Graph Homology and Stability of Coupled Oscillator Networks

TL;DR

This paper links the stability analysis of coupled oscillator networks to graph homology, providing new computational methods and explicit formulas for specific graph structures, with applications to models like Kuramoto.

Contribution

It introduces a dual homology-based approach to stability analysis, improving computational efficiency for sparse graphs and deriving explicit stability criteria for certain network topologies.

Findings

Homology-based stability calculation is effective for sparse graphs.

Explicit formulas for unstable manifold dimensions in graphs with loops.

Determination of maximum stable edge length in ring Kuramoto models.

Abstract

There are a number of models of coupled oscillator networks where the question of the stability of fixed points reduces to calculating the index of a graph Laplacian. Some examples of such models include the Kuramoto and Kuramoto--Sakaguchi equations as well as the swing equations, which govern the behavior of generators coupled in an electrical network. We show that the index calculation can be related to a dual calculation which is done on the first homology group of the graph, rather than the vertex space. We also show that this representation is computationally attractive for relatively sparse graphs, where the dimension of the first homology group is low, as is true in many applications. We also give explicit formulae for the dimension of the unstable manifold to a phase-locked solution for graphs containing one or two loops. As an application, we present some novel results for the Kuramoto model defined on a ring and compute the longest possible edge length for a stable solution.