Traditional network inference methods, such as Gaussian Graphical Models, which are built on continuity and homogeneity, face challenges when modeling discrete data and heterogeneous frameworks. Furthermore, under high-dimensionality, the parameter estimation of such models can be hindered by the notorious intractability of high-dimensional integrals. In this paper, we introduce a new and flexible device for graphical models, which accommodates diverse data types, including Gaussian, Poisson log-normal, and latent Gaussian copula models. The new device is driven by a new marginally recoverable parametric family, which can be effectively estimated without evaluating the high-dimensional integration in high-dimensional settings thanks to the marginal recoverability. We further introduce a mixture of marginally recoverable models to capture ubiquitous heterogeneous structures. We show the validity of the desirable properties of the models and the effective estimation methods, and demonstrate their advantages over the state-of-the-art network inference methods via extensive simulation studies and a gene regulatory network analysis of real single-cell RNA sequencing data.