# Mean-field and Fluctuations for Hub Dynamics in Heterogeneous Random Networks

## TL;DR

This paper analyzes how hubs behave in large, randomly connected networks with power-law degree distributions.

## Contribution

The paper provides a theoretical characterization of hub dynamics with statistically controlled fluctuations and proves fluctuation bounds over long time scales.

## Key findings

- Fluctuations in hub dynamics remain small over exponentially long time scales.
- The paper derives Berry-Esseen estimates for fluctuation statistics at fixed times.
- Results explain numerical observations like Gaussian fluctuations and system size-induced desynchronization.

## Abstract

We study a class of heterogeneous random networks, where the network degree distribution follows a power-law, and each node dynamics is a random dynamical system, interacting with neighboring nodes via a random coupling function. We characterize the hub behavior by the mean-field, subject to statistically controlled fluctuations. In particular, we prove that the fluctuations are small over exponentially long time scales and obtain Berry-Esseen estimates for the fluctuation statistics at any fixed time. Our results provide an explanation for several numerical observations, namely, a scaling relation between system size and frequency of large fluctuations, the system size induced desynchronization, and the Gaussian behavior of the fluctuations.