Robustness of the terminal behaviour of resistive electrical networks

In this paper, we consider the terminal behaviour of resistive electrical networks that are subject to changes or re-design. Specifically, we study the robustness of the effective resistance as a measure of this terminal behaviour. We give an explicit expression of how the effective resistance changes as a result of network perturbations, e.g., the addition/deletion of edges or changes in the conductance value. This expression has a clear interpretation and allows for efficient numerical evaluation. We extend this result to the total effective resistance, which can be regarded as a performance measure of consensus networks. In addition, these results are used to derive a graph-theoretical characterisation of the effective resistance.