A new density limit for unanimity in majority dynamics on random graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915189835694080 |
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| author | Kim, Jeong Han Tran, BaoLinh |
| author_facet | Kim, Jeong Han Tran, BaoLinh |
| contents | Majority dynamics is a process on a simple, undirected graph $G$ with an initial Red/Blue color for every vertex of $G$. Each day, each vertex updates its color following the majority among its neighbors, using its previous color for tie-breaking. The dynamics achieves \textit{unanimity} if every vertex has the same color after finitely many days, and such color is said to \textit{win}.
When $G$ is a $G(n,p)$ random graph, L. Tran and Vu (2019) found a codition in terms of $p$ and the initial difference $2Δ$ beteween the sizes of the Red and Blue camps, such that unanimity is achieved with probability arbitrarily close to 1. They showed that if $pΔ^2 \gg1 $, $pΔ\geq 100$, and $p\geq (1+\varepsilon) n^{-1}\log n$ for a positive constant $\varepsilon$, then unanimity occurs with probability $1 - o(1)$. If $p$ is not extremely small, namely $p > \log^{-1/16} n $, then Sah and Sawhney (2022) showed that the condition $pΔ^2 \gg 1$ is sufficient.
If $n^{-1}\log^2 n \ll p \ll n^{-1/2}\log^{1/4} n$, we show that $p^{3/2}Δ\gg n^{-1/2}\log n$ is enough. Since this condition holds if $pΔ\geq 100$ for $p$ in this range, this is an improvement of Tran's and Vu's result. For the closely related problem of finding the optimal condition for $p$ to achieve unanimity when the initial coloring is chosen uniformly at random among all possible Red/Blue assignments, our result implies a new lower bound $p \gg n^{-2/3}\log^{2/3} n$, which improves upon the previous bound of $n^{-3/5}\log n$ by Chakraborti, Kim, Lee and T. Tran (2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new density limit for unanimity in majority dynamics on random graphs Kim, Jeong Han Tran, BaoLinh Combinatorics Discrete Mathematics Mathematical Physics 05C80, 05C82, 05C85, 05C90 G.2.2; F.2.2 Majority dynamics is a process on a simple, undirected graph $G$ with an initial Red/Blue color for every vertex of $G$. Each day, each vertex updates its color following the majority among its neighbors, using its previous color for tie-breaking. The dynamics achieves \textit{unanimity} if every vertex has the same color after finitely many days, and such color is said to \textit{win}. When $G$ is a $G(n,p)$ random graph, L. Tran and Vu (2019) found a codition in terms of $p$ and the initial difference $2Δ$ beteween the sizes of the Red and Blue camps, such that unanimity is achieved with probability arbitrarily close to 1. They showed that if $pΔ^2 \gg1 $, $pΔ\geq 100$, and $p\geq (1+\varepsilon) n^{-1}\log n$ for a positive constant $\varepsilon$, then unanimity occurs with probability $1 - o(1)$. If $p$ is not extremely small, namely $p > \log^{-1/16} n $, then Sah and Sawhney (2022) showed that the condition $pΔ^2 \gg 1$ is sufficient. If $n^{-1}\log^2 n \ll p \ll n^{-1/2}\log^{1/4} n$, we show that $p^{3/2}Δ\gg n^{-1/2}\log n$ is enough. Since this condition holds if $pΔ\geq 100$ for $p$ in this range, this is an improvement of Tran's and Vu's result. For the closely related problem of finding the optimal condition for $p$ to achieve unanimity when the initial coloring is chosen uniformly at random among all possible Red/Blue assignments, our result implies a new lower bound $p \gg n^{-2/3}\log^{2/3} n$, which improves upon the previous bound of $n^{-3/5}\log n$ by Chakraborti, Kim, Lee and T. Tran (2021). |
| title | A new density limit for unanimity in majority dynamics on random graphs |
| topic | Combinatorics Discrete Mathematics Mathematical Physics 05C80, 05C82, 05C85, 05C90 G.2.2; F.2.2 |
| url | https://arxiv.org/abs/2503.07447 |