Fig. 1. Fluctuation-induced dynamic response patterns.

A sample network (A) driven by Brownian noise (B) at a single unit k = 1 exhibiting (C) complex dynamic response patterns that nonlinearly vary with frequency content of the input signal (D, F, and H) as well as with graph-theoretic distance between the input and the response unit (E, G, and I). Three response classes emerge: homogeneous responses at low frequencies (D and E), spatiotemporally irregular patterns at intermediate frequencies (F and G), and localized responses at high frequencies (H and I). To identify frequency regimes, we selected specific frequency bands [yellow in (D), (F), and (H)] from the spectrum of the original noise realization (B), and all others are displayed in purple. (C), (E), (G), and (I) display the time series of driving signal at unit k = 1 (top) and the response in the phase velocities of the units i ∈ {1, …,4} (bottom). In (C), (E), (G), and (I), the time series of the band-filtered signal are displayed together with the responses (yellow for unit 1 and green for units 2 to 4). Our theory (Eq. 5) (thin black lines) well predicts the system responses obtained from direct numerical simulations. (For details of further settings, see Materials and Methods.)