Rigorous mean-field dimensional reduction in heterogeneous network dynamics

Author(s)

Zheng Bian

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2024-08-06

Examining board members

Tiago Pereira da Silva; Edgar Matias da Silva; Ali Tahzibi; Matteo Tanzi

• Advisor: Tiago Pereira da Silva

Abstract

This thesis presents phenomenological and theoretical studies of 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. Some fundamental results from random graphs, network dynamics, Markov chains, and random dynamical systems are reviewed and reinterpreted. (AU)

• FAPESP's process: 18/26107-0 - Dynamics of complex heterogeneous networks: reduction techniques
• Grantee: Zheng Bian
• Support Opportunities: Scholarships in Brazil - Doctorate