Internal reliability and anti-reliability in dynamical networks

TL;DR

This paper introduces the concepts of internal reliability and anti-reliability in finite dynamical networks, analyzing their properties through Lyapunov exponents and applying these ideas to Kuramoto and neural network models.

Contribution

It defines and quantifies reliability in dynamical networks, providing new insights into synchronization and anti-synchronization behaviors in these systems.

Findings

Peripheral units are anti-reliable before synchronization in Kuramoto models.

Central units are reliable prior to synchronization.

Large sub-networks tend to be anti-reliable.

Abstract

We consider finite dynamical networks and define internal reliability according to 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 anti-reliable. Quantification of reliability with the transversal Lyapunov exponent allows for a straightforward analysis of different models. For a Kuramoto model of globally coupled phase oscillators with a distribution of natural frequencies, we show that prior to the onset of synchronization, peripheral in frequency units are anti-reliable, while central are reliable. For this model, reliability can be expressed via phase correlations in a sort of a fluctuation-dissipation relation. Sufficiently large sub-networks in the Kuramoto model are always anti-reliable; the same holds for a recurrent neural network, where individual units are always reliable.