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arXiv · 2311.13034

Soft random simplicial complexes

Julián David Candela
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Abstract

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.

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Julián David Candela. 2025-07-12. Soft random simplicial complexes. https://arxiv.org/abs/2311.13034

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