Limits on reconstruction of dynamical networks

TL;DR

This paper introduces an observability condition number for network systems, quantifies the limits of reconstructing network trajectories from noisy observations, and guides optimal sensor placement.

Contribution

It defines a new observability measure for network dynamics and analyzes how it affects the feasibility of trajectory reconstruction from partial noisy data.

Findings

Reconstruction becomes infeasible when the condition number is large.

The observability measure helps identify optimal nodes for observation.

Provides a framework for sensor placement to improve network monitoring.

Abstract

An observability condition number is defined for physical systems modeled by network dynamics. Assuming the dynamical equations of the network are known and a noisy trajectory is observed at a subset of the nodes, we calculate the expected distance to the nearest correct trajectory as a function of the observation noise level, and discuss how it varies over the unobserved nodes of the network. When the condition number is sufficiently large, reconstructing the trajectory from observations from the subset will be infeasible. This knowledge can be used to choose an optimal subset from which to observe a network.