Threshold graphs are globally synchronizing

TL;DR

This paper proves that threshold graphs, which can have arbitrary edge densities and extremal degree sequences, ensure global synchronization in the Kuramoto model, even with sparse connectivity.

Contribution

It demonstrates that threshold graphs guarantee global synchronization in the Kuramoto model, extending beyond dense graphs and leveraging their structural symmetries.

Findings

Threshold graphs ensure global synchronization in the Kuramoto model.

Threshold graphs can have arbitrary edge densities.

The analysis uses a phasor-geometric characterization of stationary points.

Abstract

The Kuramoto model can be formulated as a gradient flow on a nonconvex energy landscape of the form $E(\boldsymbol{\theta}) := \frac{1}{2} \sum_{1\le i,j\le n} A_{ij}\bigl(1-\cos(\theta_i-\theta_j)\bigr).$ A fundamental question is to identify graph structures for which this landscape is benign, in the sense that every second-order stationary point corresponds to a fully synchronized state. This property guarantees that all trajectories of the Kuramoto model converge to a fully synchronized state except for a measure-zero set of initial conditions, a phenomenon known as global synchronization. Existing guarantees typically require that each node be connected to a sufficiently large fraction of the other nodes, enforcing high graph density. In this work, we show that threshold graphs lie well outside this regime while still exhibiting global synchronization. In particular, threshold graphs realize arbitrary edge densities and have degree sequences that are extremal in the sense of majorization. Our analysis is based on a phasor--geometric characterization of stationary points that exploits the structural and geometric symmetries induced by threshold graphs.