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

Abstract

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 $\lambda \ge 1$ can reproduce the scaling of a power-law degree distribution. In turn, it is this large cut-off that generates non-self-averaging.

19 pages, 16 figures. arXiv admin note: text overlap with arXiv:1306.5565

References in corpus (1)

• The entropy of randomized network ensembles

Cited by in corpus (8)

• Entropy distribution and condensation in random networks with a given degree distribution
• Variational principle for scale-free network motifs
• Statistical mechanics of random geometric graphs: Geometry-induced first order phase transition
• Scale-free network clustering in hyperbolic and other random graphs
• When is a scale-free graph ultra-small?
• Characterizing the intrinsic correlations of scale-free networks
• Degree correlations in scale-free null models
• Robust subgraph counting with distribution-free random graph analysis