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

We develop a computational framework to study macroscopic dynamics in complex agent-based network models by combining Topological Data Analysis (TDA) with the Equation-Free Method, and demonstrate it on Erdős–Rényi random networks. Our approach builds a TDA filtration driven by the density of activated nodes to extract a coarse macroscopic topological observable based on persistent Betti numbers, yielding a low-dimensional yet informative representation. Within the Equation-Free setting, we construct a lifting procedure using topological information and infer a data-driven evolution law for the resulting macroscopic variable. We then use numerical bifurcation and stability analysis to characterize global behavior and identify qualitative transitions in the emergent dynamics.