  We consider graphs parameterized on a portion $X\subset\mathbb Z^d\times
\{1,\ldots, M\}^k$ of a cylindrical subset of the lattice $\mathbb Z^d\times
\mathbb Z^k$, and perform a discrete-to-continuum dimension-reduction process
for energies defined on $X$ of quadratic type. Our only assumptions are that
$X$ be connected as a graph and periodic in the first $d$-directions. We show
that, upon scaling of the domain and of the energies by a small parameter
$\varepsilon$, the scaled energies converge to a $d$-dimensional limit energy.
The main technical points are a dimension-lowering coarse-graining process and
a discrete version of the $p$-connectedness approach by Zhikov.
