We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between variables. When the errors are homoscedastic, recent work has proposed methods for inferring the graph from observational data under the assumption that the graph is acyclic (i.e., the SEM is recursive). In this work, we study the setting of homoscedastic errors but allow the graph to be cyclic (i.e., the SEM to be non-recursive). Using an algebraic approach that compares matroids derived from the parameterizations of the models, we derive sufficient conditions for when two simple directed graphs generate different distributions generically. Based on these conditions, we exhibit subclasses of graphs that allow for directed cycles, yet are generically identifiable. We also conjecture a strengthening of our graphical criterion which can be used to distinguish many more non-complete graphs.

翻译：我们考虑结构方程模型（SEM），其中每个变量是其他变量子集与随机误差的函数。每个此类模型自然关联于一个有向图，描述变量间的关系。当误差为同方差时，近期研究提出在假设图无环（即SEM为递归模型）的情况下，从观测数据推断该图的方法。本研究探讨同方差误差但允许图有环（即SEM为非递归模型）的情形。通过比较模型参数化导出的拟阵的代数方法，我们推导了两个简单有向图在一般情况下产生不同分布的充分条件。基于这些条件，我们展示了允许有向环但仍具有一般可辨识性的图子类。此外，我们推测了一个更强的图准则，可用于区分更多非完全图。