  We study the emergence of a giant component in the configuration model
subject to additional constraints on the edges. We partition a $d$-dimensional
torus into a cubic lattice with a diverging number of compartments containing
vertices and allow only local edges inside and between neighbouring
compartments. We show that, when the number of vertices per compartment grows
quickly enough, a giant component emerges under similar conditions as for the
standard configuration model. Conversely, when the compartment sizes are fixed,
our model might not feature a giant component even if the standard
configuration model does have one.
  Locally, our model resembles the configuration model, while globally, it has
properties more akin to a $d$-dimensional lattice. Nonetheless the model
remains analytically tractable using multitype branching processes with
infinite number of types and opens new potential ways to study percolation in
graphs with geometric properties.
