  We study a one parameter family of random graph models that spans a continuum
between traditional random graphs of the Erd\H{o}s-R\'enyi type, where there is
no underlying structure, and percolation models, where the possible edges are
dictated exactly by a geometry. We find that previously developed theories in
the fields of random graphs and percolation have, starting from different
directions, covered almost all the models described by our family. In
particular, the existence or not of a phase transition where a giant cluster
arises has been proved for all values of the parameter but one. We prove that
the single remaining case behaves like a random graph and has a single linearly
sized cluster when the expected vertex degree is greater than one.
