A large driver of the complexity of graph learning is the interplay between structure and features. When analyzing the expressivity of graph neural networks, however, existing approaches ignore features in favor of structure, making it nigh-impossible to assess to what extent two graphs with close features should be considered similar. We address this by developing a new (pseudo-)metric based on graph homomorphisms Inspired by concepts from metric geometry, our graph homomorphism distortion measures the minimal worst-case distortion that node features of one graph are subjected to when mapping one graph to another. We demonstrate the utility of our (pseudo-)metric by showing that i.) it can be efficiently approximated while retaining distinction capabilities, ii.) it provides a granular approach to graph distinction, and, iii.) it serves as a structural encoding, extending the concept of homomorphism-based descriptors to attributed graphs, thus improving the predictive capabilities of graph neural networks.

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