Color-avoiding percolation in edge-colored Erdős-Rényi graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916344924995584 |
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| author | Ráth, Balázs Varga, Kitti Fekete, Panna Tímea Molontay, Roland |
| author_facet | Ráth, Balázs Varga, Kitti Fekete, Panna Tímea Molontay, Roland |
| contents | We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős-Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected.
We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_12727 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Color-avoiding percolation in edge-colored Erdős-Rényi graphs Ráth, Balázs Varga, Kitti Fekete, Panna Tímea Molontay, Roland Probability Mathematical Physics We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős-Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size. |
| title | Color-avoiding percolation in edge-colored Erdős-Rényi graphs |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2208.12727 |