  Stochastic networks based on random point sets as nodes have attracted
considerable interest in many applications, particularly in communication
networks, including wireless sensor networks, peer-to-peer networks and so on.
The study of such networks generally requires the nodes to be independently and
uniformly distributed as a Poisson point process. In this work, we venture
beyond this standard paradigm and investigate the stochastic geometry of
networks obtained from \textit{directed spanning forests} (DSF) based on
randomly perturbed lattices, which have desirable statistical properties as a
models of spatially dependent point fields. In the regime of low disorder, we
show in 2D and 3D that the DSF almost surely consists of a single tree. In 2D,
we further establish that the DSF, as a collection of paths, converges under
diffusive scaling to the Brownian web.
