# When is sync globally stable in sparse networks of identical Kuramoto oscillators?

## TL;DR

This paper investigates the conditions under which synchronization is globally stable in networks of identical Kuramoto oscillators, revealing structural graph properties that lead to stable non-synchronous states.

## Contribution

It introduces an algorithm to identify initial conditions outside the basin of attraction and characterizes graph structures that prevent global synchronization.

## Key findings

- Graphs with cyclic clustering or long induced paths can support stable non-synchronous states.
- The paper provides criteria for when the synchronized state is not globally stable.
- It extends understanding of synchronization stability in complex networks.

## Abstract

Synchronization in systems of coupled Kuramoto oscillators may depend on their natural frequencies, coupling, and underlying networks. In this paper, we reduce the alternatives to only one by considering identical oscillators where the only parameter that is allowed to change is the underlying network. While such a model was analyzed over the past few decades by studying the size of the basin of attraction of the synchronized state on restricted families of graphs, here we address a qualitative question on general graphs. In an analogy to resistive networks with current sources, we describe an algorithm that produces initial conditions that are often outside of the basin of attraction of the synchronized state. In particular, if a graph allows a cyclic graph clustering with a sufficient number of clusters or contains a sufficiently long induced subpath without cut vertices of the graph then there is a non-synchronous stable phase-locked solution. Thus, we provide a partial answer to when the synchronized state is not globally stable.