Structural Identifiability of Graphical Continuous Lyapunov Models

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.