Identifying faulty edges in resistive electrical networks

2025-12-29

AI Summary

This work addresses the problem of localizing single-edge faults in resistive networks using only effective resistance measurements between node pairs, aiming to identify the faulty edge with the minimum number of measurements. Methodologically, it integrates graph-theoretic modeling, circuit theory analysis, combinatorial optimization, and rigorous derivation of effective resistance bounds—surpassing prior heuristic or non-tight approaches. For fundamental graph classes—including trees, cycles, and complete graphs—the study establishes the first provably tight upper and lower bounds on the minimum number of required measurements and demonstrates their theoretical optimality. Key contributions include: (i) analytically characterizable optimal measurement strategies for multiple graph families, with matching upper and lower bounds; and (ii) a fundamental characterization of the intrinsic relationship between graph topology and diagnosability. The results provide both a theoretical foundation and practical design principles for efficient fault diagnosis in resistive networks.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Identifying faulty edges in resistive electrical networks
• Minimizing measurements for fault detection in networks
• Proving optimal bounds for measurement strategies in graphs

Innovation

• Uses effective resistance measurements between nodes
• Minimizes number of measurements to identify faults
• Proves tight bounds for optimal measurement strategies