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Structural Identifiability of Graphical Continuous Lyapunov Models

Carlos Améndola, Tobias Boege, Benjamin Hollering, Pratik Misra

TL;DR

This work develops a complete theory of structural identifiability for graphical continuous Lyapunov models (GCLMs) under uncorrelated noise, providing Lyapunov-analogues of Verma–Pearl and Chickering results for Bayesian networks. By exploiting the Lyapunov equation and its vectorized forms, the authors derive missing-edge relations and show that model equivalence for acyclic GCLMs is determined by the skeleton and by the equivalence of induced 4-node subgraphs, with a transformational characterization via super-covered edge flips. The results demonstrate that Lyapunov models refine Bayesian-network equivalence classes and yield polynomial-time algorithms to test model equivalence and structural identifiability. The paper also develops a suite of algebraic and graph-theoretic tools, including irreducibility arguments and subgraph-based reasoning, to connect local (4-node) criteria to global (whole-graph) identifiability. The framework lays the groundwork for causal discovery in diffusion-based systems and points to future extensions to cyclic graphs and correlated noise, broadening the applicability of structural identifiability in continuous-time causal models.

Abstract

We prove two characterizations of model equivalence of acyclic graphical continuous Lyapunov models (GCLMs) with uncorrelated noise. The first result shows that two graphs are model equivalent if and only if they have the same skeleton and equivalent induced 4-node subgraphs. We also give a transformational characterization via structured edge reversals. The two theorems are Lyapunov analogues of celebrated results for Bayesian networks by Verma and Pearl, and Chickering, respectively. Our results have broad consequences for the theory of causal inference of GCLMs. First, we find that model equivalence classes of acyclic GCLMs refine the corresponding classes of Bayesian networks. Furthermore, we obtain polynomial-time algorithms to test model equivalence and structural identifiability of given directed acyclic graphs.

Structural Identifiability of Graphical Continuous Lyapunov Models

TL;DR

This work develops a complete theory of structural identifiability for graphical continuous Lyapunov models (GCLMs) under uncorrelated noise, providing Lyapunov-analogues of Verma–Pearl and Chickering results for Bayesian networks. By exploiting the Lyapunov equation and its vectorized forms, the authors derive missing-edge relations and show that model equivalence for acyclic GCLMs is determined by the skeleton and by the equivalence of induced 4-node subgraphs, with a transformational characterization via super-covered edge flips. The results demonstrate that Lyapunov models refine Bayesian-network equivalence classes and yield polynomial-time algorithms to test model equivalence and structural identifiability. The paper also develops a suite of algebraic and graph-theoretic tools, including irreducibility arguments and subgraph-based reasoning, to connect local (4-node) criteria to global (whole-graph) identifiability. The framework lays the groundwork for causal discovery in diffusion-based systems and points to future extensions to cyclic graphs and correlated noise, broadening the applicability of structural identifiability in continuous-time causal models.

Abstract

We prove two characterizations of model equivalence of acyclic graphical continuous Lyapunov models (GCLMs) with uncorrelated noise. The first result shows that two graphs are model equivalent if and only if they have the same skeleton and equivalent induced 4-node subgraphs. We also give a transformational characterization via structured edge reversals. The two theorems are Lyapunov analogues of celebrated results for Bayesian networks by Verma and Pearl, and Chickering, respectively. Our results have broad consequences for the theory of causal inference of GCLMs. First, we find that model equivalence classes of acyclic GCLMs refine the corresponding classes of Bayesian networks. Furthermore, we obtain polynomial-time algorithms to test model equivalence and structural identifiability of given directed acyclic graphs.

Paper Structure

This paper contains 15 sections, 24 theorems, 26 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Graphical Continuous Lyapunov Models
  3. Definition of the Model
  4. Vectorizations of the Lyapunov Equation
  5. Missing-edge Relations
  6. Main Results
  7. Proofs
  8. Techniques for Proving Structural Identifiability Results
  9. Distinguishing Models via Induced Subgraphs
  10. Equivalence Under Super-Covered Edge Flips
  11. Proofs of the Main Theorems
  12. Discussion
  13. Lyapunov as a causal refinement of Bayes
  14. Model-defining equations and conditional independence
  15. Cyclic models and correlated noise

Key Result

Theorem 2.4

Let $\mathcal{G} = (V, E)$ be a simple graph. Then $A_\mathcal{G}(\Sigma)$ is full rank, implying that the model $\mathcal{M}_{\mathcal{G}, C}$ is globally identifiable for all $C \in \mathrm{PD}_n$. In particular, if $\mathcal{G}$ is simple and complete then holds for any $M \in \mathbb{R}^E$ and $\Sigma \in \mathcal{M}_{\mathcal{G},C}$ solving eq:Lyapunov. Here, $A_\mathcal{G}^{ij}(\Sigma; C)$

Figures (4)

  • Figure 1: The seven fundamental types of non-identifiable Lyapunov DAG models on four vertices. The classes (\ref{['fig:NonIdent4']})--(\ref{['fig:NonIdent7']}) are trivial in that they consist of graphs whose connected components are cliques. The other equivalence classes are made up as follows: (\ref{['fig:NonIdent1']}) Twelve classes with two DAGs each; (\ref{['fig:NonIdent2']}) six classes with two DAGs each; and (\ref{['fig:NonIdent3']}) six classes with two DAGs each.
  • Figure 2: Connecting model equivalent graphs via two super-covered edge flips.
  • Figure 3: Two almost-complete DAGs that differ by a single super-covered edge and the missing edge is between two parents.
  • Figure 4: Two almost-complete DAGs that differ by a single super-covered edge and the missing edge is between two children, and $\mathcal{G}=\mathcal{G}_1\cup \mathcal{G}_2$.

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4: dettling2022identifiability
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Example 2.8
  • Definition 3.1
  • ...and 44 more