Networks grow and evolve by local events; such as the addition of new nodes and links; or rewiring of links from one node to another. We show that depending on the frequency of these processes two topologically different networks can emerge; the connectivity distribution following either a generalized power law or an exponential. We propose a continuum theory that predicts these two regimes as well as the scaling function and the exponents; in good agreement with numerical results. Finally; we use the obtained predictions to fit the connectivity distribution of the network describing the professional links between movie actors.