# Synchronization of Network-Coupled Oscillators with Uncertain Dynamics

## TL;DR

This paper develops a probabilistic framework to analyze how uncertainty in natural frequencies affects synchronization in complex networks of oscillators, revealing robustness factors and network properties that promote synchronization.

## Contribution

It extends the Synchrony Alignment Function (SAF) to account for uncertain, heterogeneous natural frequencies using probability theory, providing new insights into synchronization robustness.

## Key findings

- Uncertainty in natural frequencies can enhance synchronization robustness.
- Network properties like degree-frequency correlation promote synchronization.
- The extended SAF framework predicts synchronization behavior under uncertainty.

## Abstract

Synchronization of network-coupled dynamical units is important to a variety of natural and engineered processes including circadian rhythms, cardiac function, neural processing, and power grids. Despite this ubiquity, it remains poorly understood how complex network structures and heterogeneous local dynamics combine to either promote or inhibit synchronization. Moreover, for most real-world applications it is impossible to obtain the exact specifications of the system, and there is a lack of theory for how uncertainty affects synchronization. We address this open problem by studying the Synchrony Alignment Function (SAF), which is an objective measure for the synchronization properties of a network of heterogeneous oscillators with given natural frequencies. We extend the SAF framework to analyze network-coupled oscillators with heterogeneous natural frequencies that are drawn as a multivariate random vector. Using probability theory for quadratic forms, we obtain expressions for the expectation and variance of the SAF for given network structures. We conclude with numerical experiments that illustrate how the incorporation of uncertainty yields a more robust theoretical framework for enhancing synchronization, and we provide new perspectives for why synchronization is generically promoted by network properties including degree-frequency correlations, link directedness, and link weight delocalization.