  We study the death and restoration of collective oscillations in networks of
oscillators coupled through random-walk diffusion. Differently than the usual
diffusion coupling used to model chemical reactions, here the equilibria of the
uncoupled unit is not a solution of the coupled ensemble. Instead, the
connectivity modifies both, the original unstable fixed point and the stable
limit-cycle, making them node-dependent. Using numerical simulations in random
networks we show that, in some cases, this diffusion induced heterogeneity
stabilizes the initially unstable fixed point via a Hopf bifurcation. Further
increasing the coupling strength the oscillations can be restored as well. Upon
numerical analysis of the stability properties we conclude that this is a novel
case of amplitude death. Finally we use a heterogeneous mean-field reduction of
the system in order to proof the robustness of this phenomena upon increasing
the system size.
