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Explosive Synchronization is Discontinuous

Spontaneous explosive is an abrupt transition to collective behavior taking place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. This explosive transition was conjectured to be discontinuous. Indeed, numerical investigations reveal a hysteresis behavior associated with the transition. Here, we analyze explosive synchronization in star graphs. We show that in the thermodynamic limit the transition to (and out) collective behavior is indeed discontinuous. The discontinuous nature of the transition is related to the nonlinear behavior of the order parameter, which in the thermodynamic limit exhibits multiple fixed points. Moreover, we unravel the hysteresis behavior in terms of the graph parameters. Our numerical results show that finite size graphs are well described by our predictions.