Identifying faulty edges in resistive electrical networks
Barbara Fiedorowicz, Amitabh Basu
TL;DR
This work addresses fault detection in resistive electrical networks by querying effective resistances between node pairs to identify a potentially altered edge with the fewest measurements. It develops a measurement-graph framework and butterfly-wing measurement strategies to quantify information gained from each measurement, yielding tight upper and lower bounds for complete graphs and complete $k$-partite graphs. The main results include the exact bound $\left\lceil \frac{2n}{3} \right\rceil$ for complete graphs ($n \ge 6$) and precise bounds for bipartite, tripartite, and general $k$-partite graphs, with a unifying bound $\left\lceil \frac{n-k}{2} \right\rceil$ as the lower limit for $k$-partite structures. The findings advance discrete inverse-problem theory in electrical networks and offer actionable measurement strategies that are tight and constructive, potentially informing fault-detection design in power, VLSI, and circuit-testing contexts.
Abstract
Given a resistive electrical network, we would like to determine whether all the resistances (edges) in the network are working, and if not, identify which edge (or edges) are faulty. To make this determination, we are allowed to measure the effective resistance between certain pairs of nodes (which can be done by measuring the amount of current when one unit of voltage difference is applied at the chosen pair of nodes). The goal is to determine which edge, if any, is not working in the network using the smallest number of measurements. We prove rigorous upper and lower bounds on this optimal number of measurements for different classes of graphs. These bounds are tight for several of these classes showing that our measurement strategies are optimal.