Scale-free networks as entropy competition.

Complex networks describe several and different real-world systems consisting of a number of interacting elements. A very important characteristic of such networks is the degree distribution that strongly controls their behavior. Based on statistical mechanics, three classes of uncorrelated complex networks are identified here, depending on the role played by the connectivities amongst elements. In particular, by identifying the connectivities of a node with the number of its nearest neighbors, we show that the power law is the most probable degree distribution that both nodes and neighbors, in a reciprocal competition, assume when the respective entropy functions reach their maxima, under mutual constraint. As a result, we obtain scaling exponent values as a function of the structural characteristics of the whole network. Moreover, our approach sheds light on the exponential and Poissonian degree distributions, derived, respectively, when connectivities are thought of as degenerated connections or as half-edges. Thus, all three classes of degree distributions are derived, starting from a common principle and leading to a general and unified framework for investigating the network structure.