📝 SummarXiv

Abstract

This work studies the limitations of uniquely identifying a linear network's topology from partial measurements of its nodes. We show that the set of networks that are consistent with the measurements are related through the nullspace of the observability matrix for the true network. In doing so, we illustrate how potentially many networks are fully consistent with the measurements despite having topologies that are structurally inconsistent with each other, an often neglected consideration in the design of topology inference methods. We then provide an aggregate characterization of the space of possible networks by analytically solving for the most structurally dissimilar network. We find that when observing over 6% of nodes in random network models (e.g., Erd\H{o}s-R\'{e}nyi and Watts-Strogatz) the rate of edge misclassification drops to ~1%. Extending this discussion, we construct a family of networks that keep measurements $\epsilon$-"close" to each other, and connect the identifiability of these networks to the spectral properties of an augmented observability Gramian.