Leaders do not look back, or do they?

TL;DR

This paper investigates how adding feedback to a directed chain affects system dynamics, revealing bounds on eigenvalues, synchronization thresholds, and the emergence of rotating waves, emphasizing the impact of network topology on behavior.

Contribution

It provides new bounds on eigenvalues for directed networks, analyzes synchronization conditions, and uncovers dynamic regimes like rotating waves in oscillator networks.

Findings

Eigenvalue bounds depend on network size and structure.

Synchronization requires connection strengths above a critical threshold.

Large networks exhibit multiple dynamic regimes, including rotating waves.

Abstract

We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue $\lambda$ of this matrix, the following inequality holds: $\frac{|\Im \lambda |}{|\Re \lambda |} \leq \cot\frac{\pi}{n}$. This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower bound $\sigma_c$ for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: $\sigma_c=\frac{1}{1-\cos\left( 2\pi /n\right)} $. Numerical analysis revealed that, depending on the network size $n$, multiple dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state a rotating wave solution occurs. The effect is observed in networks exceeding a certain critical size. The emergence of a rotating wave highlights the importance of long chains and loops in networks of oscillators: the larger the size of chains and loops, the more sensitive the network dynamics becomes to removal or addition of a single connection.