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.

翻译：我们证明了具有不相关噪声的非循环图结构连续李雅普诺夫模型（GCLMs）模型等价性的两个特征刻画。第一个结果表明，两个图模型等价当且仅当它们具有相同的骨架结构和等价的诱导四节点子图。我们还通过结构化边反转给出了变换特征的描述。这两个定理分别是Verma和Pearl以及Chickering针对贝叶斯网络的经典结论在李雅普诺夫模型中的对应结果。我们的研究对GCLMs的因果推断理论具有广泛意义。首先，我们发现非循环GCLMs的模型等价类能够细化贝叶斯网络的相应等价类。此外，我们获得了多项式时间算法来检验给定有向无环图的模型等价性与结构可辨识性。