We consider a classical First-order Vector AutoRegressive (VAR(1)) model, where we interpret the autoregressive interaction matrix as influence relationships among the components of the VAR(1) process that can be encoded by a weighted directed graph. A majority of previous work studies the structural identifiability of the graph based on time series observations and therefore relies on dynamical information. In this work we assume that an equilibrium exists, and study instead the identifiability of the graph from the stationary distribution, meaning that we seek a way to reconstruct the influence graph underlying the dynamic network using only static information. We use an approach from algebraic statistics that characterizes models using the Jacobian matroids associated with the parametrization of the models, and we introduce sufficient graphical conditions under which different graphs yield distinct steady-state distributions. Additionally, we illustrate how our results could be applied to characterize networks inspired by ecological research.