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Identifying Network Structure of Linear Dynamical Systems: Observability and Edge Misclassification

Jaidev Gill, Jing Shuang Li

Abstract

This work studies the limitations of uniquely identifying the structure (i.e., topology) of a networked linear system from partial measurements of its nodal dynamics. In general, many networks can be consistent with these measurements; this is a consideration often neglected by standard network inference methods. We show that the space of these networks are related through the nullspace of the observability matrix for the true network. We establish relevant metrics to investigate this space, including an analytic characterization of the most structurally dissimilar network that can be inferred, as well as the possibility of mis-inferring presence or absence of edges. In simulations, we find that when observing over 6\% of nodes in random network models (e.g., Erd\H os-R\' enyi and Watts-Strogatz), approximately 99\% of edges are correctly classified. Extending this discussion, we construct a family of networks that keep measurements $ε$-close to each other, and connect the identifiability of these networks to the spectral properties of an augmented observability Gramian.

Identifying Network Structure of Linear Dynamical Systems: Observability and Edge Misclassification

Abstract

This work studies the limitations of uniquely identifying the structure (i.e., topology) of a networked linear system from partial measurements of its nodal dynamics. In general, many networks can be consistent with these measurements; this is a consideration often neglected by standard network inference methods. We show that the space of these networks are related through the nullspace of the observability matrix for the true network. We establish relevant metrics to investigate this space, including an analytic characterization of the most structurally dissimilar network that can be inferred, as well as the possibility of mis-inferring presence or absence of edges. In simulations, we find that when observing over 6\% of nodes in random network models (e.g., Erd\H os-R\' enyi and Watts-Strogatz), approximately 99\% of edges are correctly classified. Extending this discussion, we construct a family of networks that keep measurements -close to each other, and connect the identifiability of these networks to the spectral properties of an augmented observability Gramian.

Paper Structure

This paper contains 10 sections, 5 theorems, 43 equations, 5 figures.

Table of Contents

  1. Introduction and Motivation
  2. Conditions for Identical Measurements
  3. Dissimilar Networks
  4. Equivalence of the $\ell_{1}$ and $\ell_{2}$ Programs
  5. Illustrative Example
  6. Networks with $\epsilon$-Close Measurements
  7. Simulations
  8. Identical Measurements with Random Networks
  9. $\epsilon$-Close Measurements
  10. Conclusions

Key Result

Lemma 1

Measurements of systems sys:orig and sys:perturbed are indistinguishable, if and only if $CA^k = C(A+\Delta)^k~~\forall k \in \mathbb{Z}_{\geq 0}$.

Figures (5)

  • Figure 1: Given time series measurements of a subset of nodes in a network, inference methods often predict different networks and structures. This paper describes the set of possible networks consistent with these measurements and provides characterizations of this set.
  • Figure 2: (Left) Network described by \ref{['eq:example']}. (Right) The most structurally dissimilar network exhibiting identical measurements as the original network. Blue dashed circles indicate measured nodes.
  • Figure 3: Percentage of edges flipped as more nodes are measured for Erdős-Rényi and Watts-Strogatz random networks.
  • Figure 4: Comparison of the observed trajectories $y(t)$ and $\tilde{y}(t)$ and corresponding error $e(t)$ of the $\epsilon$-close network.
  • Figure 5: (Left) Original network. (Middle) The most structurally dissimilar network that generates similar measurements. (Right) The most structurally dissimilar network that generates identical measurements. Blue dashed circles indicate which nodes are measured.

Theorems & Definitions (16)

  • Definition 1
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Definition 2
  • Definition 3
  • ...and 6 more