Percolation on interacting, antagonistic networks

Recently, new results on percolation of interdependent networks have shown that the percolation transition can be first order. In this paper we show that, when considering antagonistic interactions between interacting networks, the percolation process might present a bistability of the equilibrium solution. To this end, we introduce antagonistic interactions for which the functionality, or activity, of a node in a network is incompatible with the functionality, of the linked nodes in the other interacting networks. In particular, we study the percolation transition in two interacting networks with purely antagonistic interaction and different topology. For two antagonistic Poisson networks of different average degree we found a large region in the phase diagram in which there is a bistability of the steady state solutions of the percolation process, i.e. we can find that either one of the two networks might percolate. For two antagonistic scale-free networks we found that there is a region in the phase diagram in which, despite the antagonistic interactions, both networks are percolating. Finally we characterize the rich phase diagram of the percolation problems on two antagonistic networks, the first one of the two being a Poisson network and the second one being a scale-free network.