  We carry out MC studies of 2D superconducting networks, in an applied
magnetic field, for square and honeycomb geometries. We consider both dilute
systems (f=1/q) and systems near full frustration (f=1/2-1/q). For the dilute
case (which models a film as q->infinity), we find two transitions: at
T_c(f)~1/q there is a depinning transition from a pinned to a floating vortex
lattice; at T_m(f)~constant the floating vortex lattice melts into an isotropic
liquid. We analyze this melting according to the Kosterlitz- Thouless theory of
dislocation mediated melting, and find that the melting is weakly first order.
For the case near full frustration, the system can be described in terms of the
density of defects in an otherwise fully frustrated vortex pattern. We find
pinned solid, floating solid, and liquid defect phases, as well as a higher
sharp transition corresponding to the disordering of the fully frustrated
background.
