FSW-GNN: A Bi-Lipschitz WL-Equivalent Graph Neural Network

Many of the most popular graph neural networks fall into the category of message-passing neural networks (MPNNs). Famously, MPNNs' ability to distinguish between graphs is limited to graphs separable by the Weisfeiler-Lemann (WL) graph isomorphism test, and the strongest MPNNs, in terms of separation power, are WL-equivalent. Recently, it was shown that the quality of separation provided by standard WL-equivalent MPNN can be very low, resulting in WL-separable graphs being mapped to very similar, hardly distinguishable features. This paper addresses this issue by seeking bi-Lipschitz continuity guarantees for MPNNs. We demonstrate that, in contrast with standard summation-based MPNNs, which lack bi-Lipschitz properties, our proposed model provides a bi-Lipschitz graph embedding with respect to two standard graph metrics. Empirically, we show that our MPNN is competitive with standard MPNNs for several graph learning tasks and is far more accurate in over-squashing long-range tasks.

On the Hölder Stability of Multiset and Graph Neural Networks

Famously, multiset neural networks based on sum-pooling can separate all distinct multisets, and as a result can be used by message passing neural networks (MPNNs) to separate all pairs of graphs that can be separated by the 1-WL graph isomorphism test. However, the quality of this separation may be very weak, to the extent that the embeddings of "separable" multisets and graphs might even be considered identical when using fixed finite precision. In this work, we propose to fully analyze the separation quality of multiset models and MPNNs via a novel adaptation of Lipschitz and H\"{o}lder continuity to parametric functions. We prove that common sum-based models are lower-H\"{o}lder continuous, with a H\"{o}lder exponent that decays rapidly with the network's depth. Our analysis leads to adversarial examples of graphs which can be separated by three 1-WL iterations, but cannot be separated in practice by standard maximally powerful MPNNs. To remedy this, we propose two novel MPNNs with improved separation quality, one of which is lower Lipschitz continuous. We show these MPNNs can easily classify our adversarial examples, and compare favorably with standard MPNNs on standard graph learning tasks.