Reconstructing dynamical interactions among species is a central problem in ecology. In practice, however, time-series observations are often unavailable, and many datasets consist only of species abundances collected independently across sites. Modeling these data as independent and identically distributed samples from a common stationary distribution raises a fundamental question: to what extent can the underlying dynamical network be recovered without temporal observations?
We address this question for stationary first-order vector autoregressive (VAR(1)) models, developing methods for network identification and reconstruction using only i.i.d. observations from the stationary distribution. Our contributions are twofold. First, we adopt an algebraic statistical approach and use Jacobian matroids associated with the model’s parametrization map to derive sufficient graphical conditions under which distinct dynamical networks induce distinguishable stationary distributions. Second, for identifiable networks, we propose a penalized estimator and establish consistency and expected risk guarantees. The penalty design is inspired by model selection methods developed by Birgé, Laurent, and Massart. We also develop a practical reconstruction procedure for applying the estimator to simulated and real data.
Together, these results clarify when dynamical network structure can be recovered from stationary observations and provide a statistically grounded reconstruction procedure.
