y  \|\right\rceil$$ where the circles indicate the interiors of the sets. Note that the distance is always an integer. Moreover, $d(I,J) = z$ if $z$ is the smallest integer such that some point in $A_I^o$ and some point in $A_J^o$ are less than $z$ away, measured in units of $1/m$, the side length of the cubes. We force the points to be in the interior to prevent “trivialities”, such as two adjacent cells being distance $0$, since they share a boundary.

We are now ready to define the $s$-graded random geometric graph. Let $G_s(\chi,r) = (V,E_s)$ have the same vertex set as the original graph. For each vertex $v$, let $I_v$ be the index in $T$ such that $v \in A_{I_v}$; there is ambiguity on the boundary of the $A_I$’s, but that set has Lebesgue measure 0, and therefore it has no vertices of $\chi$, almost surely. We say $(v,w) \in E_s$ whenever $d(I_v, I_w) \leq s$. Essentially, the $s$-graded model allows every point to wander inside a cubical “cage” of side-length $1/m$, and connects any two points that might be connected after we allow this mobility. In this framework, it is clear that $E_s$ becomes smaller as $s$ decreases. 