Quantifying the connectivity of scale-free and biological networks

Scale-free and biological networks follow a power law distribution pk∝k−α for the probability that a node is connected to k other nodes; the corresponding ranges for α (biological: 1<α<2; scale-free: 2<α 3) yield a diverging variance for the connectivity k and lack of predictability for the average connectivity. Predictability can be achieved with the Rényi, Tsallis and Landsberg–Vedral extended entropies and corresponding “disorders” for correctly chosen values of the entropy index q. Escort distributions pk∝k−αq with q>3/α also yield a nondiverging variance and predictability. It is argued that the Tsallis entropies may be the appropriate quantities for the study of scale-free and biological networks.