  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.
