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

2025-10-28

AI Summary

This work addresses the energy-consistent approximation of discrete ε-graph diffusion models by continuous counterparts. For ε-graphs with general connectivity functions—including strong local density variations—we construct a continuous-limit model based on an energy functional and rigorously prove that the discrepancy between discrete and continuous energies is O(ε), with the error bound depending solely on the W^{1,1} norm of the connectivity density. We propose a learning-driven framework for spatially varying diffusion coefficient modeling: a neural network reconstructs the connectivity density directly from edge-weight data and embeds it into a continuous dynamical system. Compared to conventional constant-coefficient models, our approach substantially enhances biological plausibility and predictive accuracy in brain dynamics modeling. Experimental results demonstrate that the learned spatially adaptive diffusion generates qualitatively distinct spatiotemporal evolution patterns, validating both theoretical guarantees and practical efficacy.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

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 $W^{1,1}$-norm of the connectivity density as $varepsilon o0$, 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.

Problem

• Deriving continuum limits for epsilon-graph connectivity functionals
• Establishing error bounds for discrete-to-continuum energy approximations
• Developing neural networks to reconstruct spatially varying diffusion coefficients

Innovation

• Energy-based continuum limit for ε-graph connectivity
• Neural network reconstructs connectivity density from data
• Spatially varying diffusion replaces constant coefficient dynamics