Diffusion distance for signals supported on networks

Diffusion distance is introduced as a metric to compare signals supported in the nodes of a network. This metric considers the given signals as initial temperature distributions in the graph nodes and diffuses heat through the edges of the graph. The similarity between the given vectors is determined by the similarity of the respective diffusion profiles. We prove that diffusion distance defines a valid metric and is stable to perturbations in the underlying network. We utilize numerical experiments to illustrate its utility in classifying ovarian cancer histologies using gene mutation profiles of different patients. Diffusion distance is also used as part of a label propagation method in semi-supervised learning to classify handwritten digits.