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    This study introduces a novel heat-based connectivity measure for analyzing multivariate time signals on graphs. This method can characterize Alzheimer's disease (AD) by measuring graph thermal diffusivity, showing lower values in AD patients.

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    Area of Science:

    • Graph theory
    • Stochastic processes
    • Signal processing

    Background:

    • Heat diffusion models heat flow from high to low temperatures.
    • Graph theory adapts heat diffusion to node-based systems.
    • Existing methods lack inherent multivariate analysis and absolute scaling factors.

    Purpose of the Study:

    • To develop a novel model for multivariate time signals on graphs by combining graph heat and stochastic heat equations.
    • To introduce a diffusion-based connectivity measure with an absolute scaling factor (graph thermal diffusivity).
    • To investigate the utility of graph thermal diffusivity in characterizing Alzheimer's disease (AD).

    Main Methods:

    • Combined graph heat equation with stochastic heat equation for multivariate time signals.
    • Developed a theoretical framework to compute diffusion-based connectivity directly from signals.
    • Applied the graph thermal diffusivity measure to two datasets, comparing AD patients and healthy controls.

    Main Results:

    • The novel model successfully computes diffusion-based connectivity from multivariate signals.
    • Graph thermal diffusivity was found to be lower in AD patients compared to healthy controls.
    • Graph thermal diffusivity correlated with Mini-Mental State Examination (MMSE) scores in AD patients.

    Conclusions:

    • The proposed heat-based connectivity measure is inherently multivariate and provides an absolute scaling factor.
    • Graph thermal diffusivity serves as a potential biomarker for characterizing Alzheimer's disease.
    • Findings suggest structural impairments in AD patients, aligning with previous research.