  The majority of popular graph kernels is based on the concept of Haussler's
$\mathcal{R}$-convolution kernel and defines graph similarities in terms of
mutual substructures. In this work, we enrich these similarity measures by
considering graph filtrations: Using meaningful orders on the set of edges,
which allow to construct a sequence of nested graphs, we can consider a graph
at multiple granularities. For one thing, this provides access to features on
different levels of resolution. Furthermore, rather than to simply compare
frequencies of features in graphs, it allows for their comparison in terms of
when and for how long they exist in the sequences. In this work, we propose a
family of graph kernels that incorporate these existence intervals of features.
While our approach can be applied to arbitrary graph features, we particularly
highlight Weisfeiler-Lehman vertex labels, leading to efficient kernels. We
show that using Weisfeiler-Lehman labels over certain filtrations strictly
increases the expressive power over the ordinary Weisfeiler-Lehman procedure in
terms of deciding graph isomorphism. In fact, this result directly yields more
powerful graph kernels based on such features and has implications to graph
neural networks due to their close relationship to the Weisfeiler-Lehman
method. We empirically validate the expressive power of our graph kernels and
show significant improvements over state-of-the-art graph kernels in terms of
predictive performance on various real-world benchmark datasets.
