# Energy Approach from varepsilon-Graph to Continuum Diffusion Model with Connectivity Functional

arXiv: https://arxiv.org/abs/2510.25114 · HTML: https://mlsanity.com/paper/2510.25114/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 29 Oct 2025.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 9 of 9)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives](https://mlsanity.com/paper/2305.08466/) (2023)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [The game theoretic p-Laplacian and semi-supervised learning with few labels](https://mlsanity.com/paper/1711.10144/) (2017)
- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
- [Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds](https://mlsanity.com/paper/2506.00171/) (2025)
- [Convergence rates for Poisson learning to a Poisson equation with measure data](https://mlsanity.com/paper/2407.06783/) (2024)
- [Optimal Control For Anti-Abeta Treatment in Alzheimer's Disease using a Reaction-Diffusion Model](https://mlsanity.com/paper/2504.07913/) (2025)

## Abstract (arXiv)

We derive an energy-based continuum limit for varepsilon-graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most O(varepsilon); the prefactor involves only the W1,1-norm of the connectivity density as varepsilon→0, so the error bound remains valid even when that density has strong local fluctuations. As an application, we introduce a neural-network procedure that reconstructs the connectivity density from edge-weight data and then embeds the resulting continuum model into a brain-dynamics framework. In this setting, the usual constant diffusion coefficient is replaced by the spatially varying coefficient produced by the learned density, yielding dynamics that differ significantly from those obtained with conventional constant-diffusion models.