Engineering mesoscale structures with distinct dynamical implications in networks of delay-coupled delay oscillators

TL;DR

This paper demonstrates that specific mesoscale subgraphs with symmetries in delay-coupled oscillator networks have distinct dynamical effects, allowing localized analysis of instabilities that influence overall system behavior.

Contribution

It introduces a method to analyze network instabilities by focusing on symmetric mesoscale subgraphs, simplifying the study of complex delay-coupled systems.

Findings

Eigenvectors localize on symmetric subgraphs

Eigenvalues become insensitive to external topology

Instabilities originate in small subgraphs but affect the entire system

Abstract

The dynamics of networks of interacting systems depends intricately on the interaction topology. When the dynamics is explored, generally the whole topology has to be considered. However, we show that there are certain mesoscale subgraphs that have precise and distinct consequences for the system-level dynamics. In particular, if meso-scale symmetries are present then eigenvectors of the Jacobian localise on the symmetric subgraph and the corresponding eigenvalues become insensitive to the topology outside the subgraph. Hence, dynamical instabilities associated with these eigenvalues can be analyzed without considering the topology outside the subgraph. While such instabilities are thus generated entirely in small network subgraphs, they generally do not remain confined to the subgraph once the instability sets in and thus have system-level consequences. Here we illustrate the analytical investigation of such instabilities in an ecological meta-population model consisting of a network of delay-coupled delay oscillators.