A Null-Space metric for the analysis of partial network observability in sensor location problems

This paper presents a new methodology and an intuitive metric for assessing the information quality of a set of (link) sensors in a network. This methodology is based on the concept of network observability, i.e. given a set of sensors covering a few links, what is the amount of information obtained on the rest of the network, where sensors are not installed. Existing approaches can efficiently find solutions for complete network observability, which means that through these methods one can identify the smallest number of links needed to fully determine the flows on the remaining set of links. Similar approaches can be found using route and OD pairs as variables. However, no comprehensive metric is found in literature that can quantify the quality of a solution in case of partial observability. This paper aims to fill this gap. This is a crucial contribution in this field, since even in small size networks the solution for full observability requires an exceedingly large amount of sensors. We tested this new methodology both on small toy networks, in order to analyze the properties of the metric and to explain and test the local search algorithms for optimal sensor positioning, and on real-sized networks. Comparison with partial observability methods proposed in other studies shows the benefits of the new methodology in finding the links that contain the largest deal of information in a network.