Sharing tea on a graph
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arXiv
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| Autores principales: | , , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914049951793152 |
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| author | Gollin, J. Pascal Hendrey, Kevin Huang, Hao Huynh, Tony Mohar, Bojan Oum, Sang-il Yang, Ningyuan Yu, Wei-Hsuan Zhu, Xuding |
| author_facet | Gollin, J. Pascal Hendrey, Kevin Huang, Hao Huynh, Tony Mohar, Bojan Oum, Sang-il Yang, Ningyuan Yu, Wei-Hsuan Zhu, Xuding |
| contents | Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert.
We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharing tea on a graph Gollin, J. Pascal Hendrey, Kevin Huang, Hao Huynh, Tony Mohar, Bojan Oum, Sang-il Yang, Ningyuan Yu, Wei-Hsuan Zhu, Xuding Combinatorics Probability 05C57, 05C90, 05C22, 91D30, 91B32, 05C63 Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$. |
| title | Sharing tea on a graph |
| topic | Combinatorics Probability 05C57, 05C90, 05C22, 91D30, 91B32, 05C63 |
| url | https://arxiv.org/abs/2405.15353 |