Graphicality conditions for general scale-free complex networks and their application to visibility graphs.

We obtain graphicality conditions for general types of scale-free networks. The same conditions obtained for uncorrelated networks are obtained in the general case. Then an upper bound relating γ, the exponent of the degree distribution, with the cutoff exponent κ, as κ<1/γ is established. This bound is valid for all networks with a well-defined power-law degree distribution in the range γ≤2. Some recent numerical research on visibility networks arising from persistent fractional Brownian motion (fBm) processes are reviewed since they do not fulfill these conditions. As a consequence, a new relationship between the exponent γ of the degree distribution and the Hurst exponent H of the fBm process, γ⪅1/H, is postulated.