Limiting behavior of a kindness model
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910318637088768 |
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| author | Lanchier, Nicolas Mercer, Max |
| author_facet | Lanchier, Nicolas Mercer, Max |
| contents | This paper is concerned with a stochastic model for the spread of kindness across a social network. Individuals are located on the vertices of a general finite connected graph, and are characterized by their kindness belief. Each individual, say $x$, interacts with each of its neighbors, say $y$, at rate one. The interactions can be kind or unkind, with kind interactions being more likely when the kindness belief of the sender $x$ is high. In addition, kind interactions increase the kindness belief of the recipient $y$, whereas unkind interactions decrease its kindness belief. The system also depends on two parameters modeling the impact of kind and unkind interactions, respectively. We prove that, when kind interactions have a larger impact than unkind interactions, the system converges to the purely kind configuration with probability tending to one exponentially fast in the large population limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limiting behavior of a kindness model Lanchier, Nicolas Mercer, Max Probability 60K35, 91D25 This paper is concerned with a stochastic model for the spread of kindness across a social network. Individuals are located on the vertices of a general finite connected graph, and are characterized by their kindness belief. Each individual, say $x$, interacts with each of its neighbors, say $y$, at rate one. The interactions can be kind or unkind, with kind interactions being more likely when the kindness belief of the sender $x$ is high. In addition, kind interactions increase the kindness belief of the recipient $y$, whereas unkind interactions decrease its kindness belief. The system also depends on two parameters modeling the impact of kind and unkind interactions, respectively. We prove that, when kind interactions have a larger impact than unkind interactions, the system converges to the purely kind configuration with probability tending to one exponentially fast in the large population limit. |
| title | Limiting behavior of a kindness model |
| topic | Probability 60K35, 91D25 |
| url | https://arxiv.org/abs/2402.02579 |