This paper presents an inference method to learn a model for complex system based on observations of the perturbed stationary points. We propose to jointly estimate the dynamics parameters and network topology through a regularized regression formulation. A distinguished feature of our approach rests on the direct modeling of rank deficient network data, which is widely found in network science but frequently ignored in the prior research. The new modeling technique allows us to provide the network identifiability condition under the scenarios of insufficient data. In the special case where the dynamics parameters are known, we show how the interplay between dynamics and sparsity in the graph structure can lead to verifiable conditions for network identifiability. Furthermore, relying only on the steady-states equations, our method avoids the necessity to record transient data, and allows to meaningfully combine data from multiple experiments. Numerical experiments are performed on examples with gene regulatory networks and opinion dynamics to justify our claims.