I will present how scale-free degree distribution and finite-scale degree distribution of networks can emerge by a statistical mechanics model in which we introduce a energetic and an entropic term having precise graphical interpretation. We will give extimation for the size of the space of distiguishable graphs having same degree distribution. This space is strongly reduced as we consider networks with power-law exponent close to 2. Results will be compared with other equilibrium and off-equilibrium models of networks.