Exact free energy functionals for non-simply-connected lattices

We consider the nearest-neighbor Ising model in thermal equilibrium on a network with no required regularity or symmetry properties. Both coupling strengths and external fields are site-dependent. The objective is to describe this system in terms of a free energy magnetization functional whose conjugate variables are the external fields. For simply connected networks, this inverse problem has a local structure. On generalizing to loops, the local structure remains if the description is expanded in an overcomplete fashion to include a collective amplitude with respect to which the free energy is stationary. For more complex connectivity, a superbond representation is developed in terms of which the system can be described by a combined auxiliary set of branch and node collective variables.