Complex networks rarely appear in isolation. From critical infrastructure, to physiology and the human brain, we observe that functionality of nodes in one system promotes or suppresses the functionality of nodes in another. This feature is studied intensively in interdependent percolation where functionality is identified with connectivity, but its application in more realistic, dynamic systems has remained elusive. Here we will discuss two new directions: process-based and dynamic interdependence. In the first, we will see how interdependent percolation can be implemented in a system where the functionality is determined by the existence of flow through a node and not by connectivity alone. Next, we will see how the basic concept of a dependency link can be implemented in a broad class of dynamic systems by linking the coupling of a node in one layer with the local order of a node in another layer. With this approach, competition or more exotic cross-network interactions are straightforward to model. When applied to oscillator networks, we find a wealth of realistic new phenomena including explosive synchronization, hysteresis, multi-stability and chaotic attractors. This simple yet powerful approach opens up a way to model and understand the interacting multi-layer networks which surround us.
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