Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent homology (PH) has gained prominence, but identifying the class of attributed graphs that PH can recognize remains open. We introduce a novel concept of color-separating sets to provide a complete resolution to this important problem. Specifically, we establish the necessary and sufficient conditions for distinguishing graphs based on the persistence of their connected components, obtained from filter functions on vertex and edge colors. Our constructions expose the limits of vertex- and edge-level PH, proving that neither category subsumes the other. Leveraging these theoretical insights, we propose RePHINE for learning topological features on graphs. RePHINE efficiently combines vertex- and edge-level PH, achieving a scheme that is provably more powerful than both. Integrating RePHINE into MP-GNNs boosts their expressive power, resulting in gains over standard PH on several benchmarks for graph classification.

We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing $\mathrm{O}(n)$- and $\mathrm{E}(n)$-equivariant models. We identify and study the *Clifford group*: a subgroup inside the Clifford algebra tailored to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, including their grade projections, constitutes an equivariant map with respect to the Clifford group, allowing us to parameterize equivariant neural network layers. An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional $n$-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.

This work studies the evaluation of explaining graph neural networks (GNNs), which is crucial to the credibility of post-hoc explainability in practical usage. Conventional evaluation metrics, and even explanation methods -- which mainly follow the paradigm of feeding the explanatory subgraph and measuring output difference -- always suffer from the notorious out-of-distribution (OOD) issue. In this work, we endeavor to confront the issue by introducing a novel evaluation metric, termed OOD-resistant Adversarial Robustness (OAR). Specifically, we draw inspiration from the notion of adversarial robustness and evaluate post-hoc explanation subgraphs by calculating their robustness under attack. On top of that, an elaborate OOD reweighting block is inserted into the pipeline to confine the evaluation process to the original data distribution. For applications involving large datasets, we further devise a Simplified version of OAR (SimOAR), which achieves a significant improvement in computational efficiency at the cost of a small amount of performance. Extensive empirical studies validate the effectiveness of our OAR and SimOAR.