  Graphical models are a succinct way to represent the structure in probability
distributions. This article analyzes the graphical model of nodal voltages in
non-radial power distribution grids. Using algebraic and structural properties
of graphical models, algorithms exactly determining topology and detecting line
changes for distribution grids are presented along with their theoretical
limitations. We show that if distribution grids have cycles/loops of size
greater than three, then nodal voltages are sufficient for efficient topology
estimation without additional assumptions on system parameters. In contrast,
line failure or change detection using nodal voltages does not require any
structural assumption. Under noisy measurements, we provide the first
non-trivial bounds on the maximum noise that the system can tolerate for
asymptotically correct topology recovery. The performance of the designed
algorithms is validated with nonlinear AC power flow samples generated by
Matpower on test grids, including scenarios with injection correlations and
system noise.
