Reducing Sensor Requirements by Relaxing the Network Metric Dimension

2025-05-16

AI Summary

In graph-based source localization, balancing sensor count and localization accuracy remains challenging. Method: We introduce the *k-relaxed metric dimension*, a novel notion permitting nodes with pairwise distance differences ≤ *k* to share identical sensor-distance profiles—thereby reducing sensor requirements. We theoretically characterize its asymptotic behavior on deterministic and random trees, and design a two-stage adaptive source localization strategy leveraging branching process modeling and random geometric graph generation. Results: Experiments on diverse real-world and synthetic graphs demonstrate that the *k*-relaxed metric dimension is substantially smaller than the classical metric dimension, yields negligible numbers of indistinguishable nodes, and enables optimal trade-offs between total sensor count and localization accuracy. This work establishes both theoretical foundations and a practical framework for low-overhead, high-robustness network monitoring and source tracing.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Intelligent Robots: Localization, Mapping, and Navigation
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Source localization in graphs involves identifying the origin of a phenomenon or event, such as an epidemic outbreak or a misinformation source, by leveraging structural graph properties. One key concept in this context is the metric dimension, which quantifies the minimum number of strategically placed sensors needed to uniquely identify all vertices based on their distances. While powerful, the traditional metric dimension imposes a stringent requirement that every vertex must be uniquely identified, often necessitating a large number of sensors. In this work, we relax the metric dimension and allow vertices at a graph distance less than k to share identical distance profiles relative to the sensors. This relaxation reduces the number of sensors needed while maintaining sufficient resolution for practical applications like source localization and network monitoring. We provide two main theoretical contributions: an analysis of the k-relaxed metric dimension in deterministic trees, revealing the interplay between structural properties and sensor placement, and an extension to random trees generated by branching processes, offering insights into stochastic settings. We also conduct numerical experiments across a variety of graph types, including random trees, random geometric graphs, and real-world networks. The results show that the relaxed metric dimension is significantly smaller than the traditional metric dimension. Furthermore, the number of vertices indistinguishable from any given target vertex always remains small. Finally, we propose and evaluate a two-step localization strategy that balances the trade-off between resolution and the number of sensors required. This strategy identifies an optimal relaxation level that minimizes the total number of sensors across both steps, providing a practical and efficient approach to source localization.

Problem

• Reduces sensor count by relaxing unique vertex identification
• Analyzes k-relaxed metric dimension in deterministic and random trees
• Proposes efficient two-step localization strategy balancing resolution and sensors

Innovation

• Relaxing metric dimension to reduce sensor count
• Analyzing k-relaxed metric dimension in trees
• Proposing two-step localization strategy for efficiency