Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms

The Euler Characteristic Transform (ECT) is an efficiently computable geometrical-topological invariant that characterizes the global shape of data. In this paper, we introduce the local Euler Characteristic Transform ($\ell$-ECT), a novel extension of the ECT designed to enhance expressivity and interpretability in graph representation learning. Unlike traditional Graph Neural Networks (GNNs), which may lose critical local details through aggregation, the $\ell$-ECT provides a lossless representation of local neighborhoods. This approach addresses key limitations in GNNs by preserving nuanced local structures while maintaining global interpretability. Moreover, we construct a rotation-invariant metric based on $\ell$-ECTs for spatial alignment of data spaces. Our method demonstrates superior performance compared to standard GNNs on various benchmarking node classification tasks, while also offering theoretical guarantees of its effectiveness.

Graphs are used to represent complex systems like social networks, molecules, or traffic maps, where the relationships between entities matter. Most current methods for analyzing graphs use so-called *message-passing neural networks* that blend information from neighboring nodes. However, this blending can wash out important local details.Our work introduces a new tool, called the Local Euler Characteristic Transform ($\ell$-ECT), which captures the *shape* and *structure* around each point in the graph, giving every node its own "fingerprint." This fingerprint is lossless and mathematically robust, meaning it retains all relevant local information without distortion!Unlike typical graph neural network methods, the $\ell$-ECT works out-of-the-box with standard machine-learning tools and performs particularly well in settings where neighboring nodes are very different—a case where traditional methods often struggle. We also show that $\ell$-ECT fingerprints can help align and compare graphs in space, even if they’ve been rotated.By combining topology with geometry and mathematical insights with practical benefits, the $\ell$-ECT offers a new, interpretable way to learn from graph data.