[论文解读] Interacting dynamical systems on networks and fractals: discrete and continuous models, mean-field limit, and convergence rates

Georgi S. Medvedev|arXiv (Cornell University)|Jan 30, 2026

Theoretical and Computational Physics被引用 0

一句话总结

这篇论文为自相似网络上的相互作用粒子系统发展了连续极限和平均场理论，建立了与图卷 IPS 的同构，并推导了分形域非局部模型的收敛性及速率结果。

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.

研究动机与目标

• Motivate the study of IPS on self-similar networks as models for heterogeneous fractal-like media and hierarchical networks.
• Derive continuum (nonlocal) evolution equations on fractal domains that describe macroscopic dynamics.
• Establish an explicit isomorphism between self-similar IPS and graphon IPS to justify limits.
• Develop convergence rate results for discrete self-similar models to their continuum/Vlasov limits.
• Bridge discrete Galerkin discretizations on fractals with nonlocal PDEs on fractal domains.

提出的方法

• Formulate IPS on self-similar networks using graphon-inspired weights and interaction functions.
• Derive the continuum nonlocal evolution equation on fractal domains and the Vlasov-type mean-field limit.
• Construct an isomorphism between self-similar sets and the unit interval to relate IPS on fractals to graphon IPS.
• Introduce generalized Lipschitz (Nikolskii–Besov) type spaces on fractals to quantify convergence rates.
• Analyze convergence of L2-projections onto piecewise-constant subspaces and obtain rate estimates.
• Discuss Galerkin discretization of nonlocal fractal problems and provide DG-method convergence framework.

实验结果

研究问题

• RQ1 How can we formulate continuum and mean-field limits for interacting particle systems on self-similar fractal domains?
• RQ2 What is the precise relationship between self-similar IPS and graphon IPS, and how does this enable limits?
• RQ3 What are the convergence rates for discretized self-similar models to their fractal-domain continuum limits?
• RQ4 How can we extend Lipschitz-type regularity to fractal domains to obtain error estimates for Galerkin discretizations?
• RQ5 Can self-similar networks be interpreted as Galerkin discretizations of nonlocal fractal PDEs?

主要发现

• An explicit isomorphism is established between self-similar IPS and graphon IPS, justifying continuum and mean-field limits on fractal domains.
• Convergence rates for discrete self-similar models are derived using a generalized Lipschitz framework on fractals.
• Generalized Lipschitz (Nicolskii–Besov) spaces on fractals are introduced to measure regularity and bound projection errors.
• Discontinuous Galerkin discretizations of nonlocal equations on fractal domains are analyzed, with optimal convergence estimates tied to kernel regularity.
• Discrete networks on self-similar sets converge to nonlocal fractal evolution equations, showing macroscopic dynamics emerge from fractal discretizations.
• Self-similar networks are interpreted as Galerkin discretizations of fractal-domain nonlocal problems, linking numerical analysis and fractal PDEs.