  Statistical systems on random networks can be formulated in terms of
partition functions expressed with integrals by regarding Feynman diagrams as
random networks. We consider the cases of random networks with bounded but
generic degrees of vertices, and show that the free energies can be exactly
evaluated in the thermodynamic limit by the Laplace method, and that the exact
expressions can in principle be obtained by solving polynomial equations for
mean fields. As demonstrations, we apply our method to the ferromagnetic Ising
models on random networks. The free energy of the ferromagnetic Ising model on
random networks with trivalent vertices is shown to exactly reproduce that of
the ferromagnetic Ising model on the Bethe lattice. We also consider the cases
with heterogeneity with mixtures of orders of vertices, and derive the known
formula of the Curie temperature.
