Heat diffusion describes the process by which heat flows from areas with higher temperatures to ones with lower temperatures. This concept was previously adapted to graph structures, whereby heat flows between nodes of a graph depending on the graph topology. Here, we combine the graph heat equation with the stochastic heat equation, which ultimately yields a model for multivariate time signals on a graph. We show theoretically how the model can be used to directly compute the diffusion-based connectivity structure from multivariate signals. Unlike other connectivity measures, our heat model-based approach is inherently multivariate and yields an absolute scaling factor, namely the graph thermal diffusivity, which captures the extent of heat-like graph propagation in the data. On two datasets, we show how the graph thermal diffusivity can be used to characterise Alzheimer's disease. We find that the graph thermal diffusivity is lower for Alzheimer's patients than healthy controls and correlates with dementia scores, suggesting structural impairment in patients in line with previous findings.

翻译：热扩散描述了热量从高温区域向低温区域流动的过程。该概念此前已被推广至图结构，图中节点间的热量流动取决于图的拓扑结构。本文结合图热方程与随机热方程，最终构建了一种适用于图上多元时间信号的模型。我们从理论上证明了如何利用该模型直接根据多元信号计算基于扩散的连通性结构。与其他连通性度量不同，我们的基于热模型的方法本质上是多元的，并能给出一个绝对缩放因子——图热扩散系数，该系数可捕捉数据中类似热量传播的图扩散程度。在两个数据集上，我们展示了图热扩散系数如何用于表征阿尔茨海默病。研究发现，阿尔茨海默病患者的图热扩散系数低于健康对照组，且该系数与痴呆评分相关，表明患者存在结构性损伤，这与先前的研究结果一致。