Network Models with couplings between link pairs are the simplest models for a class of networks with Higher Order interactions. In this paper we give an analytic, general solution to this family of Random Graph Models extending previous results obtained for specific interaction structures. We use the Hubbard-Stratonovich transform to show that such higher order models can be decoupled into families of effectively separable random graph models, which might be enhanced with auxiliary, possibly multi-valued, parameters identifying the thermodynamical phases of the system. We study the diagonalizability of the matrix of the interactions between links, that is itself an adjacency matrix on the dual network whose vertices are the nodes pairs, that can be dealt with using tools of algebraic graph theory. By orthogonalizing the interaction, we carry out the full computation of the partition functions and discuss why in some cases interactions effectively reduce to their Mean-Field versions.