By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution P (s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ ≡ αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well P (s) in the transition from α = 0, when the vertices in the network are isolated, to α = 1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

We study numerically scattering and transport statistical properties of tight-binding random networks characterized by the number of nodes N and the average connectivity α. We use a scattering approach to electronic transport and concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements <|S(mn)|(2)>, the conductance probability distribution w(T), the average conductance , the shot noise power P, and the elastic enhancement factor F by varying α from small (α→0) to large (α→1) values. We also show that all these quantities are invariant for fixed ξ=αN. Moreover, we proposes a heuristic and universal relation between <|S(mn)|(2)>, , and P and the disorder parameter ξ.

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