survival occurs for time exp(γn) with γ > 0, however we show persistence does not hold for exp(n β ) with β > 1/3. If q > 1, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time log n but the stochastic process on the same time scale dies out at time (1/3) log n.

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“…To prove Theorem 3, we will follow the approach of Huo and Durrett [20] who proved a similar result for the latent voter model on a random graph generated by the configuration model. Although the random graph has a more complicated geometry than the torus, the proof in that setting is simpler than the one given here, since on the graph random walks mix in time O(log n) rather that in time O(n 2/3 ).…”

Section: Ode Limitmentioning

confidence: 99%

The q-voter model on the torus

Agarwal,

Simper,

Durrett

2020

Preprint

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In the q-voter model, the voter at x changes its opinion at rate f q x , where f x is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if q < 1 and clustering if q > 1. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for q close to 1. More precisely, we show that if q < 1, then for any m < ∞ the process on the three-dimensional torus with n points survives for time n m , and after an initial transient phase has a density that it is always close to 1/2. If q > 1, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time log n but the stochastic process on the same time scale dies out at time (1/3) log n.

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Latent voter model on locally tree-like random graphs

Cited by 5 publications

References 26 publications

The zealot voter model

The zealot voter model

The q-voter model on the torus

The q-voter model on the torus

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