In this paper we study the emergent structure of networks in which spike-timing dependent synaptic plasticity is induced only by external random noise. We show that such noise driven Hebbian networks are able to develop a broad range of network structures, including scale-free small-world networks. The development of such network structures may provide an explanation of the role of noise and its interplay with Hebbian plasticity. We also argue that this model can be seen as a unification of the famous Watts-Strogatz and preferential attachment models of small-world and scale-free nets. Our results may support Edelman's idea on that the development of central nervous system may have evolutionary components.