Limit theorems for the number of crossings and stress in projections of the random geometric graph
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915325878992896 |
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| author | Döring, Hanna de Jonge, Lianne |
| author_facet | Döring, Hanna de Jonge, Lianne |
| contents | We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\subset \mathbb{R}^d$, $d\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03218 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit theorems for the number of crossings and stress in projections of the random geometric graph Döring, Hanna de Jonge, Lianne Probability 60F05, 60D05, 68R10 We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\subset \mathbb{R}^d$, $d\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value. |
| title | Limit theorems for the number of crossings and stress in projections of the random geometric graph |
| topic | Probability 60F05, 60D05, 68R10 |
| url | https://arxiv.org/abs/2408.03218 |