The growth of random networks as a diffusion process

Physical modeling the dynamics of network growth help in the understanding of statistical properties of real networks. We describe here a statistical model supported on the random walk theory that generates several statistical connectivity properties observed in real networks. This formalism allows us to find microscopic rules generating the scale-free connectivity either with positive or negative exponents, i.e., pk k±α as found in several biological networks. In addition we explore the microscopic foundation responsible for distributions conformed by two power-law regions with a crossover connectivity as has been described in the case of natural language. Finally, this model allows us to relate the anomalous effect of vertices with high connectivity and high frequency as a consequence of boundary conditions in the connectivity distribution.