  The recently introduced graphical continuous Lyapunov models provide a new
approach to statistical modeling of correlated multivariate data. The models
view each observation as a one-time cross-sectional snapshot of a multivariate
dynamic process in equilibrium. The covariance matrix for the data is obtained
by solving a continuous Lyapunov equation that is parametrized by the drift
matrix of the dynamic process. In this context, different statistical models
postulate different sparsity patterns in the drift matrix, and it becomes a
crucial problem to clarify whether a given sparsity assumption allows one to
uniquely recover the drift matrix parameters from the covariance matrix of the
data. We study this identifiability problem by representing sparsity patterns
by directed graphs. Our main result proves that the drift matrix is globally
identifiable if and only if the graph for the sparsity pattern is simple (i.e.,
does not contain directed two-cycles). Moreover, we present a necessary
condition for generic identifiability and provide a computational
classification of small graphs with up to 5 nodes.
