将ε-图的连通性能量推导为连续扩散模型，误差可控且可学习密度函数。
Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional
- 从ε-图出发，基于连通性函数构建连续能量模型。
- 离散与连续能量差不超过O(ε)，即使密度波动剧烈也成立。
- 可用于脑动力学建模，实现空间可变扩散系数，优于传统常数模型。
我们推导了带有通用连通性函数的ε-图的能量型连续极限。证明离散能量与其连续对应项之间的差异最多为O(ε)，其中前因子仅依赖于连通性密度的W^{1,1}范数，因此当该密度存在强局部波动时，误差界依然有效。作为应用，我们提出一种神经网络方法，从边权数据中重建连通性密度，并将所得连续模型嵌入脑动力学框架。在此设置中，通常的常数扩散系数被学习得到的空间可变系数取代，导致动力学行为显著区别于传统常数扩散模型。
原文摘要 · Abstract (English)
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 $\varepsilon\to0$, 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.
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