  We consider a variant of so called power-law random graph. A sequence of
expected degrees corresponds to a power-law degree distribution with finite
mean and infinite variance. In previous works the asymptotic picture with
number of nodes limiting to infinity has been considered. It was found that an
interesting structure appears. It has resemblance with such graphs like the
Internet graph. Some simulations have shown that a finite sized variant has
similar properties as well. Here we investigate this case in more analytical
fashion, and, with help of some simple lower bounds for large valued
expectations of relevant random variables, we can shed some light into this
issue. A new term, 'communication range random graph' is introduced to
emphasize that some further restrictions are needed to have a relevant random
graph model for a reasonable sized communication network, like the Internet. In
this case a pleasant model is obtained, giving the opportunity to understand
such networks on an intuitive level. This would be beneficial in order to
understand, say, how a particular routing works in such networks.
