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

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