# A universal order parameter for synchrony in networks of limit cycle oscillators

## TL;DR

This paper introduces a universal order parameter for complex oscillator networks that effectively measures synchronization and phase locking, overcoming limitations of previous metrics and applicable across various systems.

## Contribution

The paper proposes a new, universal order parameter that accurately captures all synchronization transitions in finite networks, accounting for network topology and stability.

## Key findings

- Order parameter increases monotonically with coupling strength.
- It diverges at the critical coupling for full phase locking.
- Applicable to biological networks and power grids.

## Abstract

We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks. First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators. We rigorously proof that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network. Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where different types of synchrony are possible, including biological networks and power grids.