Fluctuations analysis in complex networks modeled by hidden variable models. Necessity of a large cut-off in hidden-variable models

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.