  Methods for understanding classical disordered spin systems with interactions
conforming to some idealized graphical structure are well developed. The
equilibrium properties of the Sherrington-Kirkpatrick model, which has a
densely connected structure, have become well understood. Many features
generalize to sparse Erdos-Renyi graph structures above the percolation
threshold, and to Bethe lattices when appropriate boundary conditions apply. In
this paper we consider spin states subject to a combination of sparse strong
interactions with weak dense interactions, which we term a composite model. The
equilibrium properties are examined through the replica method, with exact
analysis of the high temperature paramagnetic, spin glass and ferromagnetic
phases by perturbative schemes. We present results of a replica symmetric
variational approximations where perturbative approaches fail at lower
temperature. Results demonstrate novel reentrant behaviors from spin glass to
ferromagnetic phases as temperature is lowered, including transitions from
replica symmetry broken to replica symmetric phases. The nature of high
temperature transitions is found to be sensitive to the connectivity profile in
the sparse sub-graph, with regular connectivity a discontinuous transition from
the paramagnetic to ferromagnetic phases is apparent.
