Collective excitability in highly diluted random networks of oscillators

Collective excitable phenomena in a system of coupled elements may arise either when the single nodes display excitable dynamics or not. The latter case, more surprisingly, is possible when the entire ensemble of nodes is subject to a global feedback depending self-consistently on the level of synchronization of the network. In this case a global excitable response to an external stimulus can be observed, which corresponds, at the microscopic level, to a transient partial synchronization of the nodes. Mathematically, this is related to hidden geometric structures that organize the mean field trajectories in the phase-space. These events have been observed in two paradigmatic classes of globally-coupled oscillators, namely, the Kuramoto model with and without inertia. In this paper we analyze the robustness of the collective excitability in highly-diluted random networks, by gradually decreasing the percentage of coupled nodes. We consider global and partial stimulation protocols and we characterize the response with respect to that achievable in the corresponding fully-coupled network. Our findings demonstrate a remarkable robustness of the collective excitability, which we expect to inspire new research in the study of emergent phenomena in networks of interacting elements at the mesoscopic scale.