Data-driven macroscopic dynamics of complex networks using Topological Data Analysis and the Equation-Free Method

TL;DR

This paper introduces a novel computational framework combining Topological Data Analysis and the Equation-Free Method to analyze the large-scale dynamics of complex networks, demonstrated on Erdős–Rényi networks, revealing key topological features and dynamic behaviors.

Contribution

The work develops a new integrated approach using topological data analysis and the Equation-Free Method for macroscopic analysis of complex network dynamics, including a novel lifting procedure and evolution law.

Findings

Effective reduction of data dimensionality via persistent Betti numbers

Successful application to bifurcation and stability analysis

Identification of qualitative transitions in network dynamics

Abstract

In this work, we present a computational framework for exploring and analyzing the macroscopic dynamics of complex agent-based network models by integrating Topological Data Analysis with the Equation-Free Method. To demonstrate the effectiveness of our method, we apply it to Erd\H{o}s--R\'enyi-type random networks. Central to our approach is a Topological Data Analysis-based filtration process driven by the density of activated network nodes (agents), from which we extract a coarse-grained macroscopic topological observable. This observable is defined via persistent Betti numbers, thus requiring significantly reduced data dimensionality while retaining essential topological features. Subsequently, within the Equation-Free Method framework, we show firstly that a \textit{lifting procedure} can be achieved using topological properties and secondly, a data-driven evolution law that governs the dynamics of this macroscopic variable. Finally, we perform a numerical bifurcation and stability analysis to investigate the global behavior and qualitative transitions of the emergent macroscopic dynamics.