Leaders do not look back, or do they?
Alexander N. Gorban, Nick Jarman, Erik Steur, Cees van Leeuwen, Ivan, Tyukin

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 of this matrix, the following inequality holds: . 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
