  We consider networks made of parallel lanes along which particles hop
according to driven diffusive dynamics. The particles also hop transversely
from lane to lane, hence indirectly coupling their longitudinal dynamics. We
present a general method for constructing the phase diagram of these systems
which reveals that in many cases their physics reduce to that of single-lane
systems. The reduction to an effective single-lane description legitimizes, for
instance, the use of a single TASEP to model the hopping of molecular motors
along the many tracks of a single microtubule. Then, we show how, in quasi-2D
settings, new phenomena emerge due to the presence of non-zero transverse
currents, leading, for instance, to strong `shear localisation' along the
network.
