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The robustness of coupled networks against node failure has been of interest in the past several years, while most of the researches have considered a very strong node-coupling method, i.e., once a node fails, its dependency partner in the other network will fail immediately. However, this scenario cannot cover all the dependency situations in real world, and in most cases, some nodes cannot go so far as to fail due to theirs self-sustaining ability in case of the failures of their dependency partners. In this paper, we use the percolation framework to study the robustness of interdependent networks with weak node-coupling strength across networks analytically and numerically, where the node-coupling strength is controlled by an introduced parameter α. If a node fails, each link of its dependency partner will be removed with a probability 1 − α. By tuning the fraction of initial preserved nodes  p , we find a rich phase diagram in the plane  p− α, with a crossover point at which a first-order percolation transition changes to a second-order percolation transition.

Cascading failures in coupled networks: The critical role of node-coupling strength across networks