  It is becoming more and more clear that complex networks present remarkable
large fluctuations. These fluctuations may manifest differently according to
the given model. In this paper we re-consider hidden variable models which turn
out to be more analytically treatable and for which we have recently shown
clear evidence of non-self averaging; the density of a motif being subject to
possible uncontrollable fluctuations in the infinite size limit. Here we
provide full detailed calculations and we show that large fluctuations are only
due to the node hidden variables variability while, in ensembles where these
are frozen, fluctuations are negligible in the thermodynamic limit, and equal
the fluctuations of classical random graphs. A special attention is paid to the
choice of the cut-off: we show that in hidden-variable models, only a cut-off
growing as $N^\lambda$ with $\lambda\geq 1$ can reproduce the scaling of a
power-law degree distribution. In turn, it is this large cut-off that generates
non-self-averaging.
