Graphical notation reveals topological stability criteria for collective dynamics in complex networks

TL;DR

This paper introduces a graphical notation that simplifies the analysis of spectral properties in complex networks, revealing topological stability criteria applicable across various systems, including the Kuramoto model.

Contribution

It presents a novel graphical notation for spectral analysis and derives universal topological stability criteria for networks of coupled dynamical units.

Findings

Stability requires the Coates graph to contain a spanning tree of positive elements.

The notation allows concise topological interpretation of spectral properties.

Applicable to systems like the Kuramoto model for local stability analysis.

Abstract

We propose a graphical notation by which certain spectral properties of complex systems can be rewritten concisely and interpreted topologically. Applying this notation to analyze the stability of a class of networks of coupled dynamical units, we reveal stability criteria on all scales. In particular, we show that in systems such as the Kuramoto model the Coates graph of the Jacobian matrix must contain a spanning tree of positive elements for the system to be locally stable.