Identifying faulty edges in resistive electrical networks

TL;DR

This paper develops optimal measurement strategies to identify faulty edges in resistive electrical networks by measuring effective resistance between node pairs, providing tight bounds for various graph classes.

Contribution

It establishes rigorous upper and lower bounds on the minimal number of measurements needed to detect faulty edges, proving the optimality of the strategies for several graph classes.

Findings

Derived tight bounds for measurement numbers in different graph classes

Proposed optimal measurement strategies for fault detection

Validated bounds and strategies through theoretical proofs

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.