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Robustness of coupled oscillator networks with heterogeneous natural frequencies.

Robustness of coupled oscillator networks against local degradation of oscillators has been intensively studied in this decade. The oscillation behavior on the whole network is typically reduced with an increase in the fraction of degraded (inactive) oscillators. The critical fraction of inactive oscillators, at which a transition from an oscillatory to a quiescent state occurs, has been used as a measure for the network robustness. The larger (smaller) this measure is, the more robust (fragile) the oscillatory behavior on the network is. Most previous studies have used oscillators with identical natural frequencies, for which the oscillators are necessarily synchronized and thereby the analysis is simple. In contrast, we focus on the effect of heterogeneity in the natural frequencies on the network robustness. First, we analytically derive the robustness measure for the coupled oscillator models with heterogeneous natural frequencies under some conditions. Then, we show that increasing the heterogeneity in natural frequencies makes the network fragile. Moreover, we discuss the optimal parameter condition to maximize the network robustness.