Cascading failures in interdependent networks with finite functional components

We present a cascading failure model of two interdependent networks in which functional nodes belong to components of size greater than or equal to s. We find theoretically and via simulation that in complex networks with random dependency links the transition is first order for s >= 3 and continuous for s = 2. We also study interdependent lattices with a distance constraint r in the dependency links and find that increasing r moves the system from a regime without a phase transition to one with a second-order transition. As r continues to increase, the system collapses in a first-order transition. Each regime is associated with a different structure of domain formation of functional nodes.