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

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.