  This work studies the achievable secure rate per source-destination pair in
wireless networks. First, a path loss model is considered, where the legitimate
and eavesdropper nodes are assumed to be placed according to Poisson point
processes with intensities $\lambda$ and $\lambda_e$, respectively. It is shown
that, as long as $\lambda_e/\lambda=o((\log n)^{-2})$, almost all of the nodes
achieve a perfectly secure rate of $\Omega(\frac{1}{\sqrt{n}})$ for the
extended and dense network models. Therefore, under these assumptions, securing
the network does not entail a loss in the per-node throughput. The
achievability argument is based on a novel multi-hop forwarding scheme where
randomization is added in every hop to ensure maximal ambiguity at the
eavesdropper(s). Secondly, an ergodic fading model with $n$ source-destination
pairs and $n_e$ eavesdroppers is considered. Employing the ergodic interference
alignment scheme with an appropriate secrecy pre-coding, each user is shown to
achieve a constant positive secret rate for sufficiently large $n$. Remarkably,
the scheme does not require eavesdropper CSI (only the statistical knowledge is
assumed) and the secure throughput per node increases as we add more legitimate
users to the network in this setting. Finally, the effect of eavesdropper
collusion on the performance of the proposed schemes is characterized.
