  Recently, the percolation transition has been characterized on interacting
networks both in presence of interdependent and antagonistic interactions. Here
we characterize the phase diagram of the percolation transition in two Poisson
interdependent networks with a percentage q of antagonistic nodes. We show that
this system can present a bistability of the steady state solutions, and both
first, and second order phase transitions. In particular, we observe a
bistability of the solutions in some regions of the phase space also for a
small fraction of antagonistic interactions 0<q<0.4. Moreover, we show that a
fraction q>q_c=2/3 of antagonistic interactions is necessary to strongly reduce
the region in phase-space in which both networks are percolating. This last
result suggests that interdependent networks are robust to the presence of
antagonistic interactions. Our approach can be extended to multiple networks,
and to complex boolean rules for regulating the percolation phase transition.
