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 (AD). We find that the graph thermal diffusivity is lower for AD patients than healthy controls and correlates with mini mental state examination (MMSE) 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。该概念此前被推广至图结构，其中热量在图节点之间流动，其流动方式取决于图的拓扑结构。本文我们将图热方程与随机热方程相结合，最终建立了一个针对图上多变量时间信号的模型。我们从理论上证明了该模型如何能够直接用于从多变量信号中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变量的，并且会产生一个绝对尺度因子，即图热扩散率，它能够捕捉数据中类热图传播的程度。在两个数据集上，我们展示了如何利用图热扩散率来表征阿尔茨海默病（AD）。我们发现，阿尔茨海默病患者的图热扩散率低于健康对照组，并且与简易精神状态检查（MMSE）评分相关，这与先前研究结果中患者存在结构性损伤的发现一致。