Percolation on Interdependent Networks with a Fraction of Antagonistic Interactions

Recently, the percolation transition has been characterized on interacting networks both in presence of interdependent interactions and in presence of 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 discontinuous and continuous 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.