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We propose two multiscale comparisons of graphs using heat diffusion,allowing to compare graphs without node correspondence or even with differentsizes. These multiscale comparisons lead to the definition ofLipschitz-continuous empirical processes indexed by a real parameter. Thestatistical properties of empirical means of such processes are studied in thegeneral case. Under mild assumptions, we prove a functional central limittheorem, as well as a Gaussian approximation with a rate depending only on thesample size. Once applied to our processes, these results allow to analyze datasets of pairs of graphs. We design consistent confidence bands around empiricalmeans and consistent two-sample tests, using bootstrap methods. Theirperformances are evaluated by simulations on synthetic data sets.

Heat diffusion distance processes: a statistically founded method to analyze graph data sets