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We introduce the diffusion and superposition distances as two metrics tocompare signals supported in the nodes of a network. Both metrics consider thegiven vectors as initial temperature distributions and diffuse heat trough theedges of the graph. The similarity between the given vectors is determined bythe similarity of the respective diffusion profiles. The superposition distancecomputes the instantaneous difference between the diffused signals andintegrates the difference over time. The diffusion distance determines adistance between the integrals of the diffused signals. We prove that bothdistances define valid metrics and that they are stable to perturbations in theunderlying network. We utilize numerical experiments to illustrate theirutility in classifying signals in a synthetic network as well as in classifyingovarian cancer histologies using gene mutation profiles of different patients.We also reinterpret diffusion as a transformation of interrelated featurespaces and use it as preprocessing tool for learning. We use diffusion toincrease the accuracy of handwritten digit classification.

Diffusion and Superposition Distances for Signals Supported on Networks