Mean-field reduction in sparse networks

• Autor(a) principal: Mendonça, Hans Muller Junho de
• Data de Publicação: 2023
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/55/55134/tde-10012024-105940/
• topic: Dense networks; Redes densas; Redes esparsas; Sincronização; Sparse networks; Synchronization

Synchronization is a phenomenon observed in various scientific fields, ranging from mechanical and biological systems to social behavior. The Kuramoto model, developed in the 1970s and 1980s, revolutionized the understanding of spontaneous synchronization in large systems of interacting elements. In this model, synchronization is quantified using the order parameter, which represents the centroid of points distributed on the unit circle. The Kuramoto model revealed the existence of three distinct states: asynchronous, partially synchronous, and completely synchronous. While the classic Kuramoto model assumes an all-to-all network configuration, most real-world networks are sparse. Understanding synchronization in sparse networks and the effects of finite system sizes on synchronization is a challenging research problem. To address this problem, we adopt a dynamical system framework using Moebius maps on the complex unit circle. We investigate the transition to synchronization in both dense and sparse complex networks, where systems evolve through maps instead of ordinary differential equations. We explore the effects of finite system sizes on synchronization phenomena and examine the scaling behavior of the mean time to synchronization. Surprisingly, we discover that the incoherent state can be meta-stable for certain coupling strengths and link densities, challenging conventional assumptions. By analyzing mean-field equations, we construct a bifurcation diagram for infinitely large networks and observe the presence of chaotic transients with exponentially distributed escape times. This suggests that the system experiences transient periods of asynchrony before reaching a synchronized state. Our research provides a comprehensive understanding of synchronization in complex networks, shedding light on the behavior of real-world systems. It contributes valuable insights into the dynamics of finite-sized networks and challenges existing assumptions. Our findings have implications for network dynamics and enhance our understanding of synchronization phenomena in diverse systems.