  We study a random graph model which combines properties of the edge
percolation model on Z^d and a classical random graph G(n,c/n). We show that
this model, being a homogeneous random graph, has a natural relation to the
so-called "rank 1 case" of inhomogeneous random graphs. This allows us to use
the newly developed theory of inhomogeneous random graphs to describe
completely the phase diagram in the case d=1. The phase transition is similar
to the classical random graph, it is of the second order. We also find the
scaled size of the largest connected component above the phase transition.
