  Large real-life complex networks are often modeled by various random graph
constructions and hundreds of further references therein. In many cases it is
not at all clear how the modeling strength of differently generated random
graph model classes relate to each other. We would like to systematically
investigate such issues. Our approach was originally motivated to capture
properties of the random network topology of wireless communication networks.
We started some investigations, but here we elevate it to a more general level
that makes it possible to compare the strength of different classes of random
network models. Specially, we introduce various classes of random graph models
that are significantly more general than the ones that are usually treated in
the literature, and show relationships among them. One of our main results is
that no random graph model can fall in the following three classes at the same
time: (1) random graph models with bounded expected degrees; (2) random graph
models that are asymptotically almost connected; (3) an abstracted version of
geometric random graph models with two mild restrictions that we call locality
and name invariance. In other words, in a mildly restricted, but still very
general, class of generalized geometric-style models the requirements of
bounded expected degrees and asymptotic almost connectivity are incompatible.
