Soft random simplicial complexes
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912477494640640 |
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| author | Candela, Julián David |
| author_facet | Candela, Julián David |
| contents | A soft random graph $G(n,r,p)$ can be obtained from the random geometric graph $G(n,r)$ by keeping every edge in $G(n,r)$ with probability $p$. This random graph is a particular case of the soft random graph model introduced by Penrose, in which the probability between 2 vertices is a function that depends on the distance between them. In this article, we define models for random simplicial complexes built over the soft random graph $G(n,r,p)$, which also present randomness in all other dimensions. Furthermore, we study the homology of those random simplicial complexes in different regimes of $n,r$, and $p$ by giving asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_13034 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Soft random simplicial complexes Candela, Julián David Algebraic Topology Combinatorics A soft random graph $G(n,r,p)$ can be obtained from the random geometric graph $G(n,r)$ by keeping every edge in $G(n,r)$ with probability $p$. This random graph is a particular case of the soft random graph model introduced by Penrose, in which the probability between 2 vertices is a function that depends on the distance between them. In this article, we define models for random simplicial complexes built over the soft random graph $G(n,r,p)$, which also present randomness in all other dimensions. Furthermore, we study the homology of those random simplicial complexes in different regimes of $n,r$, and $p$ by giving asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. |
| title | Soft random simplicial complexes |
| topic | Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2311.13034 |