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 $s\ge 3$ 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.

• Received 22 March 2016
• Corrected 29 November 2016

DOI:https://doi.org/10.1103/PhysRevE.94.042304