Stochastic Graph Heat Modelling for Diffusion-based Connectivity Retrieval.

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.Clinical relevance- This study introduces a novel heat-based connectivity measure, which allows to characterise Alzheimer's disease in terms of the graph thermal diffusivity.