  We study a random graph model which is a superposition of the bond
percolation model on $Z^d$ with probability $p$ of an edge, and a classical
random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous}
random graph, has a natural relation to the so-called "rank 1 case" of {\it
inhomogeneous} random graphs. This allows us to use the newly developed theory
of inhomogeneous random graphs to describe the phase diagram on the set of
parameters $c\geq 0$ and $0 \leq p<p_c$, where $p_c=p_c(d)$ is the critical
probability for the bond percolation on $Z^d$. 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.
