A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios

TL;DR

This paper develops a unified framework for analyzing the stability of complex networks with diverse edge types, using matrix phases and asymmetry ratios, leading to less conservative stability conditions.

Contribution

It introduces a comprehensive formulation for Laplacian couplings that accommodates directed, multi-dimensional, and adaptive edges, expanding the scope of stability analysis.

Findings

Unified stability conditions for power grids, diffusion, and Kuramoto models.

Matrix phases effectively capture network stability properties.

Asymmetry Rayleigh Ratio quantifies edge asymmetry impact.

Abstract

We study the properties and stability of networks with arbitrary Laplacian coupling. Classic approaches to studying networked systems require unrealistic assumptions, including homogeneous node dynamics, one-dimensional and undirected edges, or constant edge weights. We develop a unified formulation of Laplacian-style couplings that drops these assumptions, providing a unified notion for the edge weights of adaptive, directed, and multi-dimensional edges. We show that the recently developed theory of matrix phases can capture essential stability properties of the network and its edges. We quantify the impact of the asymmetry of the higher-dimensional edge dynamics on the system's phase properties by introducing the Asymmetry Rayleigh Ratio. These theoretical advances allow us to derive new sufficient stability conditions for AC power grids, directed diffusion, and the Kuramoto-Sakaguchi model. The resulting conditions are less conservative than the specific results known for these systems.