Majority dynamics on sparse random graphs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | |
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| _version_ | 1866929237226684416 |
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| author | Chakraborti, Debsoumya Kim, Jeong Han Lee, Joonkyung Tran, Tuan |
| author_facet | Chakraborti, Debsoumya Kim, Jeong Han Lee, Joonkyung Tran, Tuan |
| contents | Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $\pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnel, Tamuz and Tan conjectured that, in the Erdős--Rényi random graph $G(n,p)$, the random initial $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=ω(1/n)$.
This conjecture was first confirmed for $p\geqλn^{-1/2}$ 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 $λ' n^{-3/5}\log n \leq p \leq λn^{-1/2}$ with a large constant $λ'>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_12709 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Majority dynamics on sparse random graphs Chakraborti, Debsoumya Kim, Jeong Han Lee, Joonkyung Tran, Tuan Combinatorics Probability 05C80, 60C05 Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $\pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnel, Tamuz and Tan conjectured that, in the Erdős--Rényi random graph $G(n,p)$, the random initial $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=ω(1/n)$. This conjecture was first confirmed for $p\geqλn^{-1/2}$ 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 $λ' n^{-3/5}\log n \leq p \leq λn^{-1/2}$ with a large constant $λ'>0$. |
| title | Majority dynamics on sparse random graphs |
| topic | Combinatorics Probability 05C80, 60C05 |
| url | https://arxiv.org/abs/2105.12709 |