Internal reliability and antireliability in dynamical networks

We consider finite dynamical networks and define internal reliability based on the synchronization properties of a replicated unit or a set of units. If the states of the replicated units coincide with their prototypes, they are reliable; otherwise, if their states differ, they are antireliable. Quantification of reliability using the transversal Lyapunov exponent enables a straightforward analysis of different models. For a Kuramoto model of globally attractively coupled phase oscillators with a distribution of natural frequencies, we show that before the onset of synchronization peripheral frequency units (at the edges of the distribution) are antireliable, while central ones (with natural frequencies close to the mean frequency) are reliable. For repulsive coupling, the complementary configuration occurs, with central antireliable units and peripheral reliable ones. For this model, reliability can be expressed through phase correlations in a sort of fluctuation-dissipation relation. Sufficiently large subnetworks in the Kuramoto model are always antireliable; the same holds for a recurrent neural network, where individual units are always reliable. Furthermore, we show that other coupled oscillator models (Winfree-type coupled phase oscillators, coupled rotators, and coupled Stuart-Landau oscillators) demonstrate patterns of reliability and antireliability similar to those of the Kuramoto setup.