Percolation in interdependent and interconnected networks: Abrupt change from second- to first-order transitions

Robustness of two coupled networks systems has been studied separately only for dependency coupling and only for connectivity coupling. Here we study, using a percolation approach, a more realistic coupled networks system where both interdependent and interconnected links exist. We find rich and unusual phase-transition phenomena including hybrid transition of mixed first and second order, i.e., discontinuities like in a first-order transition of the giant component followed by a continuous decrease to zero like in a second-order transition. Moreover, we find unusual discontinuous changes from second-order to first-order transition as a function of the dependency coupling between the two networks.