Avalanche Collapse of Interdependent Network
  We reveal the nature of the avalanche collapse of the giant viable component in multiplex networks under perturbations such as random damage. Specifically, we identify latent critical clusters associated with the avalanches of random damage. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge. 
G. J. Baxter, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes
cond-mat.stat-mech math-ph math.MP math.PR