  We analyze random resistor networks through a study of lattice Green's
functions in arbitrary dimensions. We develop a systematic disorder
perturbation expansion to describe the weak disorder regime of such a system.
We use this formulation to compute ensemble averaged nodal voltages and bond
currents in a hierarchical fashion. We verify the validity of this expansion
with direct numerical simulations of a square lattice with resistances at each
bond exponentially distributed. Additionally, we construct a formalism to
recursively obtain the exact Green's functions for finitely many disordered
bonds. We provide explicit expressions for lattices with up to four disordered
bonds, which can be used to predict nodal voltage distributions for arbitrarily
large disorder strengths. Finally, we introduce a novel order parameter that
measures the overlap between the bond current and the optimal path (the path of
least resistance), for a given resistance configuration, which helps to
characterize the weak and strong disorder regimes of the system.
