Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Georgi S. Medvedev

TL;DR

This paper extends the graphon IPS framework to self-similar fractal domains, establishing a continuum limit and mean-field (Vlasov) description for interacting particle systems on fractal sets via an explicit isomorphism with graphon IPS. It introduces a Galerkin discretization and a rate-of-convergence theory based on generalized Lipschitz spaces on fractals, enabling optimal convergence estimates for nonlocal equations on fractal domains. The results demonstrate that macroscopic dynamics on fractal domains emerge as limits of discretizations of fractal sets, and they provide a rigorous link between fractal IPS and their graphon counterparts. The work opens avenues for nonlinear nonlocal diffusion on fractals and synchronization phenomena on hierarchical networks, with implications for porous media, neural and network dynamics, and numerical methods on fractal geometries.

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.

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

TL;DR

This paper extends the graphon IPS framework to self-similar fractal domains, establishing a continuum limit and mean-field (Vlasov) description for interacting particle systems on fractal sets via an explicit isomorphism with graphon IPS. It introduces a Galerkin discretization and a rate-of-convergence theory based on generalized Lipschitz spaces on fractals, enabling optimal convergence estimates for nonlocal equations on fractal domains. The results demonstrate that macroscopic dynamics on fractal domains emerge as limits of discretizations of fractal sets, and they provide a rigorous link between fractal IPS and their graphon counterparts. The work opens avenues for nonlinear nonlocal diffusion on fractals and synchronization phenomena on hierarchical networks, with implications for porous media, neural and network dynamics, and numerical methods on fractal geometries.

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.
Paper Structure (27 sections, 13 theorems, 151 equations, 8 figures)

This paper contains 27 sections, 13 theorems, 151 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Graphon interacting particle systems
  3. Models and examples
  4. Continuum limit
  5. Mean-field limit
  6. Convergence rate
  7. Background on fractals
  8. Iterated Function Systems
  9. Symbolic space
  10. Self-similar measures
  11. Self-similar IPS
  12. The model
  13. Galerkin discretization
  14. W-random self-similar networks and continuum limit
  15. Isomorphisms
  16. ...and 12 more sections

Key Result

Theorem 2.7

Med14aMed14b Let $u(t,x)$ be the solution of the IVP for cKM subject to $u(0,\cdot)=g\in L^2([0,1])$. Likewise, solve the IVPs for KM and W-KM, respectively, and satisfy Then for $v\in\{ u^n, \bar{u}^n\}$ Here, in case $v=\bar{u}^n$ estimate approx-heat holds almost surely with respect the probability measure used in the construction of random graphs Bernoulli. Recall that the norm in $C(0,T; L^

Figures (8)

  • Figure 1: a, b, c) Three consecutive prefractals. d) SG.
  • Figure 2: Examples of attractors of IFS: the three-level SG, $SG_3$, the pentagasket, and the hexagasket (see Str06 for more details).
  • Figure 3: a$W$ takes values $1-p$ and $p$ over the black and white regions respectively. b, c Pixel plots of the deterministic weighted network \ref{['weights']} and random graph \ref{['Bernoulli']}.
  • Figure 4: An example of a fat fractal. a, b) The first two steps of the construction of golden gasket. c) Approximation of the golden gasket.
  • Figure 5: The root and first two levels of a tree representing SG.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Example 2.10
  • ...and 31 more