Interacting dynamical systems on networks and fractals: discrete and continuous models, mean-field limit, and convergence rates

TL;DR

This paper develops a mean-field theory for interacting particle systems on fractal networks, establishing continuum limits, convergence rates, and connections to graphon models, with applications to physical and real-world hierarchical networks.

Contribution

It introduces a novel framework linking self-similar fractal networks with graphon-based models, extending mean-field theory to fractal domains and deriving optimal convergence rates.

Findings

Established an explicit isomorphism between self-similar IPS and graphon IPS.

Derived optimal convergence rates for discrete models on fractals.

Extended Lipschitz spaces to fractal domains for error estimates.

Abstract

We develop a continuum limit and mean-field theory for interacting particle systems (IPS) on self-similar networks, a new class of discrete models whose large-scale behavior gives rise to nonlocal evolution equations on fractal domains. This work extends the graphon-based framework for IPS, used to derive continuum and mean-field limits in the non-exchangeable setting, to situations where the spatial domain is fractal rather than Euclidean. The motivation arises from both physical models naturally formulated on fractals and real-world networks exhibiting hierarchical or quasi-self-similar structure. Our analysis relies on tools from fractal geometry, including Iterated Function Systems and self-similar measures. A central result is an explicit isomorphism between self-similar IPS and graphon IPS, which allows us to justify the continuum and mean-field limits in the self-similar setting. This connection reveals that macroscopic dynamics on fractal domains emerge naturally as limits of dynamics on appropriate discretizations of fractal sets. Another contribution of the paper is the derivation of optimal convergence rates for the discrete self-similar models. We introduce a scale of generalized Lipschitz spaces on fractals, extending the Nikolskii-Besov spaces used in the Euclidean setting, and obtain convergence estimates for discontinuous Galerkin approximations of nonlocal equations posed on fractal domains. These results apply to kernels with minimal regularity addressing models relevant in applications.