Reducing Sensor Requirements by Relaxing the Network Metric Dimension

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. For graphs with loops, we use a greedy algorithm to obtain an upper-bound on the relaxed metric dimension. 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.