Triple Point in Correlated Interdependent Networks

Many real-world networks depend on other networks, often in non-trivial ways, to maintain their functionality. These interdependent "networks of networks" are often extremely fragile. When a fraction $1-p$ of nodes in one network randomly fails, the damage propagates to nodes in networks that are interdependent and a dynamic failure cascade occurs that affects the entire system. We present dynamic equations for two interdependent networks that allow us to reproduce the failure cascade for an arbitrary pattern of interdependency. We study the "rich club" effect found in many real interdependent network systems in which the high-degree nodes are extremely interdependent, correlating a fraction $\alpha$ of the higher degree nodes on each network. We find a rich phase diagram in the plane $p-\alpha$, with a triple point reminiscent of the triple point of liquids that separates a non-functional phase from two functional phases.