Central limit theorems for Soft random simplicial complexes

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 $p$. The soft random simplicial complexes is a model for random simplicial complexes built over the soft random graph $G(n,r,p)$. This new model depends on a probability vector $\rho$ which allows the simplicial complexes to present randomness in all dimensions. In this article, we use a normal approximation theorem to prove central limit theorems for the number of $k$-faces and for the Euler's characteristic for soft random simplicial complexes.