  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.
