Graph Filtration Kernels

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.