$I$. Using this parameter, we can compute the expected number of edges in the $s$-graded random geometric graph easily: $$\begin{aligned}
\tilde{\mu}_s &:= \mathbb{E}(|E_s|)\label{musdef}\\
&= \displaystyle \sum_{I \in T} \mathbb{E}\biggl[{X_I \choose 2}\biggr] + \frac{1}{2}  \sum_{J\, : \, 0 < d(I,J) \leq s} \mathbb{E}(X_I)
\mathbb{E}(X_J)\nonumber \\
&= \frac{|N_I| m^d \mathcal{D}^2}{2} = \frac{|N_I|n^2}{2m^d}\, . \nonumber\end{aligned}$$ As before, we are interested in conditioning the $s$-graded model on the event $\{|E_s| > (1 +\tilde{\delta}) \mu_s\}$. Again, the appropriate geometric notion is that of the largest set of diameter $s$. We call a set of indices a $\textit{maximal clique set}$ if it is a subset of $T$ with diameter $s$ that achieves the maximal cardinality of all such sets. We define $$\label{tausdef}
\tilde{\tau}_s := \max \{ |\mathfrak{I}| : \mathfrak{I} \subset T, \text{diam}(\mathfrak{I}) \leq s \} \, ,$$ i.e. $\tau_s$ is the cardinality of a maximal clique set.

We can now state the equivalent to Theorem \[thm1\] for the $s$-graded model:

\[thm2\] Fix an integer $s$ and let $G_s(\chi,r)$ be an $s$-graded random geometric graph, with $ n^{(2-\delta^*)/d} \leq r \leq n^{-\delta^*/d}$ for some fixed $\delta^*>0$. Fix $\tilde{\delta}>0$ and $\tilde{\varepsilon}>0$. Let $F$ be the event that the following happen: (a) There is a maximal clique set $\mathfrak{P}$ 