  We investigate thermodynamic phase transitions of the joint presence of spin
glass (SG) and random field (RF) using a random graph model that allows us to
deal with the quenched disorder. Therefore, the connectivity becomes a
controllable parameter in our theory, allowing us to answer what the
differences are between this description and the mean-field theory i.e., the
fully connected theory. We have considered the random network random field
Ising model where the spin exchange interaction as well as the RF are random
variables following a Gaussian distribution. The results were found within the
replica symmetric (RS) approximation, whose stability is obtained using the
two-replica method. This also puts our work in the context of a broader
discussion, which is the RS stability as a function of the connectivity. In
particular, our results show that for small connectivity there is a region at
zero temperature where the RS solution remains stable above a given value of
the magnetic field no matter the strength of RF. Consequently, our results show
important differences with the crossover between the RF and SG regimes
predicted by the fully connected theory.
