# Fluctuation of Dynamical Robustness in a Networked Oscillators System

## TL;DR

This paper investigates how the interplay between active and inactive oscillators affects the robustness of networked systems, revealing the influence of network topology and inactivation strategies on critical fluctuations.

## Contribution

It analytically links dynamical robustness to cross link density and explores how network structure impacts fluctuation behavior, supported by numerical validation.

## Key findings

- Fluctuation of critical points depends on cross link density.
- Heterogeneous networks exhibit more fluctuation than homogeneous ones.
- Low-degree nodes are key to dynamical robustness.

## Abstract

In this work, we study the dynamical robustness in a system consisting of both active and inactive oscillators. We analytically show that the dynamical robustness of such system is determined by the cross link density between active and inactive subpopulations, which depends on the specific process of inactivation. It is the multi-valued dependence of the cross link density on the control parameter, i.e., the ratio of inactive oscillators in the system, that leads to the fluctuation of the critical points. We further investigate how different network topologies and inactivation strategies affect the fluctuation. Our results explain why the fluctuation is more obvious in heterogeneous networks than in homogeneous ones, and why the low-degree nodes are crucial in terms of dynamical robustness. The analytical results are supported by numerical verifications.