  We study the problem of finding the resistors in a resistor network from
measurements of the power dissipated by the resistors under different loads. We
give sufficient conditions for local uniqueness, i.e. conditions that guarantee
that the linearization of this non-linear inverse problem admits a unique
solution. Our method is inspired by a method to study local uniqueness of
inverse problems with internal functionals in the continuum, where the inverse
problem is reformulated as a redundant system of differential equations. We use
our method to derive local uniqueness conditions for other discrete inverse
problems with internal functionals including a discrete analogue of the inverse
Schr\"odinger problem and problems where the resistors are replaced by
impedances and dissipated power at the zero and a positive frequency are
available. Moreover, we show that the dissipated power measurements can be
obtained from measurements of thermal noise induced currents.
