  This paper studies the observability radius of network systems, which
measures the robustness of a network to perturbations of the edges. We consider
linear networks, where the dynamics are described by a weighted adjacency
matrix, and dedicated sensors are positioned at a subset of nodes. We allow for
perturbations of certain edge weights, with the objective of preventing
observability of some modes of the network dynamics. To comply with the network
setting, our work considers perturbations with a desired sparsity structure,
thus extending the classic literature on the observability radius of linear
systems. The paper proposes two sets of results. First, we propose an
optimization framework to determine a perturbation with smallest Frobenius norm
that renders a desired mode unobservable from the existing sensor nodes.
Second, we study the expected observability radius of networks with given
structure and random edge weights. We provide fundamental robustness bounds
dependent on the connectivity properties of the network and we analytically
characterize optimal perturbations of line and star networks, showing that line
networks are inherently more robust than star networks.
