in comparison to $q$. This completes the proof.

In the last section, we prove the large deviation principle. We use the first stipulation of \[thm1\] and the $s$-graded model to compute the upper bound. The lower bound is derived directly from the Poisson Point Process.

The $s$-Graded Model Approximates the Random Geometric Graph
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Before proving Theorem \[thm2\], we show that it implies Theorem \[thm1\]. To do so, we first define three operations to go between subsets of $T$, the natural objects in the $s$-graded model, and subsets of $[0,1]^d$. For $\mathfrak{B} \subset T$, let $$\mathfrak{U}(\mathfrak{B}) := \displaystyle \bigcup_{I \in \mathfrak{B}} A_I$$ the set associated with the index set $\mathfrak{A}$. In the other direction, we cannot be as exact; for $K \subset [0,1]^d$, let $\mathfrak{R}(K)$ and $\mathfrak{O}(K)$ be the outer and inner hulls of $K$, defined as the minimal and maximal subsets of $T$ such that $$\mathfrak{U}\left(\mathfrak{R}(K)\right) \subset K \subset \mathfrak{U}\left(\mathfrak{O}(K)\right)$$ With these operators in place, we show that the $s$-graded parameters are good approximations of their respective properties in the usual random geometric graph:

\[lmm3\] Let $G(\chi,r) = (V,E)$ be the usual random geometric graph, and $G_s(\chi,r) = (V,E_s)$ be the $s$-graded model. As before, $\mu = \mathbb{E}(|E|)$, $\tilde{\mu}_s 