  Drawing inspiration from real world interacting systems we study a system
consisting of two networks that exhibit antagonistic and dependent
interactions. By antagonistic and dependent interactions, we mean, that a
proportion of functional nodes in a network cause failure of nodes in the
other, while failure of nodes in the other results in failure of links in the
first. As opposed to interdependent networks, which can exhibit first order
phase transitions, we find that the phase transitions in such networks are
continuous. Our analysis shows that, compared to an isolated network, the
system is more robust against random attacks. Surprisingly, we observe a region
in the parameter space where the giant connected components of both networks
start oscillating. Furthermore, we find that for Erdos-Renyi and scale free
networks the system oscillates only when the dependency and antagonism between
the two networks is very high. We believe that this study can further our
understanding of real world interacting systems.
