Lydia Mezrag

Path Laplacian Encodings for Directed Graphs

Michael Perlmutter

Bastian Rieck

Directed graphs naturally model many real-world systems in which interactions are asymmetric, such as citation networks, web graphs, and inf… (see more)ormation-flow networks. However, graph learning methods commonly rely on message passing with symmetrized graph representations or positional encodings that only partially exploit edge directionality. We introduce PathLapPE, a novel spectral positional encoding (PE) derived from the path Laplacian on directed graphs. PathLapPE provides node- and edge-level features that encode directional higher-order structure and can be incorporated into standard graph learning architectures. Empirical results on node- and graph-level benchmark tasks show that PathLapPE yields consistent improvements across several architectures, especially when combined with direction-aware message passing. Compared with magnetic Laplacian positional encodings, a widely studied spectral positional encoding for directed graphs, PathLapPE does not require additional fine-tuning of directionality hyperparameters while offering competitive runtime and performance.

2026-10-02

arXiv (preprint)

doi.org

arxiv.org

Geometry-Aware Edge Pooling for Graph Neural Networks

Katharina Limbeck

Lydia Mezrag

Guy Wolf

Bastian Rieck

Graph Neural Networks (GNNs) have shown significant success for graph-based tasks. Motivated by the prevalence of large datasets in real-wor… (see more)ld applications, pooling layers are crucial components of GNNs. By reducing the size of input graphs, pooling enables faster training and potentially better generalisation. However, existing pooling operations often optimise for the learning task at the expense of discarding fundamental graph structures, thus reducing interpretability. This leads to unreliable performance across dataset types, downstream tasks and pooling ratios. Addressing these concerns, we propose novel graph pooling layers for structure-aware pooling via edge collapses. Our methods leverage diffusion geometry and iteratively reduce a graph's size while preserving both its metric structure and its structural diversity. We guide pooling using magnitude, an isometry-invariant diversity measure, which permits us to control the fidelity of the pooling process. Further, we use the spread of a metric space as a faster and more stable alternative ensuring computational efficiency. Empirical results demonstrate that our methods (i) achieve top performance compared to alternative pooling layers across a range of diverse graph classification tasks, (ii) preserve key spectral properties of the input graphs, and (iii) retain high accuracy across varying pooling ratios.

2025-12-02

Conference on Neural Information Processing Systems (Accept (poster))

doi.org

openreview.net

Principal Curvatures Estimation with Applications to Single Cell Data

Yanlei Zhang

Lydia Mezrag

Xingzhi Sun

Charles Xu

Kincaid MacDonald

Dhananjay Bhaskar

Smita Krishnaswamy