  A new graph dual formalism is presented for the analysis of line outages in
electricity networks. The dual formalism is based on a consideration of the
flows around closed cycles in the network. After some exposition of the theory
is presented, a new formula for the computation of Line Outage Distribution
Factors (LODFs) is derived, which is not only computationally faster than
existing methods, but also generalizes easily for multiple line outages and
arbitrary changes to line series reactance. In addition, the dual formalism
provides new physical insight for how the effects of line outages propagate
through the network. For example, in a planar network a single line outage can
be shown to induce monotonically decreasing flow changes, which are
mathematically equivalent to an electrostatic dipole field.
