Clique percolation

Derenyi, Palla and Vicsek introduced the following dependent percolation
model, in the context of finding communities in networks. Starting with a
random graph $G$ generated by some rule, form an auxiliary graph $G'$ whose
vertices are the $k$-cliques of $G$, in which two vertices are joined if the
corresponding cliques share $k-1$ vertices. They considered in particular the
case where $G=G(n,p)$, and found heuristically the threshold for a giant
component to appear in $G'$. Here we give a rigorous proof of this result, as
well as many extensions. The model turns out to be very interesting due to the
essential global dependence present in $G'$.