Mean-field and fluctuations for hub dynamics in heterogeneous random networks

TL;DR

This paper analyzes the behavior of hub nodes in heterogeneous random networks, showing that their fluctuations are small over long times and explaining phenomena like desynchronization and fluctuation scaling.

Contribution

It provides the first mathematical characterization of mean-field fluctuations in networks with power-law degree distributions, including Berry-Esseen estimates.

Findings

Fluctuations are small over exponentially long time scales.

Established the scaling relation between system size and fluctuation frequency.

Explained system size induced desynchronization.

Abstract

In a class of heterogeneous random networks, where each node dynamics is a random dynamical system, interacting with neighbor nodes via a random coupling function, we characterize the hub behavior as 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 a mathematical explanation for several numerical observations, including the scaling relation between system size and frequency of large fluctuations, as well as system size induced desynchronization. To our best knowledge, these are the first characterizations of mean-field fluctuations on networks with a degree distribution that follows a power-law, a common feature for many realistic systems.