Probabilistic reduction principle in complex network dynamics

• Grant number: 21/11091-3
• Support type: Scholarships abroad - Research Internship - Doctorate
• Effective date (Start): December 20, 2021
• Effective date (End): December 19, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal researcher: Tiago Pereira da Silva
• Grantee: Zheng Bian
• Supervisor abroad: Jeroen Steven Willibrord Lamb
• Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Research place: Imperial College London, England
• Associated to the scholarship: 18/26107-0 - Dynamics of complex heterogeneous networks: reduction techniques, BP.DR

Abstract

This proposal aims to develop a rigorous dimension reduction principle for dynamical systems coupled on complex networks by comparing the low-degree-node dynamics against a reference random dynamical system (RDS). A recent breakthrough due to Pereira, van Strien, and Tanzi proved the dimension reduction for a special class of heterogeneously coupled maps, relying crucially on global hyperbolicity properties of the network dynamics. In particular, this means that their result does not apply in power-law graphs, which constitute important examples of real-world networks. Our strategy removes the restrictive conditions of global hyperbolicity, and introduces noise to dynamics on the low-degree nodes. Specifically, we propose the following objectives.1- Merge a probabilistic argument and a topological one.2- Large-scale numerics for heterogeneously coupled hyperbolic systems.These objectives will, on the one hand, significantly extend the dimension reduction results to realistic networks including power-law and hierarchical graph models, and on the other hand, pro- vide theoretical framework for applications including reconstruction problems where the isolated dynamics are high-dimensional and possess both expanding and contracting directions.