The Observability Radius of Networks

TL;DR

This paper investigates the robustness of network systems to edge perturbations by analyzing the observability radius, proposing optimization methods, and characterizing the effects of network structure on robustness.

Contribution

It introduces an optimization framework for minimal perturbations to disrupt observability and analyzes the expected observability radius considering network topology and randomness.

Findings

Line networks are more robust than star networks.

Optimal perturbations can be characterized analytically for specific network structures.

Fundamental bounds relate network connectivity to robustness against perturbations.

Abstract

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.