Percolation in Interdependent and Interconnected Networks: Abrupt Change   from Second to First Order Transition
  Robustness of two coupled networks system has been studied only for dependency coupling (S. Buldyrev et. al., Nature, 2010) and only for connectivity coupling (E. A. Leicht and R. M. D'Souza, arxiv:09070894). Here we study, using a percolation approach, a more realistic coupled networks system where both interdependent and interconnected links exist. We find a rich and unusual phase transition phenomena including hybrid transition of mixed first and second order i.e., discontinuities like a first order transition of the giant component followed by a continuous decrease to zero like 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. 
Yanqing Hu, Baruch Ksherim, Reuven Cohen, Shlomo Havlin
physics.soc-ph cs.SI