Networks of noisy oscillators with correlated degree and frequency dispersion

We investigate how correlations between the diversity of the connectivity of networks andthe dynamics at their nodes affect the macroscopic behavior. In particular, we study thesynchronization transition of coupled stochastic phase oscillators that represent the nodedynamics. Crucially in our work, the variability in the number of connections of the nodesis correlated with the width of the frequency distribution of the oscillators. Bynumerical simulations on Erdös-Rényi networks, where the frequencies of the oscillatorsare Gaussian distributed, we make the counterintuitive observation that an increase in thestrength of the correlation is accompanied by an increase in the critical couplingstrength for the onset of synchronization. We further observe that the critical couplingcan solely depend on the average number of connections or even completely lose itsdependence on the network connectivity. Only beyond this state, a weighted mean-fieldapproximation breaks down. If noise is present, the correlations have to be stronger toyield similar observations.