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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2405.00363 |
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| _version_ | 1866914070886612992 |
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| author | Garetto, Michele Leonardi, Emilio Torrisi, Giovanni Luca |
| author_facet | Garetto, Michele Leonardi, Emilio Torrisi, Giovanni Luca |
| contents | We extend classical bootstrap percolation by introducing two concurrent, competing processes on an Erdős--Rényi random graph $G(n,p_n)$. Each node can assume one of three states: red, black, or white. The process begins with $a_R^{(n)}$ randomly selected active red seeds and $a_B^{(n)}$ randomly selected active black seeds, while all other nodes start as white and inactive. White nodes activate according to independent Poisson clocks with rate 1. Upon activation, a white node evaluates its neighborhood: if its red (black) active neighbors exceed its black (red) active neighbors by at least a fixed threshold $r \geq 2$, the node permanently becomes red (black) and active. Model's key parameters are $r$ (fixed), $n$ (tending to $\infty$), $a_R^{(n)}$, $a_B^{(n)}$, and $p_n$. We investigate the final sizes of the active red ($A^{*(n)}_R$) and black ($A^{*(n)}_B$) node sets across different parameter regimes. For each regime, we determine the relevant time scale and provide detailed characterization of asymptotic dynamics of the two concurrent activation processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Competing bootstrap processes on the random graph $G(n,p)$ Garetto, Michele Leonardi, Emilio Torrisi, Giovanni Luca Probability Dynamical Systems We extend classical bootstrap percolation by introducing two concurrent, competing processes on an Erdős--Rényi random graph $G(n,p_n)$. Each node can assume one of three states: red, black, or white. The process begins with $a_R^{(n)}$ randomly selected active red seeds and $a_B^{(n)}$ randomly selected active black seeds, while all other nodes start as white and inactive. White nodes activate according to independent Poisson clocks with rate 1. Upon activation, a white node evaluates its neighborhood: if its red (black) active neighbors exceed its black (red) active neighbors by at least a fixed threshold $r \geq 2$, the node permanently becomes red (black) and active. Model's key parameters are $r$ (fixed), $n$ (tending to $\infty$), $a_R^{(n)}$, $a_B^{(n)}$, and $p_n$. We investigate the final sizes of the active red ($A^{*(n)}_R$) and black ($A^{*(n)}_B$) node sets across different parameter regimes. For each regime, we determine the relevant time scale and provide detailed characterization of asymptotic dynamics of the two concurrent activation processes. |
| title | Competing bootstrap processes on the random graph $G(n,p)$ |
| topic | Probability Dynamical Systems |
| url | https://arxiv.org/abs/2405.00363 |