Synchronisation Properties of Trees in the Kuramoto Model

TL;DR

This paper analyzes the synchronization properties of tree networks in the Kuramoto model, deriving explicit formulas and bounds for the critical coupling needed for synchronization.

Contribution

It provides the first closed-form expression for the critical coupling in tree networks and tight bounds for various classes of trees and frequency distributions.

Findings

Closed-form expression for critical coupling in trees

Tight bounds for different tree classes and distributions

Frequency rearrangements can bound critical coupling

Abstract

We consider the Kuramoto model of coupled oscillators, specifically the case of tree networks, for which we prove a simple closed-form expression for the critical coupling. For several classes of tree, and for both uniform and Gaussian vertex frequency distributions, we provide tight closed form bounds and empirical expressions for the expected value of the critical coupling. We also provide several bounds on the expected value of the critical coupling for all trees. Finally, we show that for a given set of vertex frequencies, there is a rearrangement of oscillator frequencies for which the critical coupling is bounded by the spread of frequencies.