Explosive transitions to synchronization in networked phase oscillators

We introduce a condition for an ensemble of networked phase oscillators to feature an abrupt, first-order phase transition from an unsynchronized to a synchronized state. This condition is met in a very wide spectrum of situations, and for various oscillators' initial frequency distributions. We show that the occurrence of such transitions is always accompanied by the spontaneous emergence of frequency-degree correlations in random network architectures. We also discuss ways to relax the condition, and to further extend the possibility for the first-order transition to occur, and illustrate how to engineer magnetic-like states of synchronization. Our findings thus indicate how to search for abrupt transitions in real-world applications.