A statistical mechanics approach for scale-free networks and finite-scale networks.

We present a statistical mechanics approach for the description of complex networks. We first define an energy and an entropy, associated with a degree distribution, which have a geometrical interpretation. Next we evaluate the distribution that extremizes the free energy of the network. We find two important limiting cases: a scale-free degree distribution and a finite-scale degree distribution. The size of the space of allowed simple networks given these distributions is evaluated in the large network limit. Results are compared with simulations of algorithms generating these networks.